# Properties

 Label 1200.2.o.b.1199.3 Level $1200$ Weight $2$ Character 1200.1199 Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1199.3 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1199 Dual form 1200.2.o.b.1199.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.73205i q^{3} -3.46410 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q+1.73205i q^{3} -3.46410 q^{7} -3.00000 q^{9} -3.46410 q^{11} +4.00000i q^{13} +6.00000 q^{17} -3.46410i q^{19} -6.00000i q^{21} -3.46410i q^{23} -5.19615i q^{27} -6.00000i q^{29} -3.46410i q^{31} -6.00000i q^{33} +4.00000i q^{37} -6.92820 q^{39} -12.0000i q^{41} -6.92820 q^{43} -3.46410i q^{47} +5.00000 q^{49} +10.3923i q^{51} -6.00000 q^{53} +6.00000 q^{57} +3.46410 q^{59} -10.0000 q^{61} +10.3923 q^{63} +6.92820 q^{67} +6.00000 q^{69} -13.8564 q^{71} -2.00000i q^{73} +12.0000 q^{77} +10.3923i q^{79} +9.00000 q^{81} +10.3923i q^{83} +10.3923 q^{87} -13.8564i q^{91} +6.00000 q^{93} -10.0000i q^{97} +10.3923 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} + 24 q^{17} + 20 q^{49} - 24 q^{53} + 24 q^{57} - 40 q^{61} + 24 q^{69} + 48 q^{77} + 36 q^{81} + 24 q^{93}+O(q^{100})$$ 4 * q - 12 * q^9 + 24 * q^17 + 20 * q^49 - 24 * q^53 + 24 * q^57 - 40 * q^61 + 24 * q^69 + 48 * q^77 + 36 * q^81 + 24 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.73205i 1.00000i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.46410 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ − 3.46410i − 0.794719i −0.917663 0.397360i $$-0.869927\pi$$
0.917663 0.397360i $$-0.130073\pi$$
$$20$$ 0 0
$$21$$ − 6.00000i − 1.30931i
$$22$$ 0 0
$$23$$ − 3.46410i − 0.722315i −0.932505 0.361158i $$-0.882382\pi$$
0.932505 0.361158i $$-0.117618\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.19615i − 1.00000i
$$28$$ 0 0
$$29$$ − 6.00000i − 1.11417i −0.830455 0.557086i $$-0.811919\pi$$
0.830455 0.557086i $$-0.188081\pi$$
$$30$$ 0 0
$$31$$ − 3.46410i − 0.622171i −0.950382 0.311086i $$-0.899307\pi$$
0.950382 0.311086i $$-0.100693\pi$$
$$32$$ 0 0
$$33$$ − 6.00000i − 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 0 0
$$39$$ −6.92820 −1.10940
$$40$$ 0 0
$$41$$ − 12.0000i − 1.87409i −0.349215 0.937043i $$-0.613552\pi$$
0.349215 0.937043i $$-0.386448\pi$$
$$42$$ 0 0
$$43$$ −6.92820 −1.05654 −0.528271 0.849076i $$-0.677159\pi$$
−0.528271 + 0.849076i $$0.677159\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 3.46410i − 0.505291i −0.967559 0.252646i $$-0.918699\pi$$
0.967559 0.252646i $$-0.0813007\pi$$
$$48$$ 0 0
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ 10.3923i 1.45521i
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ 0 0
$$59$$ 3.46410 0.450988 0.225494 0.974245i $$-0.427600\pi$$
0.225494 + 0.974245i $$0.427600\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 10.3923 1.30931
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.92820 0.846415 0.423207 0.906033i $$-0.360904\pi$$
0.423207 + 0.906033i $$0.360904\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −13.8564 −1.64445 −0.822226 0.569160i $$-0.807268\pi$$
−0.822226 + 0.569160i $$0.807268\pi$$
$$72$$ 0 0
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ 10.3923i 1.16923i 0.811312 + 0.584613i $$0.198754\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 10.3923i 1.14070i 0.821401 + 0.570352i $$0.193193\pi$$
−0.821401 + 0.570352i $$0.806807\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.3923 1.11417
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ − 13.8564i − 1.45255i
$$92$$ 0 0
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 0 0
$$99$$ 10.3923 1.04447
$$100$$ 0 0
$$101$$ − 6.00000i − 0.597022i −0.954406 0.298511i $$-0.903510\pi$$
0.954406 0.298511i $$-0.0964900\pi$$
$$102$$ 0 0
$$103$$ −10.3923 −1.02398 −0.511992 0.858990i $$-0.671092\pi$$
−0.511992 + 0.858990i $$0.671092\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 3.46410i − 0.334887i −0.985882 0.167444i $$-0.946449\pi$$
0.985882 0.167444i $$-0.0535512\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −6.92820 −0.657596
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 12.0000i − 1.10940i
$$118$$ 0 0
$$119$$ −20.7846 −1.90532
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 20.7846 1.87409
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.3923 −0.922168 −0.461084 0.887357i $$-0.652539\pi$$
−0.461084 + 0.887357i $$0.652539\pi$$
$$128$$ 0 0
$$129$$ − 12.0000i − 1.05654i
$$130$$ 0 0
$$131$$ 17.3205 1.51330 0.756650 0.653820i $$-0.226835\pi$$
0.756650 + 0.653820i $$0.226835\pi$$
$$132$$ 0 0
$$133$$ 12.0000i 1.04053i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 10.3923i 0.881464i 0.897639 + 0.440732i $$0.145281\pi$$
−0.897639 + 0.440732i $$0.854719\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ − 13.8564i − 1.15873i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.66025i 0.714286i
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ 3.46410i 0.281905i 0.990016 + 0.140952i $$0.0450164\pi$$
−0.990016 + 0.140952i $$0.954984\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 16.0000i 1.27694i 0.769647 + 0.638470i $$0.220432\pi$$
−0.769647 + 0.638470i $$0.779568\pi$$
$$158$$ 0 0
$$159$$ − 10.3923i − 0.824163i
$$160$$ 0 0
$$161$$ 12.0000i 0.945732i
$$162$$ 0 0
$$163$$ −13.8564 −1.08532 −0.542659 0.839953i $$-0.682582\pi$$
−0.542659 + 0.839953i $$0.682582\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 3.46410i − 0.268060i −0.990977 0.134030i $$-0.957208\pi$$
0.990977 0.134030i $$-0.0427919\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 10.3923i 0.794719i
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000i 0.450988i
$$178$$ 0 0
$$179$$ 24.2487 1.81243 0.906217 0.422813i $$-0.138957\pi$$
0.906217 + 0.422813i $$0.138957\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ − 17.3205i − 1.28037i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −20.7846 −1.51992
$$188$$ 0 0
$$189$$ 18.0000i 1.30931i
$$190$$ 0 0
$$191$$ 20.7846 1.50392 0.751961 0.659208i $$-0.229108\pi$$
0.751961 + 0.659208i $$0.229108\pi$$
$$192$$ 0 0
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ − 24.2487i − 1.71895i −0.511182 0.859473i $$-0.670792\pi$$
0.511182 0.859473i $$-0.329208\pi$$
$$200$$ 0 0
$$201$$ 12.0000i 0.846415i
$$202$$ 0 0
$$203$$ 20.7846i 1.45879i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 10.3923i 0.722315i
$$208$$ 0 0
$$209$$ 12.0000i 0.830057i
$$210$$ 0 0
$$211$$ − 17.3205i − 1.19239i −0.802839 0.596196i $$-0.796678\pi$$
0.802839 0.596196i $$-0.203322\pi$$
$$212$$ 0 0
$$213$$ − 24.0000i − 1.64445i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000i 0.814613i
$$218$$ 0 0
$$219$$ 3.46410 0.234082
$$220$$ 0 0
$$221$$ 24.0000i 1.61441i
$$222$$ 0 0
$$223$$ −17.3205 −1.15987 −0.579934 0.814664i $$-0.696921\pi$$
−0.579934 + 0.814664i $$0.696921\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.46410i 0.229920i 0.993370 + 0.114960i $$0.0366741\pi$$
−0.993370 + 0.114960i $$0.963326\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 20.7846i 1.36753i
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −18.0000 −1.16923
$$238$$ 0 0
$$239$$ −13.8564 −0.896296 −0.448148 0.893959i $$-0.647916\pi$$
−0.448148 + 0.893959i $$0.647916\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 0 0
$$243$$ 15.5885i 1.00000i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 13.8564 0.881662
$$248$$ 0 0
$$249$$ −18.0000 −1.14070
$$250$$ 0 0
$$251$$ −3.46410 −0.218652 −0.109326 0.994006i $$-0.534869\pi$$
−0.109326 + 0.994006i $$0.534869\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 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
$$259$$ − 13.8564i − 0.860995i
$$260$$ 0 0
$$261$$ 18.0000i 1.11417i
$$262$$ 0 0
$$263$$ − 3.46410i − 0.213606i −0.994280 0.106803i $$-0.965939\pi$$
0.994280 0.106803i $$-0.0340614\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 6.00000i − 0.365826i −0.983129 0.182913i $$-0.941447\pi$$
0.983129 0.182913i $$-0.0585527\pi$$
$$270$$ 0 0
$$271$$ − 10.3923i − 0.631288i −0.948878 0.315644i $$-0.897780\pi$$
0.948878 0.315644i $$-0.102220\pi$$
$$272$$ 0 0
$$273$$ 24.0000 1.45255
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 4.00000i − 0.240337i −0.992754 0.120168i $$-0.961657\pi$$
0.992754 0.120168i $$-0.0383434\pi$$
$$278$$ 0 0
$$279$$ 10.3923i 0.622171i
$$280$$ 0 0
$$281$$ 12.0000i 0.715860i 0.933748 + 0.357930i $$0.116517\pi$$
−0.933748 + 0.357930i $$0.883483\pi$$
$$282$$ 0 0
$$283$$ 6.92820 0.411839 0.205919 0.978569i $$-0.433982\pi$$
0.205919 + 0.978569i $$0.433982\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 41.5692i 2.45375i
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 17.3205 1.01535
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 18.0000i 1.04447i
$$298$$ 0 0
$$299$$ 13.8564 0.801337
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ 10.3923 0.597022
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −6.92820 −0.395413 −0.197707 0.980261i $$-0.563349\pi$$
−0.197707 + 0.980261i $$0.563349\pi$$
$$308$$ 0 0
$$309$$ − 18.0000i − 1.02398i
$$310$$ 0 0
$$311$$ −13.8564 −0.785725 −0.392862 0.919597i $$-0.628515\pi$$
−0.392862 + 0.919597i $$0.628515\pi$$
$$312$$ 0 0
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ 20.7846i 1.16371i
$$320$$ 0 0
$$321$$ 6.00000 0.334887
$$322$$ 0 0
$$323$$ − 20.7846i − 1.15649i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 17.3205i − 0.957826i
$$328$$ 0 0
$$329$$ 12.0000i 0.661581i
$$330$$ 0 0
$$331$$ 10.3923i 0.571213i 0.958347 + 0.285606i $$0.0921950\pi$$
−0.958347 + 0.285606i $$0.907805\pi$$
$$332$$ 0 0
$$333$$ − 12.0000i − 0.657596i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 34.0000i − 1.85210i −0.377403 0.926049i $$-0.623183\pi$$
0.377403 0.926049i $$-0.376817\pi$$
$$338$$ 0 0
$$339$$ 10.3923i 0.564433i
$$340$$ 0 0
$$341$$ 12.0000i 0.649836i
$$342$$ 0 0
$$343$$ 6.92820 0.374088
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 10.3923i − 0.557888i −0.960307 0.278944i $$-0.910016\pi$$
0.960307 0.278944i $$-0.0899844\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 20.7846 1.10940
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 36.0000i − 1.90532i
$$358$$ 0 0
$$359$$ 6.92820 0.365657 0.182828 0.983145i $$-0.441475\pi$$
0.182828 + 0.983145i $$0.441475\pi$$
$$360$$ 0 0
$$361$$ 7.00000 0.368421
$$362$$ 0 0
$$363$$ 1.73205i 0.0909091i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.46410 0.180825 0.0904123 0.995904i $$-0.471182\pi$$
0.0904123 + 0.995904i $$0.471182\pi$$
$$368$$ 0 0
$$369$$ 36.0000i 1.87409i
$$370$$ 0 0
$$371$$ 20.7846 1.07908
$$372$$ 0 0
$$373$$ − 16.0000i − 0.828449i −0.910175 0.414224i $$-0.864053\pi$$
0.910175 0.414224i $$-0.135947\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 24.2487i 1.24557i 0.782392 + 0.622786i $$0.213999\pi$$
−0.782392 + 0.622786i $$0.786001\pi$$
$$380$$ 0 0
$$381$$ − 18.0000i − 0.922168i
$$382$$ 0 0
$$383$$ − 17.3205i − 0.885037i −0.896759 0.442518i $$-0.854085\pi$$
0.896759 0.442518i $$-0.145915\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 20.7846 1.05654
$$388$$ 0 0
$$389$$ 6.00000i 0.304212i 0.988364 + 0.152106i $$0.0486055\pi$$
−0.988364 + 0.152106i $$0.951394\pi$$
$$390$$ 0 0
$$391$$ − 20.7846i − 1.05112i
$$392$$ 0 0
$$393$$ 30.0000i 1.51330i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 16.0000i 0.803017i 0.915855 + 0.401508i $$0.131514\pi$$
−0.915855 + 0.401508i $$0.868486\pi$$
$$398$$ 0 0
$$399$$ −20.7846 −1.04053
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 13.8564 0.690237
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 13.8564i − 0.686837i
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ − 31.1769i − 1.53784i
$$412$$ 0 0
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −18.0000 −0.881464
$$418$$ 0 0
$$419$$ 10.3923 0.507697 0.253849 0.967244i $$-0.418303\pi$$
0.253849 + 0.967244i $$0.418303\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ 10.3923i 0.505291i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 34.6410 1.67640
$$428$$ 0 0
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ −6.92820 −0.333720 −0.166860 0.985981i $$-0.553363\pi$$
−0.166860 + 0.985981i $$0.553363\pi$$
$$432$$ 0 0
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −12.0000 −0.574038
$$438$$ 0 0
$$439$$ 3.46410i 0.165333i 0.996577 + 0.0826663i $$0.0263436\pi$$
−0.996577 + 0.0826663i $$0.973656\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ 0 0
$$443$$ 17.3205i 0.822922i 0.911427 + 0.411461i $$0.134981\pi$$
−0.911427 + 0.411461i $$0.865019\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −10.3923 −0.491539
$$448$$ 0 0
$$449$$ − 36.0000i − 1.69895i −0.527633 0.849473i $$-0.676920\pi$$
0.527633 0.849473i $$-0.323080\pi$$
$$450$$ 0 0
$$451$$ 41.5692i 1.95742i
$$452$$ 0 0
$$453$$ −6.00000 −0.281905
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ 0 0
$$459$$ − 31.1769i − 1.45521i
$$460$$ 0 0
$$461$$ − 18.0000i − 0.838344i −0.907907 0.419172i $$-0.862320\pi$$
0.907907 0.419172i $$-0.137680\pi$$
$$462$$ 0 0
$$463$$ 10.3923 0.482971 0.241486 0.970404i $$-0.422365\pi$$
0.241486 + 0.970404i $$0.422365\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 24.2487i 1.12210i 0.827783 + 0.561048i $$0.189602\pi$$
−0.827783 + 0.561048i $$0.810398\pi$$
$$468$$ 0 0
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −27.7128 −1.27694
$$472$$ 0 0
$$473$$ 24.0000 1.10352
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 18.0000 0.824163
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 0 0
$$483$$ −20.7846 −0.945732
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −17.3205 −0.784867 −0.392434 0.919780i $$-0.628367\pi$$
−0.392434 + 0.919780i $$0.628367\pi$$
$$488$$ 0 0
$$489$$ − 24.0000i − 1.08532i
$$490$$ 0 0
$$491$$ −31.1769 −1.40699 −0.703497 0.710698i $$-0.748379\pi$$
−0.703497 + 0.710698i $$0.748379\pi$$
$$492$$ 0 0
$$493$$ − 36.0000i − 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 48.0000 2.15309
$$498$$ 0 0
$$499$$ − 3.46410i − 0.155074i −0.996989 0.0775372i $$-0.975294\pi$$
0.996989 0.0775372i $$-0.0247057\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ 0 0
$$503$$ 10.3923i 0.463370i 0.972791 + 0.231685i $$0.0744239\pi$$
−0.972791 + 0.231685i $$0.925576\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 5.19615i − 0.230769i
$$508$$ 0 0
$$509$$ − 6.00000i − 0.265945i −0.991120 0.132973i $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 0 0
$$511$$ 6.92820i 0.306486i
$$512$$ 0 0
$$513$$ −18.0000 −0.794719
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 12.0000i 0.527759i
$$518$$ 0 0
$$519$$ 10.3923i 0.456172i
$$520$$ 0 0
$$521$$ − 12.0000i − 0.525730i −0.964833 0.262865i $$-0.915333\pi$$
0.964833 0.262865i $$-0.0846673\pi$$
$$522$$ 0 0
$$523$$ −34.6410 −1.51475 −0.757373 0.652983i $$-0.773517\pi$$
−0.757373 + 0.652983i $$0.773517\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 20.7846i − 0.905392i
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ −10.3923 −0.450988
$$532$$ 0 0
$$533$$ 48.0000 2.07911
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 42.0000i 1.81243i
$$538$$ 0 0
$$539$$ −17.3205 −0.746047
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ − 3.46410i − 0.148659i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.7846 −0.888686 −0.444343 0.895857i $$-0.646563\pi$$
−0.444343 + 0.895857i $$0.646563\pi$$
$$548$$ 0 0
$$549$$ 30.0000 1.28037
$$550$$ 0 0
$$551$$ −20.7846 −0.885454
$$552$$ 0 0
$$553$$ − 36.0000i − 1.53088i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ − 27.7128i − 1.17213i
$$560$$ 0 0
$$561$$ − 36.0000i − 1.51992i
$$562$$ 0 0
$$563$$ 3.46410i 0.145994i 0.997332 + 0.0729972i $$0.0232564\pi$$
−0.997332 + 0.0729972i $$0.976744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −31.1769 −1.30931
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 24.2487i 1.01478i 0.861717 + 0.507388i $$0.169389\pi$$
−0.861717 + 0.507388i $$0.830611\pi$$
$$572$$ 0 0
$$573$$ 36.0000i 1.50392i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ −3.46410 −0.143963
$$580$$ 0 0
$$581$$ − 36.0000i − 1.49353i
$$582$$ 0 0
$$583$$ 20.7846 0.860811
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 31.1769i − 1.28681i −0.765526 0.643404i $$-0.777521\pi$$
0.765526 0.643404i $$-0.222479\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ − 10.3923i − 0.427482i
$$592$$ 0 0
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 42.0000 1.71895
$$598$$ 0 0
$$599$$ −20.7846 −0.849236 −0.424618 0.905373i $$-0.639592\pi$$
−0.424618 + 0.905373i $$0.639592\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ −20.7846 −0.846415
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 31.1769 1.26543 0.632716 0.774384i $$-0.281940\pi$$
0.632716 + 0.774384i $$0.281940\pi$$
$$608$$ 0 0
$$609$$ −36.0000 −1.45879
$$610$$ 0 0
$$611$$ 13.8564 0.560570
$$612$$ 0 0
$$613$$ 32.0000i 1.29247i 0.763139 + 0.646234i $$0.223657\pi$$
−0.763139 + 0.646234i $$0.776343\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ − 45.0333i − 1.81004i −0.425367 0.905021i $$-0.639855\pi$$
0.425367 0.905021i $$-0.360145\pi$$
$$620$$ 0 0
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −20.7846 −0.830057
$$628$$ 0 0
$$629$$ 24.0000i 0.956943i
$$630$$ 0 0
$$631$$ − 3.46410i − 0.137904i −0.997620 0.0689519i $$-0.978035\pi$$
0.997620 0.0689519i $$-0.0219655\pi$$
$$632$$ 0 0
$$633$$ 30.0000 1.19239
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 20.0000i 0.792429i
$$638$$ 0 0
$$639$$ 41.5692 1.64445
$$640$$ 0 0
$$641$$ − 24.0000i − 0.947943i −0.880540 0.473972i $$-0.842820\pi$$
0.880540 0.473972i $$-0.157180\pi$$
$$642$$ 0 0
$$643$$ 41.5692 1.63933 0.819665 0.572843i $$-0.194160\pi$$
0.819665 + 0.572843i $$0.194160\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 45.0333i − 1.77044i −0.465170 0.885221i $$-0.654007\pi$$
0.465170 0.885221i $$-0.345993\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −20.7846 −0.814613
$$652$$ 0 0
$$653$$ −42.0000 −1.64359 −0.821794 0.569785i $$-0.807026\pi$$
−0.821794 + 0.569785i $$0.807026\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ −31.1769 −1.21448 −0.607240 0.794518i $$-0.707723\pi$$
−0.607240 + 0.794518i $$0.707723\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 0 0
$$663$$ −41.5692 −1.61441
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −20.7846 −0.804783
$$668$$ 0 0
$$669$$ − 30.0000i − 1.15987i
$$670$$ 0 0
$$671$$ 34.6410 1.33730
$$672$$ 0 0
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ 0 0
$$679$$ 34.6410i 1.32940i
$$680$$ 0 0
$$681$$ −6.00000 −0.229920
$$682$$ 0 0
$$683$$ − 17.3205i − 0.662751i −0.943499 0.331375i $$-0.892487\pi$$
0.943499 0.331375i $$-0.107513\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 24.2487i 0.925146i
$$688$$ 0 0
$$689$$ − 24.0000i − 0.914327i
$$690$$ 0 0
$$691$$ − 31.1769i − 1.18603i −0.805193 0.593013i $$-0.797938\pi$$
0.805193 0.593013i $$-0.202062\pi$$
$$692$$ 0 0
$$693$$ −36.0000 −1.36753
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 72.0000i − 2.72719i
$$698$$ 0 0
$$699$$ 10.3923i 0.393073i
$$700$$ 0 0
$$701$$ − 42.0000i − 1.58632i −0.609015 0.793159i $$-0.708435\pi$$
0.609015 0.793159i $$-0.291565\pi$$
$$702$$ 0 0
$$703$$ 13.8564 0.522604
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 20.7846i 0.781686i
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ − 31.1769i − 1.16923i
$$712$$ 0 0
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 24.0000i − 0.896296i
$$718$$ 0 0
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 0 0
$$721$$ 36.0000 1.34071
$$722$$ 0 0
$$723$$ − 45.0333i − 1.67481i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −3.46410 −0.128476 −0.0642382 0.997935i $$-0.520462\pi$$
−0.0642382 + 0.997935i $$0.520462\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −41.5692 −1.53749
$$732$$ 0 0
$$733$$ 28.0000i 1.03420i 0.855924 + 0.517102i $$0.172989\pi$$
−0.855924 + 0.517102i $$0.827011\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −24.0000 −0.884051
$$738$$ 0 0
$$739$$ 10.3923i 0.382287i 0.981562 + 0.191144i $$0.0612196\pi$$
−0.981562 + 0.191144i $$0.938780\pi$$
$$740$$ 0 0
$$741$$ 24.0000i 0.881662i
$$742$$ 0 0
$$743$$ 38.1051i 1.39794i 0.715150 + 0.698971i $$0.246358\pi$$
−0.715150 + 0.698971i $$0.753642\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 31.1769i − 1.14070i
$$748$$ 0 0
$$749$$ 12.0000i 0.438470i
$$750$$ 0 0
$$751$$ 51.9615i 1.89610i 0.318117 + 0.948051i $$0.396950\pi$$
−0.318117 + 0.948051i $$0.603050\pi$$
$$752$$ 0 0
$$753$$ − 6.00000i − 0.218652i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 20.0000i 0.726912i 0.931611 + 0.363456i $$0.118403\pi$$
−0.931611 + 0.363456i $$0.881597\pi$$
$$758$$ 0 0
$$759$$ −20.7846 −0.754434
$$760$$ 0 0
$$761$$ − 12.0000i − 0.435000i −0.976060 0.217500i $$-0.930210\pi$$
0.976060 0.217500i $$-0.0697902\pi$$
$$762$$ 0 0
$$763$$ 34.6410 1.25409
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 13.8564i 0.500326i
$$768$$ 0 0
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ − 31.1769i − 1.12281i
$$772$$ 0 0
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 24.0000 0.860995
$$778$$ 0 0
$$779$$ −41.5692 −1.48937
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 0 0
$$783$$ −31.1769 −1.11417
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 34.6410 1.23482 0.617409 0.786642i $$-0.288182\pi$$
0.617409 + 0.786642i $$0.288182\pi$$
$$788$$ 0 0
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ −20.7846 −0.739016
$$792$$ 0 0
$$793$$ − 40.0000i − 1.42044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ − 20.7846i − 0.735307i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 6.92820i 0.244491i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10.3923 0.365826
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 24.2487i 0.851487i 0.904844 + 0.425744i $$0.139987\pi$$
−0.904844 + 0.425744i $$0.860013\pi$$
$$812$$ 0 0
$$813$$ 18.0000 0.631288
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24.0000i 0.839654i
$$818$$ 0 0
$$819$$ 41.5692i 1.45255i
$$820$$ 0 0
$$821$$ − 6.00000i − 0.209401i −0.994504 0.104701i $$-0.966612\pi$$
0.994504 0.104701i $$-0.0333885\pi$$
$$822$$ 0 0
$$823$$ 45.0333 1.56976 0.784881 0.619646i $$-0.212724\pi$$
0.784881 + 0.619646i $$0.212724\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 38.1051i − 1.32504i −0.749042 0.662522i $$-0.769486\pi$$
0.749042 0.662522i $$-0.230514\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 6.92820 0.240337
$$832$$ 0 0
$$833$$ 30.0000 1.03944
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −18.0000 −0.622171
$$838$$ 0 0
$$839$$ 48.4974 1.67432 0.837158 0.546960i $$-0.184215\pi$$
0.837158 + 0.546960i $$0.184215\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ −20.7846 −0.715860
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.46410 −0.119028
$$848$$ 0 0
$$849$$ 12.0000i 0.411839i
$$850$$ 0 0
$$851$$ 13.8564 0.474991
$$852$$ 0 0
$$853$$ 8.00000i 0.273915i 0.990577 + 0.136957i $$0.0437323\pi$$
−0.990577 + 0.136957i $$0.956268\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ 51.9615i 1.77290i 0.462820 + 0.886452i $$0.346838\pi$$
−0.462820 + 0.886452i $$0.653162\pi$$
$$860$$ 0 0
$$861$$ −72.0000 −2.45375
$$862$$ 0 0
$$863$$ 51.9615i 1.76879i 0.466738 + 0.884395i $$0.345429\pi$$
−0.466738 + 0.884395i $$0.654571\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 32.9090i 1.11765i
$$868$$ 0 0
$$869$$ − 36.0000i − 1.22122i
$$870$$ 0 0
$$871$$ 27.7128i 0.939013i
$$872$$ 0 0
$$873$$ 30.0000i 1.01535i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 32.0000i − 1.08056i −0.841484 0.540282i $$-0.818318\pi$$
0.841484 0.540282i $$-0.181682\pi$$
$$878$$ 0 0
$$879$$ − 10.3923i − 0.350524i
$$880$$ 0 0
$$881$$ 48.0000i 1.61716i 0.588386 + 0.808581i $$0.299764\pi$$
−0.588386 + 0.808581i $$0.700236\pi$$
$$882$$ 0 0
$$883$$ 13.8564 0.466305 0.233153 0.972440i $$-0.425096\pi$$
0.233153 + 0.972440i $$0.425096\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.2487i 0.814192i 0.913385 + 0.407096i $$0.133459\pi$$
−0.913385 + 0.407096i $$0.866541\pi$$
$$888$$ 0 0
$$889$$ 36.0000 1.20740
$$890$$ 0 0
$$891$$ −31.1769 −1.04447
$$892$$ 0 0
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 24.0000i 0.801337i
$$898$$ 0 0
$$899$$ −20.7846 −0.693206
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 41.5692i 1.38334i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 41.5692 1.38028 0.690142 0.723674i $$-0.257548\pi$$
0.690142 + 0.723674i $$0.257548\pi$$
$$908$$ 0 0
$$909$$ 18.0000i 0.597022i
$$910$$ 0 0
$$911$$ 34.6410 1.14771 0.573854 0.818958i $$-0.305448\pi$$
0.573854 + 0.818958i $$0.305448\pi$$
$$912$$ 0 0
$$913$$ − 36.0000i − 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −60.0000 −1.98137
$$918$$ 0 0
$$919$$ − 45.0333i − 1.48551i −0.669562 0.742756i $$-0.733518\pi$$
0.669562 0.742756i $$-0.266482\pi$$
$$920$$ 0 0
$$921$$ − 12.0000i − 0.395413i
$$922$$ 0 0
$$923$$ − 55.4256i − 1.82436i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 31.1769 1.02398
$$928$$ 0 0
$$929$$ 12.0000i 0.393707i 0.980433 + 0.196854i $$0.0630724\pi$$
−0.980433 + 0.196854i $$0.936928\pi$$
$$930$$ 0 0
$$931$$ − 17.3205i − 0.567657i
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 38.0000i − 1.24141i −0.784046 0.620703i $$-0.786847\pi$$
0.784046 0.620703i $$-0.213153\pi$$
$$938$$ 0 0
$$939$$ −24.2487 −0.791327
$$940$$ 0 0
$$941$$ 54.0000i 1.76035i 0.474650 + 0.880175i $$0.342575\pi$$
−0.474650 + 0.880175i $$0.657425\pi$$
$$942$$ 0 0
$$943$$ −41.5692 −1.35368
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 31.1769i 1.01311i 0.862207 + 0.506557i $$0.169082\pi$$
−0.862207 + 0.506557i $$0.830918\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ − 31.1769i − 1.01098i
$$952$$ 0 0
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −36.0000 −1.16371
$$958$$ 0 0
$$959$$ 62.3538 2.01351
$$960$$ 0 0
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ 10.3923i 0.334887i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 51.9615 1.67097 0.835485 0.549513i $$-0.185187\pi$$
0.835485 + 0.549513i $$0.185187\pi$$
$$968$$ 0 0
$$969$$ 36.0000 1.15649
$$970$$ 0 0
$$971$$ −3.46410 −0.111168 −0.0555842 0.998454i $$-0.517702\pi$$
−0.0555842 + 0.998454i $$0.517702\pi$$
$$972$$ 0 0
$$973$$ − 36.0000i − 1.15411i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 30.0000 0.957826
$$982$$ 0 0
$$983$$ − 58.8897i − 1.87829i −0.343520 0.939145i $$-0.611619\pi$$
0.343520 0.939145i $$-0.388381\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −20.7846 −0.661581
$$988$$ 0 0
$$989$$ 24.0000i 0.763156i
$$990$$ 0 0
$$991$$ 17.3205i 0.550204i 0.961415 + 0.275102i $$0.0887116\pi$$
−0.961415 + 0.275102i $$0.911288\pi$$
$$992$$ 0 0
$$993$$ −18.0000 −0.571213
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 52.0000i 1.64686i 0.567420 + 0.823428i $$0.307941\pi$$
−0.567420 + 0.823428i $$0.692059\pi$$
$$998$$ 0 0
$$999$$ 20.7846 0.657596
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 1200.2.o.b.1199.3 4
3.2 odd 2 1200.2.o.a.1199.1 4
4.3 odd 2 inner 1200.2.o.b.1199.2 4
5.2 odd 4 240.2.h.b.191.3 yes 4
5.3 odd 4 1200.2.h.m.1151.2 4
5.4 even 2 1200.2.o.a.1199.2 4
12.11 even 2 1200.2.o.a.1199.4 4
15.2 even 4 240.2.h.b.191.2 yes 4
15.8 even 4 1200.2.h.m.1151.4 4
15.14 odd 2 inner 1200.2.o.b.1199.4 4
20.3 even 4 1200.2.h.m.1151.3 4
20.7 even 4 240.2.h.b.191.1 4
20.19 odd 2 1200.2.o.a.1199.3 4
40.27 even 4 960.2.h.d.191.4 4
40.37 odd 4 960.2.h.d.191.2 4
60.23 odd 4 1200.2.h.m.1151.1 4
60.47 odd 4 240.2.h.b.191.4 yes 4
60.59 even 2 inner 1200.2.o.b.1199.1 4
120.77 even 4 960.2.h.d.191.3 4
120.107 odd 4 960.2.h.d.191.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.h.b.191.1 4 20.7 even 4
240.2.h.b.191.2 yes 4 15.2 even 4
240.2.h.b.191.3 yes 4 5.2 odd 4
240.2.h.b.191.4 yes 4 60.47 odd 4
960.2.h.d.191.1 4 120.107 odd 4
960.2.h.d.191.2 4 40.37 odd 4
960.2.h.d.191.3 4 120.77 even 4
960.2.h.d.191.4 4 40.27 even 4
1200.2.h.m.1151.1 4 60.23 odd 4
1200.2.h.m.1151.2 4 5.3 odd 4
1200.2.h.m.1151.3 4 20.3 even 4
1200.2.h.m.1151.4 4 15.8 even 4
1200.2.o.a.1199.1 4 3.2 odd 2
1200.2.o.a.1199.2 4 5.4 even 2
1200.2.o.a.1199.3 4 20.19 odd 2
1200.2.o.a.1199.4 4 12.11 even 2
1200.2.o.b.1199.1 4 60.59 even 2 inner
1200.2.o.b.1199.2 4 4.3 odd 2 inner
1200.2.o.b.1199.3 4 1.1 even 1 trivial
1200.2.o.b.1199.4 4 15.14 odd 2 inner