# Properties

 Label 1200.2.o.a.1199.4 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: $$x^{4} - x^{2} + 1$$ 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 1199.4 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1199 Dual form 1200.2.o.a.1199.2

## $q$-expansion

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

## 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 $$-0.272814\pi$$
0.654654 + 0.755929i $$0.272814\pi$$
$$8$$ 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 $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ 3.46410i 0.794719i 0.917663 + 0.397360i $$0.130073\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 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 $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ 3.46410i 0.622171i 0.950382 + 0.311086i $$0.100693\pi$$
−0.950382 + 0.311086i $$0.899307\pi$$
$$32$$ 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 $$0.386448\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 0 0
$$43$$ 6.92820 1.05654 0.528271 0.849076i $$-0.322841\pi$$
0.528271 + 0.849076i $$0.322841\pi$$
$$44$$ 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 $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 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 $$-0.639096\pi$$
−0.423207 + 0.906033i $$0.639096\pi$$
$$68$$ 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 $$-0.801246\pi$$
0.811312 0.584613i $$-0.198754\pi$$
$$80$$ 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 $$0.0964900\pi$$
−0.954406 + 0.298511i $$0.903510\pi$$
$$102$$ 0 0
$$103$$ 10.3923 1.02398 0.511992 0.858990i $$-0.328908\pi$$
0.511992 + 0.858990i $$0.328908\pi$$
$$104$$ 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 $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 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 $$-0.347461\pi$$
0.461084 + 0.887357i $$0.347461\pi$$
$$128$$ 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 $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ − 10.3923i − 0.881464i −0.897639 0.440732i $$-0.854719\pi$$
0.897639 0.440732i $$-0.145281\pi$$
$$140$$ 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 $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 0 0
$$151$$ − 3.46410i − 0.281905i −0.990016 0.140952i $$-0.954984\pi$$
0.990016 0.140952i $$-0.0450164\pi$$
$$152$$ 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 $$-0.317418\pi$$
0.542659 + 0.839953i $$0.317418\pi$$
$$164$$ 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 $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 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 $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 24.2487i 1.71895i 0.511182 + 0.859473i $$0.329208\pi$$
−0.511182 + 0.859473i $$0.670792\pi$$
$$200$$ 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 $$0.203322\pi$$
−0.802839 + 0.596196i $$0.796678\pi$$
$$212$$ 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 $$-0.303079\pi$$
0.579934 + 0.814664i $$0.303079\pi$$
$$224$$ 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 $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 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 $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 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 $$0.0585527\pi$$
−0.983129 + 0.182913i $$0.941447\pi$$
$$270$$ 0 0
$$271$$ 10.3923i 0.631288i 0.948878 + 0.315644i $$0.102220\pi$$
−0.948878 + 0.315644i $$0.897780\pi$$
$$272$$ 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 $$-0.883483\pi$$
0.933748 0.357930i $$-0.116517\pi$$
$$282$$ 0 0
$$283$$ −6.92820 −0.411839 −0.205919 0.978569i $$-0.566018\pi$$
−0.205919 + 0.978569i $$0.566018\pi$$
$$284$$ 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 $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 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 $$-0.436651\pi$$
0.197707 + 0.980261i $$0.436651\pi$$
$$308$$ 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 $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 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 $$-0.907805\pi$$
0.958347 0.285606i $$-0.0921950\pi$$
$$332$$ 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 $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 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 $$-0.528818\pi$$
−0.0904123 + 0.995904i $$0.528818\pi$$
$$368$$ 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 $$-0.786001\pi$$
0.782392 0.622786i $$-0.213999\pi$$
$$380$$ 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 $$-0.951394\pi$$
0.988364 0.152106i $$-0.0486055\pi$$
$$390$$ 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 $$-0.973656\pi$$
0.996577 0.0826663i $$-0.0263436\pi$$
$$440$$ 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 $$0.323080\pi$$
−0.527633 + 0.849473i $$0.676920\pi$$
$$450$$ 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 $$0.137680\pi$$
−0.907907 + 0.419172i $$0.862320\pi$$
$$462$$ 0 0
$$463$$ −10.3923 −0.482971 −0.241486 0.970404i $$-0.577635\pi$$
−0.241486 + 0.970404i $$0.577635\pi$$
$$464$$ 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 $$-0.371633\pi$$
0.392434 + 0.919780i $$0.371633\pi$$
$$488$$ 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 $$0.0247057\pi$$
−0.996989 + 0.0775372i $$0.975294\pi$$
$$500$$ 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 $$0.0424523\pi$$
−0.991120 + 0.132973i $$0.957548\pi$$
$$510$$ 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 $$0.0846673\pi$$
−0.964833 + 0.262865i $$0.915333\pi$$
$$522$$ 0 0
$$523$$ 34.6410 1.51475 0.757373 0.652983i $$-0.226483\pi$$
0.757373 + 0.652983i $$0.226483\pi$$
$$524$$ 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 $$-0.353437\pi$$
0.444343 + 0.895857i $$0.353437\pi$$
$$548$$ 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 $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 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 $$-0.830611\pi$$
0.861717 0.507388i $$-0.169389\pi$$
$$572$$ 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 $$-0.168981\pi$$
0.862367 + 0.506284i $$0.168981\pi$$
$$594$$ 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 $$-0.718060\pi$$
−0.632716 + 0.774384i $$0.718060\pi$$
$$608$$ 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 $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 45.0333i 1.81004i 0.425367 + 0.905021i $$0.360145\pi$$
−0.425367 + 0.905021i $$0.639855\pi$$
$$620$$ 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 $$0.0219655\pi$$
−0.997620 + 0.0689519i $$0.978035\pi$$
$$632$$ 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 $$0.157180\pi$$
−0.880540 + 0.473972i $$0.842820\pi$$
$$642$$ 0 0
$$643$$ −41.5692 −1.63933 −0.819665 0.572843i $$-0.805840\pi$$
−0.819665 + 0.572843i $$0.805840\pi$$
$$644$$ 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 $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ 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 $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ 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 $$0.202062\pi$$
−0.805193 + 0.593013i $$0.797938\pi$$
$$692$$ 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 $$0.291565\pi$$
−0.609015 + 0.793159i $$0.708435\pi$$
$$702$$ 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 $$-0.479538\pi$$
0.0642382 + 0.997935i $$0.479538\pi$$
$$728$$ 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 $$-0.938780\pi$$
0.981562 0.191144i $$-0.0612196\pi$$
$$740$$ 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 $$-0.603050\pi$$
0.318117 0.948051i $$-0.396950\pi$$
$$752$$ 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 $$0.0697902\pi$$
−0.976060 + 0.217500i $$0.930210\pi$$
$$762$$ 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 $$-0.923318\pi$$
−0.971123 + 0.238581i $$0.923318\pi$$
$$774$$ 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 $$-0.711818\pi$$
−0.617409 + 0.786642i $$0.711818\pi$$
$$788$$ 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 $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 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 $$-0.860013\pi$$
0.904844 0.425744i $$-0.139987\pi$$
$$812$$ 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 $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ 0 0
$$823$$ −45.0333 −1.56976 −0.784881 0.619646i $$-0.787276\pi$$
−0.784881 + 0.619646i $$0.787276\pi$$
$$824$$ 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 $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ − 51.9615i − 1.77290i −0.462820 0.886452i $$-0.653162\pi$$
0.462820 0.886452i $$-0.346838\pi$$
$$860$$ 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 $$-0.700236\pi$$
0.588386 0.808581i $$-0.299764\pi$$
$$882$$ 0 0
$$883$$ −13.8564 −0.466305 −0.233153 0.972440i $$-0.574904\pi$$
−0.233153 + 0.972440i $$0.574904\pi$$
$$884$$ 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 $$-0.742452\pi$$
−0.690142 + 0.723674i $$0.742452\pi$$
$$908$$ 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 $$0.266482\pi$$
−0.669562 + 0.742756i $$0.733518\pi$$
$$920$$ 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 $$-0.936928\pi$$
0.980433 0.196854i $$-0.0630724\pi$$
$$930$$ 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 $$-0.657425\pi$$
0.474650 0.880175i $$-0.342575\pi$$
$$942$$ 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 $$-0.405833\pi$$
0.291539 + 0.956559i $$0.405833\pi$$
$$954$$ 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 $$-0.814813\pi$$
−0.835485 + 0.549513i $$0.814813\pi$$
$$968$$ 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 $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 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 $$-0.911288\pi$$
0.961415 0.275102i $$-0.0887116\pi$$
$$992$$ 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.a.1199.4 4
3.2 odd 2 1200.2.o.b.1199.2 4
4.3 odd 2 inner 1200.2.o.a.1199.1 4
5.2 odd 4 240.2.h.b.191.4 yes 4
5.3 odd 4 1200.2.h.m.1151.1 4
5.4 even 2 1200.2.o.b.1199.1 4
12.11 even 2 1200.2.o.b.1199.3 4
15.2 even 4 240.2.h.b.191.1 4
15.8 even 4 1200.2.h.m.1151.3 4
15.14 odd 2 inner 1200.2.o.a.1199.3 4
20.3 even 4 1200.2.h.m.1151.4 4
20.7 even 4 240.2.h.b.191.2 yes 4
20.19 odd 2 1200.2.o.b.1199.4 4
40.27 even 4 960.2.h.d.191.3 4
40.37 odd 4 960.2.h.d.191.1 4
60.23 odd 4 1200.2.h.m.1151.2 4
60.47 odd 4 240.2.h.b.191.3 yes 4
60.59 even 2 inner 1200.2.o.a.1199.2 4
120.77 even 4 960.2.h.d.191.4 4
120.107 odd 4 960.2.h.d.191.2 4

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