# Properties

 Label 4608.2.d.g.2305.3 Level $4608$ Weight $2$ Character 4608.2305 Analytic conductor $36.795$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1536) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2305.3 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 4608.2305 Dual form 4608.2.d.g.2305.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.41421i q^{5} -1.41421 q^{7} +O(q^{10})$$ $$q+1.41421i q^{5} -1.41421 q^{7} +2.00000i q^{11} -2.00000 q^{17} +4.00000i q^{19} +2.82843 q^{23} +3.00000 q^{25} +9.89949i q^{29} -7.07107 q^{31} -2.00000i q^{35} -8.48528i q^{37} +6.00000 q^{41} -8.00000i q^{43} -2.82843 q^{47} -5.00000 q^{49} +1.41421i q^{53} -2.82843 q^{55} +12.0000i q^{59} -14.1421i q^{61} +8.00000i q^{67} -14.1421 q^{71} +8.00000 q^{73} -2.82843i q^{77} -4.24264 q^{79} -6.00000i q^{83} -2.82843i q^{85} +2.00000 q^{89} -5.65685 q^{95} -14.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q - 8 q^{17} + 12 q^{25} + 24 q^{41} - 20 q^{49} + 32 q^{73} + 8 q^{89} - 56 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$$.

 $$n$$ $$2053$$ $$3583$$ $$4097$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.41421i 0.632456i 0.948683 + 0.316228i $$0.102416\pi$$
−0.948683 + 0.316228i $$0.897584\pi$$
$$6$$ 0 0
$$7$$ −1.41421 −0.534522 −0.267261 0.963624i $$-0.586119\pi$$
−0.267261 + 0.963624i $$0.586119\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.00000i 0.603023i 0.953463 + 0.301511i $$0.0974911\pi$$
−0.953463 + 0.301511i $$0.902509\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 9.89949i 1.83829i 0.393919 + 0.919145i $$0.371119\pi$$
−0.393919 + 0.919145i $$0.628881\pi$$
$$30$$ 0 0
$$31$$ −7.07107 −1.27000 −0.635001 0.772512i $$-0.719000\pi$$
−0.635001 + 0.772512i $$0.719000\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 2.00000i − 0.338062i
$$36$$ 0 0
$$37$$ − 8.48528i − 1.39497i −0.716599 0.697486i $$-0.754302\pi$$
0.716599 0.697486i $$-0.245698\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ −5.00000 −0.714286
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.41421i 0.194257i 0.995272 + 0.0971286i $$0.0309658\pi$$
−0.995272 + 0.0971286i $$0.969034\pi$$
$$54$$ 0 0
$$55$$ −2.82843 −0.381385
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 12.0000i 1.56227i 0.624364 + 0.781133i $$0.285358\pi$$
−0.624364 + 0.781133i $$0.714642\pi$$
$$60$$ 0 0
$$61$$ − 14.1421i − 1.81071i −0.424650 0.905357i $$-0.639603\pi$$
0.424650 0.905357i $$-0.360397\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.1421 −1.67836 −0.839181 0.543852i $$-0.816965\pi$$
−0.839181 + 0.543852i $$0.816965\pi$$
$$72$$ 0 0
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 2.82843i − 0.322329i
$$78$$ 0 0
$$79$$ −4.24264 −0.477334 −0.238667 0.971101i $$-0.576710\pi$$
−0.238667 + 0.971101i $$0.576710\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ − 2.82843i − 0.306786i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −5.65685 −0.580381
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.89949i 0.985037i 0.870302 + 0.492518i $$0.163924\pi$$
−0.870302 + 0.492518i $$0.836076\pi$$
$$102$$ 0 0
$$103$$ −12.7279 −1.25412 −0.627060 0.778971i $$-0.715742\pi$$
−0.627060 + 0.778971i $$0.715742\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ 11.3137i 1.08366i 0.840489 + 0.541828i $$0.182268\pi$$
−0.840489 + 0.541828i $$0.817732\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 4.00000i 0.373002i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 2.82843 0.259281
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 11.3137i 1.01193i
$$126$$ 0 0
$$127$$ −4.24264 −0.376473 −0.188237 0.982124i $$-0.560277\pi$$
−0.188237 + 0.982124i $$0.560277\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ − 5.65685i − 0.490511i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −14.0000 −1.16264
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 4.24264i − 0.347571i −0.984784 0.173785i $$-0.944400\pi$$
0.984784 0.173785i $$-0.0555999\pi$$
$$150$$ 0 0
$$151$$ −4.24264 −0.345261 −0.172631 0.984987i $$-0.555227\pi$$
−0.172631 + 0.984987i $$0.555227\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 10.0000i − 0.803219i
$$156$$ 0 0
$$157$$ − 14.1421i − 1.12867i −0.825547 0.564333i $$-0.809134\pi$$
0.825547 0.564333i $$-0.190866\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −11.3137 −0.875481 −0.437741 0.899101i $$-0.644221\pi$$
−0.437741 + 0.899101i $$0.644221\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 15.5563i − 1.18273i −0.806405 0.591364i $$-0.798590\pi$$
0.806405 0.591364i $$-0.201410\pi$$
$$174$$ 0 0
$$175$$ −4.24264 −0.320713
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.00000i 0.298974i 0.988764 + 0.149487i $$0.0477622\pi$$
−0.988764 + 0.149487i $$0.952238\pi$$
$$180$$ 0 0
$$181$$ 11.3137i 0.840941i 0.907306 + 0.420471i $$0.138135\pi$$
−0.907306 + 0.420471i $$0.861865\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ − 4.00000i − 0.292509i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −22.6274 −1.63726 −0.818631 0.574320i $$-0.805267\pi$$
−0.818631 + 0.574320i $$0.805267\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 7.07107i 0.503793i 0.967754 + 0.251896i $$0.0810542\pi$$
−0.967754 + 0.251896i $$0.918946\pi$$
$$198$$ 0 0
$$199$$ −24.0416 −1.70427 −0.852133 0.523325i $$-0.824691\pi$$
−0.852133 + 0.523325i $$0.824691\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 14.0000i − 0.982607i
$$204$$ 0 0
$$205$$ 8.48528i 0.592638i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 11.3137 0.771589
$$216$$ 0 0
$$217$$ 10.0000 0.678844
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −12.7279 −0.852325 −0.426162 0.904647i $$-0.640135\pi$$
−0.426162 + 0.904647i $$0.640135\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 22.0000i 1.46019i 0.683345 + 0.730096i $$0.260525\pi$$
−0.683345 + 0.730096i $$0.739475\pi$$
$$228$$ 0 0
$$229$$ 16.9706i 1.12145i 0.828003 + 0.560723i $$0.189477\pi$$
−0.828003 + 0.560723i $$0.810523\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ − 4.00000i − 0.260931i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 22.6274 1.46365 0.731823 0.681495i $$-0.238670\pi$$
0.731823 + 0.681495i $$0.238670\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 7.07107i − 0.451754i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 6.00000i − 0.378717i −0.981908 0.189358i $$-0.939359\pi$$
0.981908 0.189358i $$-0.0606408\pi$$
$$252$$ 0 0
$$253$$ 5.65685i 0.355643i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 12.0000i 0.745644i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −28.2843 −1.74408 −0.872041 0.489432i $$-0.837204\pi$$
−0.872041 + 0.489432i $$0.837204\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 21.2132i − 1.29339i −0.762748 0.646696i $$-0.776150\pi$$
0.762748 0.646696i $$-0.223850\pi$$
$$270$$ 0 0
$$271$$ −24.0416 −1.46043 −0.730213 0.683220i $$-0.760579\pi$$
−0.730213 + 0.683220i $$0.760579\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000i 0.361814i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.48528 −0.500870
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 9.89949i − 0.578335i −0.957279 0.289167i $$-0.906622\pi$$
0.957279 0.289167i $$-0.0933784\pi$$
$$294$$ 0 0
$$295$$ −16.9706 −0.988064
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 11.3137i 0.652111i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ − 32.0000i − 1.82634i −0.407583 0.913168i $$-0.633628\pi$$
0.407583 0.913168i $$-0.366372\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 22.6274 1.28308 0.641542 0.767088i $$-0.278295\pi$$
0.641542 + 0.767088i $$0.278295\pi$$
$$312$$ 0 0
$$313$$ 4.00000 0.226093 0.113047 0.993590i $$-0.463939\pi$$
0.113047 + 0.993590i $$0.463939\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 9.89949i − 0.556011i −0.960579 0.278006i $$-0.910327\pi$$
0.960579 0.278006i $$-0.0896734\pi$$
$$318$$ 0 0
$$319$$ −19.7990 −1.10853
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 32.0000i 1.75888i 0.476011 + 0.879440i $$0.342082\pi$$
−0.476011 + 0.879440i $$0.657918\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −11.3137 −0.618134
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 14.1421i − 0.765840i
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 26.0000i − 1.39575i −0.716218 0.697877i $$-0.754128\pi$$
0.716218 0.697877i $$-0.245872\pi$$
$$348$$ 0 0
$$349$$ 25.4558i 1.36262i 0.731995 + 0.681310i $$0.238589\pi$$
−0.731995 + 0.681310i $$0.761411\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ − 20.0000i − 1.06149i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −19.7990 −1.04495 −0.522475 0.852654i $$-0.674991\pi$$
−0.522475 + 0.852654i $$0.674991\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 11.3137i 0.592187i
$$366$$ 0 0
$$367$$ 4.24264 0.221464 0.110732 0.993850i $$-0.464680\pi$$
0.110732 + 0.993850i $$0.464680\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 2.00000i − 0.103835i
$$372$$ 0 0
$$373$$ 19.7990i 1.02515i 0.858642 + 0.512576i $$0.171309\pi$$
−0.858642 + 0.512576i $$0.828691\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 20.0000i − 1.02733i −0.857991 0.513665i $$-0.828287\pi$$
0.857991 0.513665i $$-0.171713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5.65685 0.289052 0.144526 0.989501i $$-0.453834\pi$$
0.144526 + 0.989501i $$0.453834\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 24.0416i − 1.21896i −0.792802 0.609480i $$-0.791378\pi$$
0.792802 0.609480i $$-0.208622\pi$$
$$390$$ 0 0
$$391$$ −5.65685 −0.286079
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 6.00000i − 0.301893i
$$396$$ 0 0
$$397$$ − 2.82843i − 0.141955i −0.997478 0.0709773i $$-0.977388\pi$$
0.997478 0.0709773i $$-0.0226118\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 16.9706 0.841200
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 16.9706i − 0.835067i
$$414$$ 0 0
$$415$$ 8.48528 0.416526
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 30.0000i 1.46560i 0.680446 + 0.732798i $$0.261786\pi$$
−0.680446 + 0.732798i $$0.738214\pi$$
$$420$$ 0 0
$$421$$ 5.65685i 0.275698i 0.990453 + 0.137849i $$0.0440189\pi$$
−0.990453 + 0.137849i $$0.955981\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 20.0000i 0.967868i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.82843 −0.136241 −0.0681203 0.997677i $$-0.521700\pi$$
−0.0681203 + 0.997677i $$0.521700\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 11.3137i 0.541208i
$$438$$ 0 0
$$439$$ 4.24264 0.202490 0.101245 0.994862i $$-0.467717\pi$$
0.101245 + 0.994862i $$0.467717\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 30.0000i 1.42534i 0.701498 + 0.712672i $$0.252515\pi$$
−0.701498 + 0.712672i $$0.747485\pi$$
$$444$$ 0 0
$$445$$ 2.82843i 0.134080i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 12.0000i 0.565058i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12.7279i 0.592798i 0.955064 + 0.296399i $$0.0957859\pi$$
−0.955064 + 0.296399i $$0.904214\pi$$
$$462$$ 0 0
$$463$$ 35.3553 1.64310 0.821551 0.570135i $$-0.193109\pi$$
0.821551 + 0.570135i $$0.193109\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.0000i 1.01804i 0.860755 + 0.509019i $$0.169992\pi$$
−0.860755 + 0.509019i $$0.830008\pi$$
$$468$$ 0 0
$$469$$ − 11.3137i − 0.522419i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 12.0000i 0.550598i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −25.4558 −1.16311 −0.581554 0.813508i $$-0.697555\pi$$
−0.581554 + 0.813508i $$0.697555\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 19.7990i − 0.899026i
$$486$$ 0 0
$$487$$ 4.24264 0.192252 0.0961262 0.995369i $$-0.469355\pi$$
0.0961262 + 0.995369i $$0.469355\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 28.0000i 1.26362i 0.775122 + 0.631811i $$0.217688\pi$$
−0.775122 + 0.631811i $$0.782312\pi$$
$$492$$ 0 0
$$493$$ − 19.7990i − 0.891702i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 20.0000 0.897123
$$498$$ 0 0
$$499$$ 20.0000i 0.895323i 0.894203 + 0.447661i $$0.147743\pi$$
−0.894203 + 0.447661i $$0.852257\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2.82843 0.126113 0.0630567 0.998010i $$-0.479915\pi$$
0.0630567 + 0.998010i $$0.479915\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 24.0416i − 1.06563i −0.846233 0.532813i $$-0.821135\pi$$
0.846233 0.532813i $$-0.178865\pi$$
$$510$$ 0 0
$$511$$ −11.3137 −0.500489
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 18.0000i − 0.793175i
$$516$$ 0 0
$$517$$ − 5.65685i − 0.248788i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 0 0
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 14.1421 0.616041
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −5.65685 −0.244567
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 10.0000i − 0.430730i
$$540$$ 0 0
$$541$$ 28.2843i 1.21604i 0.793923 + 0.608018i $$0.208035\pi$$
−0.793923 + 0.608018i $$0.791965\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −16.0000 −0.685365
$$546$$ 0 0
$$547$$ − 16.0000i − 0.684111i −0.939680 0.342055i $$-0.888877\pi$$
0.939680 0.342055i $$-0.111123\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −39.5980 −1.68693
$$552$$ 0 0
$$553$$ 6.00000 0.255146
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 38.1838i 1.61790i 0.587879 + 0.808949i $$0.299963\pi$$
−0.587879 + 0.808949i $$0.700037\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.00000i 0.252870i 0.991975 + 0.126435i $$0.0403535\pi$$
−0.991975 + 0.126435i $$0.959647\pi$$
$$564$$ 0 0
$$565$$ − 8.48528i − 0.356978i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ − 28.0000i − 1.17176i −0.810397 0.585882i $$-0.800748\pi$$
0.810397 0.585882i $$-0.199252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.48528 0.353861
$$576$$ 0 0
$$577$$ −4.00000 −0.166522 −0.0832611 0.996528i $$-0.526534\pi$$
−0.0832611 + 0.996528i $$0.526534\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 8.48528i 0.352029i
$$582$$ 0 0
$$583$$ −2.82843 −0.117141
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ 0 0
$$589$$ − 28.2843i − 1.16543i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 4.00000i 0.163984i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 36.7696 1.50236 0.751182 0.660096i $$-0.229484\pi$$
0.751182 + 0.660096i $$0.229484\pi$$
$$600$$ 0 0
$$601$$ −44.0000 −1.79480 −0.897399 0.441221i $$-0.854546\pi$$
−0.897399 + 0.441221i $$0.854546\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 9.89949i 0.402472i
$$606$$ 0 0
$$607$$ −4.24264 −0.172203 −0.0861017 0.996286i $$-0.527441\pi$$
−0.0861017 + 0.996286i $$0.527441\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 8.48528i − 0.342717i −0.985209 0.171359i $$-0.945184\pi$$
0.985209 0.171359i $$-0.0548157\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 46.0000 1.85189 0.925945 0.377658i $$-0.123271\pi$$
0.925945 + 0.377658i $$0.123271\pi$$
$$618$$ 0 0
$$619$$ − 36.0000i − 1.44696i −0.690344 0.723481i $$-0.742541\pi$$
0.690344 0.723481i $$-0.257459\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2.82843 −0.113319
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 16.9706i 0.676661i
$$630$$ 0 0
$$631$$ 15.5563 0.619288 0.309644 0.950852i $$-0.399790\pi$$
0.309644 + 0.950852i $$0.399790\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 6.00000i − 0.238103i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 0 0
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −14.1421 −0.555985 −0.277992 0.960583i $$-0.589669\pi$$
−0.277992 + 0.960583i $$0.589669\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 29.6985i 1.16219i 0.813835 + 0.581096i $$0.197376\pi$$
−0.813835 + 0.581096i $$0.802624\pi$$
$$654$$ 0 0
$$655$$ 16.9706 0.663095
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 44.0000i − 1.71400i −0.515319 0.856998i $$-0.672327\pi$$
0.515319 0.856998i $$-0.327673\pi$$
$$660$$ 0 0
$$661$$ − 19.7990i − 0.770091i −0.922897 0.385046i $$-0.874186\pi$$
0.922897 0.385046i $$-0.125814\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ 28.0000i 1.08416i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 28.2843 1.09190
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 24.0416i 0.923995i 0.886881 + 0.461997i $$0.152867\pi$$
−0.886881 + 0.461997i $$0.847133\pi$$
$$678$$ 0 0
$$679$$ 19.7990 0.759815
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 26.0000i 0.994862i 0.867503 + 0.497431i $$0.165723\pi$$
−0.867503 + 0.497431i $$0.834277\pi$$
$$684$$ 0 0
$$685$$ − 14.1421i − 0.540343i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 8.00000i − 0.304334i −0.988355 0.152167i $$-0.951375\pi$$
0.988355 0.152167i $$-0.0486252\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 4.24264i − 0.160242i −0.996785 0.0801212i $$-0.974469\pi$$
0.996785 0.0801212i $$-0.0255307\pi$$
$$702$$ 0 0
$$703$$ 33.9411 1.28011
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 14.0000i − 0.526524i
$$708$$ 0 0
$$709$$ 33.9411i 1.27469i 0.770580 + 0.637343i $$0.219966\pi$$
−0.770580 + 0.637343i $$0.780034\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −20.0000 −0.749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 25.4558 0.949343 0.474671 0.880163i $$-0.342567\pi$$
0.474671 + 0.880163i $$0.342567\pi$$
$$720$$ 0 0
$$721$$ 18.0000 0.670355
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 29.6985i 1.10297i
$$726$$ 0 0
$$727$$ 24.0416 0.891655 0.445827 0.895119i $$-0.352910\pi$$
0.445827 + 0.895119i $$0.352910\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 16.0000i 0.591781i
$$732$$ 0 0
$$733$$ 11.3137i 0.417881i 0.977928 + 0.208941i $$0.0670016\pi$$
−0.977928 + 0.208941i $$0.932998\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 12.0000i 0.441427i 0.975339 + 0.220714i $$0.0708386\pi$$
−0.975339 + 0.220714i $$0.929161\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 28.2843 1.03765 0.518825 0.854881i $$-0.326370\pi$$
0.518825 + 0.854881i $$0.326370\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 5.65685i − 0.206697i
$$750$$ 0 0
$$751$$ 12.7279 0.464448 0.232224 0.972662i $$-0.425400\pi$$
0.232224 + 0.972662i $$0.425400\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 6.00000i − 0.218362i
$$756$$ 0 0
$$757$$ − 50.9117i − 1.85042i −0.379459 0.925208i $$-0.623890\pi$$
0.379459 0.925208i $$-0.376110\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ − 16.0000i − 0.579239i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 15.5563i 0.559523i 0.960070 + 0.279761i $$0.0902554\pi$$
−0.960070 + 0.279761i $$0.909745\pi$$
$$774$$ 0 0
$$775$$ −21.2132 −0.762001
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000i 0.859889i
$$780$$ 0 0
$$781$$ − 28.2843i − 1.01209i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 20.0000 0.713831
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8.48528 0.301702
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 1.41421i − 0.0500940i −0.999686 0.0250470i $$-0.992026\pi$$
0.999686 0.0250470i $$-0.00797354\pi$$
$$798$$ 0 0
$$799$$ 5.65685 0.200125
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 16.0000i 0.564628i
$$804$$ 0 0
$$805$$ − 5.65685i − 0.199378i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ − 28.0000i − 0.983213i −0.870817 0.491606i $$-0.836410\pi$$
0.870817 0.491606i $$-0.163590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 22.6274 0.792604
$$816$$ 0 0
$$817$$ 32.0000 1.11954
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 9.89949i 0.345495i 0.984966 + 0.172747i $$0.0552644\pi$$
−0.984966 + 0.172747i $$0.944736\pi$$
$$822$$ 0 0
$$823$$ 21.2132 0.739446 0.369723 0.929142i $$-0.379453\pi$$
0.369723 + 0.929142i $$0.379453\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 0 0
$$829$$ − 11.3137i − 0.392941i −0.980510 0.196471i $$-0.937052\pi$$
0.980510 0.196471i $$-0.0629480\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 10.0000 0.346479
$$834$$ 0 0
$$835$$ − 16.0000i − 0.553703i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 14.1421 0.488241 0.244120 0.969745i $$-0.421501\pi$$
0.244120 + 0.969745i $$0.421501\pi$$
$$840$$ 0 0
$$841$$ −69.0000 −2.37931
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 18.3848i 0.632456i
$$846$$ 0 0
$$847$$ −9.89949 −0.340151
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 24.0000i − 0.822709i
$$852$$ 0 0
$$853$$ 14.1421i 0.484218i 0.970249 + 0.242109i $$0.0778391\pi$$
−0.970249 + 0.242109i $$0.922161\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ 0 0
$$859$$ − 8.00000i − 0.272956i −0.990643 0.136478i $$-0.956422\pi$$
0.990643 0.136478i $$-0.0435784\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −22.6274 −0.770246 −0.385123 0.922865i $$-0.625841\pi$$
−0.385123 + 0.922865i $$0.625841\pi$$
$$864$$ 0 0
$$865$$ 22.0000 0.748022
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 8.48528i − 0.287843i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 16.0000i − 0.540899i
$$876$$ 0 0
$$877$$ − 42.4264i − 1.43264i −0.697773 0.716319i $$-0.745826\pi$$
0.697773 0.716319i $$-0.254174\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22.0000 0.741199 0.370599 0.928793i $$-0.379152\pi$$
0.370599 + 0.928793i $$0.379152\pi$$
$$882$$ 0 0
$$883$$ 48.0000i 1.61533i 0.589643 + 0.807664i $$0.299269\pi$$
−0.589643 + 0.807664i $$0.700731\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 50.9117 1.70945 0.854724 0.519083i $$-0.173727\pi$$
0.854724 + 0.519083i $$0.173727\pi$$
$$888$$ 0 0
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 11.3137i − 0.378599i
$$894$$ 0 0
$$895$$ −5.65685 −0.189088
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 70.0000i − 2.33463i
$$900$$ 0 0
$$901$$ − 2.82843i − 0.0942286i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16.0000 −0.531858
$$906$$ 0 0
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −5.65685 −0.187420 −0.0937100 0.995600i $$-0.529873\pi$$
−0.0937100 + 0.995600i $$0.529873\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.9706i 0.560417i
$$918$$ 0 0
$$919$$ 21.2132 0.699759 0.349880 0.936795i $$-0.386223\pi$$
0.349880 + 0.936795i $$0.386223\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 25.4558i − 0.836983i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ − 20.0000i − 0.655474i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 5.65685 0.184999
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1.41421i 0.0461020i 0.999734 + 0.0230510i $$0.00733802\pi$$
−0.999734 + 0.0230510i $$0.992662\pi$$
$$942$$ 0 0
$$943$$ 16.9706 0.552638
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ 0 0
$$955$$ − 32.0000i − 1.03550i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 14.1421 0.456673
$$960$$ 0 0
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 5.65685i − 0.182101i
$$966$$ 0 0
$$967$$ 52.3259 1.68269 0.841344 0.540500i $$-0.181765\pi$$
0.841344 + 0.540500i $$0.181765\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 22.0000i − 0.706014i −0.935621 0.353007i $$-0.885159\pi$$
0.935621 0.353007i $$-0.114841\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −50.0000 −1.59964 −0.799821 0.600239i $$-0.795072\pi$$
−0.799821 + 0.600239i $$0.795072\pi$$
$$978$$ 0 0
$$979$$ 4.00000i 0.127841i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −50.9117 −1.62383 −0.811915 0.583775i $$-0.801575\pi$$
−0.811915 + 0.583775i $$0.801575\pi$$
$$984$$ 0 0
$$985$$ −10.0000 −0.318626
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 22.6274i − 0.719510i
$$990$$ 0 0
$$991$$ 46.6690 1.48249 0.741246 0.671234i $$-0.234235\pi$$
0.741246 + 0.671234i $$0.234235\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 34.0000i − 1.07787i
$$996$$ 0 0
$$997$$ − 31.1127i − 0.985349i −0.870214 0.492675i $$-0.836019\pi$$
0.870214 0.492675i $$-0.163981\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 4608.2.d.g.2305.3 4
3.2 odd 2 1536.2.d.d.769.3 4
4.3 odd 2 inner 4608.2.d.g.2305.4 4
8.3 odd 2 inner 4608.2.d.g.2305.2 4
8.5 even 2 inner 4608.2.d.g.2305.1 4
12.11 even 2 1536.2.d.d.769.1 4
16.3 odd 4 4608.2.a.h.1.2 2
16.5 even 4 4608.2.a.h.1.1 2
16.11 odd 4 4608.2.a.j.1.1 2
16.13 even 4 4608.2.a.j.1.2 2
24.5 odd 2 1536.2.d.d.769.2 4
24.11 even 2 1536.2.d.d.769.4 4
48.5 odd 4 1536.2.a.j.1.2 yes 2
48.11 even 4 1536.2.a.c.1.2 yes 2
48.29 odd 4 1536.2.a.c.1.1 2
48.35 even 4 1536.2.a.j.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.c.1.1 2 48.29 odd 4
1536.2.a.c.1.2 yes 2 48.11 even 4
1536.2.a.j.1.1 yes 2 48.35 even 4
1536.2.a.j.1.2 yes 2 48.5 odd 4
1536.2.d.d.769.1 4 12.11 even 2
1536.2.d.d.769.2 4 24.5 odd 2
1536.2.d.d.769.3 4 3.2 odd 2
1536.2.d.d.769.4 4 24.11 even 2
4608.2.a.h.1.1 2 16.5 even 4
4608.2.a.h.1.2 2 16.3 odd 4
4608.2.a.j.1.1 2 16.11 odd 4
4608.2.a.j.1.2 2 16.13 even 4
4608.2.d.g.2305.1 4 8.5 even 2 inner
4608.2.d.g.2305.2 4 8.3 odd 2 inner
4608.2.d.g.2305.3 4 1.1 even 1 trivial
4608.2.d.g.2305.4 4 4.3 odd 2 inner