# Properties

 Label 1200.2.o.f.1199.2 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1199.2 Root $$1.22474 + 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1199 Dual form 1200.2.o.f.1199.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.22474 + 1.22474i) q^{3} +2.44949 q^{7} -3.00000i q^{9} +O(q^{10})$$ $$q+(-1.22474 + 1.22474i) q^{3} +2.44949 q^{7} -3.00000i q^{9} -4.89898 q^{11} +2.00000i q^{13} +6.00000 q^{17} +4.89898i q^{19} +(-3.00000 + 3.00000i) q^{21} -2.44949i q^{23} +(3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(6.00000 - 6.00000i) q^{33} +2.00000i q^{37} +(-2.44949 - 2.44949i) q^{39} -6.00000i q^{41} +2.44949 q^{43} +12.2474i q^{47} -1.00000 q^{49} +(-7.34847 + 7.34847i) q^{51} -6.00000 q^{53} +(-6.00000 - 6.00000i) q^{57} -9.79796 q^{59} +8.00000 q^{61} -7.34847i q^{63} +7.34847 q^{67} +(3.00000 + 3.00000i) q^{69} -4.89898 q^{71} +14.0000i q^{73} -12.0000 q^{77} -4.89898i q^{79} -9.00000 q^{81} +7.34847i q^{83} +12.0000i q^{89} +4.89898i q^{91} +(-12.0000 - 12.0000i) q^{93} +10.0000i q^{97} +14.6969i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 24q^{17} - 12q^{21} + 24q^{33} - 4q^{49} - 24q^{53} - 24q^{57} + 32q^{61} + 12q^{69} - 48q^{77} - 36q^{81} - 48q^{93} + O(q^{100})$$

## 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
$$3$$ −1.22474 + 1.22474i −0.707107 + 0.707107i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.44949 0.925820 0.462910 0.886405i $$-0.346805\pi$$
0.462910 + 0.886405i $$0.346805\pi$$
$$8$$ 0 0
$$9$$ 3.00000i 1.00000i
$$10$$ 0 0
$$11$$ −4.89898 −1.47710 −0.738549 0.674200i $$-0.764489\pi$$
−0.738549 + 0.674200i $$0.764489\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 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
$$19$$ 4.89898i 1.12390i 0.827170 + 0.561951i $$0.189949\pi$$
−0.827170 + 0.561951i $$0.810051\pi$$
$$20$$ 0 0
$$21$$ −3.00000 + 3.00000i −0.654654 + 0.654654i
$$22$$ 0 0
$$23$$ 2.44949i 0.510754i −0.966842 0.255377i $$-0.917800\pi$$
0.966842 0.255377i $$-0.0821996\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.67423 + 3.67423i 0.707107 + 0.707107i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 9.79796i 1.75977i 0.475191 + 0.879883i $$0.342379\pi$$
−0.475191 + 0.879883i $$0.657621\pi$$
$$32$$ 0 0
$$33$$ 6.00000 6.00000i 1.04447 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −2.44949 2.44949i −0.392232 0.392232i
$$40$$ 0 0
$$41$$ 6.00000i 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ 0 0
$$43$$ 2.44949 0.373544 0.186772 0.982403i $$-0.440197\pi$$
0.186772 + 0.982403i $$0.440197\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.2474i 1.78647i 0.449586 + 0.893237i $$0.351571\pi$$
−0.449586 + 0.893237i $$0.648429\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −7.34847 + 7.34847i −1.02899 + 1.02899i
$$52$$ 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 6.00000i −0.794719 0.794719i
$$58$$ 0 0
$$59$$ −9.79796 −1.27559 −0.637793 0.770208i $$-0.720152\pi$$
−0.637793 + 0.770208i $$0.720152\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 0 0
$$63$$ 7.34847i 0.925820i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.34847 0.897758 0.448879 0.893592i $$-0.351823\pi$$
0.448879 + 0.893592i $$0.351823\pi$$
$$68$$ 0 0
$$69$$ 3.00000 + 3.00000i 0.361158 + 0.361158i
$$70$$ 0 0
$$71$$ −4.89898 −0.581402 −0.290701 0.956814i $$-0.593888\pi$$
−0.290701 + 0.956814i $$0.593888\pi$$
$$72$$ 0 0
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ 4.89898i 0.551178i −0.961276 0.275589i $$-0.911127\pi$$
0.961276 0.275589i $$-0.0888729\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ 7.34847i 0.806599i 0.915068 + 0.403300i $$0.132137\pi$$
−0.915068 + 0.403300i $$0.867863\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.0000i 1.27200i 0.771690 + 0.635999i $$0.219412\pi$$
−0.771690 + 0.635999i $$0.780588\pi$$
$$90$$ 0 0
$$91$$ 4.89898i 0.513553i
$$92$$ 0 0
$$93$$ −12.0000 12.0000i −1.24434 1.24434i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ 14.6969i 1.47710i
$$100$$ 0 0
$$101$$ 12.0000i 1.19404i 0.802225 + 0.597022i $$0.203650\pi$$
−0.802225 + 0.597022i $$0.796350\pi$$
$$102$$ 0 0
$$103$$ −12.2474 −1.20678 −0.603388 0.797447i $$-0.706183\pi$$
−0.603388 + 0.797447i $$0.706183\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.44949i 0.236801i −0.992966 0.118401i $$-0.962223\pi$$
0.992966 0.118401i $$-0.0377767\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ −2.44949 2.44949i −0.232495 0.232495i
$$112$$ 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
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ 14.6969 1.34727
$$120$$ 0 0
$$121$$ 13.0000 1.18182
$$122$$ 0 0
$$123$$ 7.34847 + 7.34847i 0.662589 + 0.662589i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.34847 0.652071 0.326036 0.945357i $$-0.394287\pi$$
0.326036 + 0.945357i $$0.394287\pi$$
$$128$$ 0 0
$$129$$ −3.00000 + 3.00000i −0.264135 + 0.264135i
$$130$$ 0 0
$$131$$ −4.89898 −0.428026 −0.214013 0.976831i $$-0.568653\pi$$
−0.214013 + 0.976831i $$0.568653\pi$$
$$132$$ 0 0
$$133$$ 12.0000i 1.04053i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 4.89898i 0.415526i 0.978179 + 0.207763i $$0.0666183\pi$$
−0.978179 + 0.207763i $$0.933382\pi$$
$$140$$ 0 0
$$141$$ −15.0000 15.0000i −1.26323 1.26323i
$$142$$ 0 0
$$143$$ 9.79796i 0.819346i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.22474 1.22474i 0.101015 0.101015i
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i −0.675556 0.737309i $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ 0 0
$$151$$ 19.5959i 1.59469i −0.603522 0.797347i $$-0.706236\pi$$
0.603522 0.797347i $$-0.293764\pi$$
$$152$$ 0 0
$$153$$ 18.0000i 1.45521i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 0 0
$$159$$ 7.34847 7.34847i 0.582772 0.582772i
$$160$$ 0 0
$$161$$ 6.00000i 0.472866i
$$162$$ 0 0
$$163$$ 7.34847 0.575577 0.287788 0.957694i $$-0.407080\pi$$
0.287788 + 0.957694i $$0.407080\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17.1464i 1.32683i −0.748251 0.663415i $$-0.769106\pi$$
0.748251 0.663415i $$-0.230894\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 14.6969 1.12390
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000 12.0000i 0.901975 0.901975i
$$178$$ 0 0
$$179$$ −9.79796 −0.732334 −0.366167 0.930549i $$-0.619330\pi$$
−0.366167 + 0.930549i $$0.619330\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ −9.79796 + 9.79796i −0.724286 + 0.724286i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −29.3939 −2.14949
$$188$$ 0 0
$$189$$ 9.00000 + 9.00000i 0.654654 + 0.654654i
$$190$$ 0 0
$$191$$ 14.6969 1.06343 0.531717 0.846922i $$-0.321547\pi$$
0.531717 + 0.846922i $$0.321547\pi$$
$$192$$ 0 0
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 14.6969i 1.04184i −0.853606 0.520919i $$-0.825589\pi$$
0.853606 0.520919i $$-0.174411\pi$$
$$200$$ 0 0
$$201$$ −9.00000 + 9.00000i −0.634811 + 0.634811i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −7.34847 −0.510754
$$208$$ 0 0
$$209$$ 24.0000i 1.66011i
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 6.00000 6.00000i 0.411113 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 24.0000i 1.62923i
$$218$$ 0 0
$$219$$ −17.1464 17.1464i −1.15865 1.15865i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 2.44949 0.164030 0.0820150 0.996631i $$-0.473864\pi$$
0.0820150 + 0.996631i $$0.473864\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.44949i 0.162578i 0.996691 + 0.0812892i $$0.0259037\pi$$
−0.996691 + 0.0812892i $$0.974096\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 14.6969 14.6969i 0.966988 0.966988i
$$232$$ 0 0
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 6.00000 + 6.00000i 0.389742 + 0.389742i
$$238$$ 0 0
$$239$$ 9.79796 0.633777 0.316889 0.948463i $$-0.397362\pi$$
0.316889 + 0.948463i $$0.397362\pi$$
$$240$$ 0 0
$$241$$ 4.00000 0.257663 0.128831 0.991667i $$-0.458877\pi$$
0.128831 + 0.991667i $$0.458877\pi$$
$$242$$ 0 0
$$243$$ 11.0227 11.0227i 0.707107 0.707107i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9.79796 −0.623429
$$248$$ 0 0
$$249$$ −9.00000 9.00000i −0.570352 0.570352i
$$250$$ 0 0
$$251$$ 24.4949 1.54610 0.773052 0.634343i $$-0.218729\pi$$
0.773052 + 0.634343i $$0.218729\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 4.89898i 0.304408i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2.44949i 0.151042i −0.997144 0.0755210i $$-0.975938\pi$$
0.997144 0.0755210i $$-0.0240620\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −14.6969 14.6969i −0.899438 0.899438i
$$268$$ 0 0
$$269$$ 18.0000i 1.09748i 0.835993 + 0.548740i $$0.184892\pi$$
−0.835993 + 0.548740i $$0.815108\pi$$
$$270$$ 0 0
$$271$$ 9.79796i 0.595184i −0.954693 0.297592i $$-0.903817\pi$$
0.954693 0.297592i $$-0.0961834\pi$$
$$272$$ 0 0
$$273$$ −6.00000 6.00000i −0.363137 0.363137i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000i 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ 29.3939 1.75977
$$280$$ 0 0
$$281$$ 6.00000i 0.357930i 0.983855 + 0.178965i $$0.0572749\pi$$
−0.983855 + 0.178965i $$0.942725\pi$$
$$282$$ 0 0
$$283$$ 26.9444 1.60168 0.800839 0.598880i $$-0.204387\pi$$
0.800839 + 0.598880i $$0.204387\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 14.6969i 0.867533i
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −12.2474 12.2474i −0.717958 0.717958i
$$292$$ 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
$$297$$ −18.0000 18.0000i −1.04447 1.04447i
$$298$$ 0 0
$$299$$ 4.89898 0.283315
$$300$$ 0 0
$$301$$ 6.00000 0.345834
$$302$$ 0 0
$$303$$ −14.6969 14.6969i −0.844317 0.844317i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.44949 0.139800 0.0698999 0.997554i $$-0.477732\pi$$
0.0698999 + 0.997554i $$0.477732\pi$$
$$308$$ 0 0
$$309$$ 15.0000 15.0000i 0.853320 0.853320i
$$310$$ 0 0
$$311$$ 24.4949 1.38898 0.694489 0.719503i $$-0.255630\pi$$
0.694489 + 0.719503i $$0.255630\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 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
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 3.00000 + 3.00000i 0.167444 + 0.167444i
$$322$$ 0 0
$$323$$ 29.3939i 1.63552i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4.89898 4.89898i 0.270914 0.270914i
$$328$$ 0 0
$$329$$ 30.0000i 1.65395i
$$330$$ 0 0
$$331$$ 19.5959i 1.07709i −0.842597 0.538545i $$-0.818974\pi$$
0.842597 0.538545i $$-0.181026\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 0 0
$$339$$ −7.34847 + 7.34847i −0.399114 + 0.399114i
$$340$$ 0 0
$$341$$ 48.0000i 2.59935i
$$342$$ 0 0
$$343$$ −19.5959 −1.05808
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.34847i 0.394486i −0.980355 0.197243i $$-0.936801\pi$$
0.980355 0.197243i $$-0.0631989\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −7.34847 + 7.34847i −0.392232 + 0.392232i
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −18.0000 + 18.0000i −0.952661 + 0.952661i
$$358$$ 0 0
$$359$$ 9.79796 0.517116 0.258558 0.965996i $$-0.416753\pi$$
0.258558 + 0.965996i $$0.416753\pi$$
$$360$$ 0 0
$$361$$ −5.00000 −0.263158
$$362$$ 0 0
$$363$$ −15.9217 + 15.9217i −0.835672 + 0.835672i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −7.34847 −0.383587 −0.191793 0.981435i $$-0.561430\pi$$
−0.191793 + 0.981435i $$0.561430\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ −14.6969 −0.763027
$$372$$ 0 0
$$373$$ 26.0000i 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 14.6969i 0.754931i 0.926024 + 0.377466i $$0.123204\pi$$
−0.926024 + 0.377466i $$0.876796\pi$$
$$380$$ 0 0
$$381$$ −9.00000 + 9.00000i −0.461084 + 0.461084i
$$382$$ 0 0
$$383$$ 2.44949i 0.125163i 0.998040 + 0.0625815i $$0.0199333\pi$$
−0.998040 + 0.0625815i $$0.980067\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7.34847i 0.373544i
$$388$$ 0 0
$$389$$ 18.0000i 0.912636i 0.889817 + 0.456318i $$0.150832\pi$$
−0.889817 + 0.456318i $$0.849168\pi$$
$$390$$ 0 0
$$391$$ 14.6969i 0.743256i
$$392$$ 0 0
$$393$$ 6.00000 6.00000i 0.302660 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ 0 0
$$399$$ −14.6969 14.6969i −0.735767 0.735767i
$$400$$ 0 0
$$401$$ 24.0000i 1.19850i 0.800561 + 0.599251i $$0.204535\pi$$
−0.800561 + 0.599251i $$0.795465\pi$$
$$402$$ 0 0
$$403$$ −19.5959 −0.976142
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.79796i 0.485667i
$$408$$ 0 0
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 0 0
$$411$$ 7.34847 7.34847i 0.362473 0.362473i
$$412$$ 0 0
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −6.00000 6.00000i −0.293821 0.293821i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 0 0
$$423$$ 36.7423 1.78647
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 19.5959 0.948313
$$428$$ 0 0
$$429$$ 12.0000 + 12.0000i 0.579365 + 0.579365i
$$430$$ 0 0
$$431$$ −24.4949 −1.17988 −0.589939 0.807448i $$-0.700848\pi$$
−0.589939 + 0.807448i $$0.700848\pi$$
$$432$$ 0 0
$$433$$ 2.00000i 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 12.0000 0.574038
$$438$$ 0 0
$$439$$ 24.4949i 1.16908i −0.811366 0.584539i $$-0.801275\pi$$
0.811366 0.584539i $$-0.198725\pi$$
$$440$$ 0 0
$$441$$ 3.00000i 0.142857i
$$442$$ 0 0
$$443$$ 17.1464i 0.814651i −0.913283 0.407326i $$-0.866461\pi$$
0.913283 0.407326i $$-0.133539\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 22.0454 + 22.0454i 1.04271 + 1.04271i
$$448$$ 0 0
$$449$$ 6.00000i 0.283158i −0.989927 0.141579i $$-0.954782\pi$$
0.989927 0.141579i $$-0.0452178\pi$$
$$450$$ 0 0
$$451$$ 29.3939i 1.38410i
$$452$$ 0 0
$$453$$ 24.0000 + 24.0000i 1.12762 + 1.12762i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ 0 0
$$459$$ 22.0454 + 22.0454i 1.02899 + 1.02899i
$$460$$ 0 0
$$461$$ 12.0000i 0.558896i −0.960161 0.279448i $$-0.909849\pi$$
0.960161 0.279448i $$-0.0901514\pi$$
$$462$$ 0 0
$$463$$ 2.44949 0.113837 0.0569187 0.998379i $$-0.481872\pi$$
0.0569187 + 0.998379i $$0.481872\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 17.1464i 0.793442i 0.917939 + 0.396721i $$0.129852\pi$$
−0.917939 + 0.396721i $$0.870148\pi$$
$$468$$ 0 0
$$469$$ 18.0000 0.831163
$$470$$ 0 0
$$471$$ 12.2474 + 12.2474i 0.564333 + 0.564333i
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 18.0000i 0.824163i
$$478$$ 0 0
$$479$$ −29.3939 −1.34304 −0.671520 0.740986i $$-0.734358\pi$$
−0.671520 + 0.740986i $$0.734358\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ 7.34847 + 7.34847i 0.334367 + 0.334367i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −22.0454 −0.998973 −0.499486 0.866322i $$-0.666478\pi$$
−0.499486 + 0.866322i $$0.666478\pi$$
$$488$$ 0 0
$$489$$ −9.00000 + 9.00000i −0.406994 + 0.406994i
$$490$$ 0 0
$$491$$ −14.6969 −0.663264 −0.331632 0.943409i $$-0.607599\pi$$
−0.331632 + 0.943409i $$0.607599\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ 4.89898i 0.219308i −0.993970 0.109654i $$-0.965026\pi$$
0.993970 0.109654i $$-0.0349744\pi$$
$$500$$ 0 0
$$501$$ 21.0000 + 21.0000i 0.938211 + 0.938211i
$$502$$ 0 0
$$503$$ 36.7423i 1.63826i −0.573608 0.819130i $$-0.694457\pi$$
0.573608 0.819130i $$-0.305543\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −11.0227 + 11.0227i −0.489535 + 0.489535i
$$508$$ 0 0
$$509$$ 24.0000i 1.06378i −0.846813 0.531891i $$-0.821482\pi$$
0.846813 0.531891i $$-0.178518\pi$$
$$510$$ 0 0
$$511$$ 34.2929i 1.51703i
$$512$$ 0 0
$$513$$ −18.0000 + 18.0000i −0.794719 + 0.794719i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 60.0000i 2.63880i
$$518$$ 0 0
$$519$$ −22.0454 + 22.0454i −0.967686 + 0.967686i
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 2.44949 0.107109 0.0535544 0.998565i $$-0.482945\pi$$
0.0535544 + 0.998565i $$0.482945\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 58.7878i 2.56083i
$$528$$ 0 0
$$529$$ 17.0000 0.739130
$$530$$ 0 0
$$531$$ 29.3939i 1.27559i
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.0000 12.0000i 0.517838 0.517838i
$$538$$ 0 0
$$539$$ 4.89898 0.211014
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ −12.2474 + 12.2474i −0.525588 + 0.525588i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 41.6413 1.78045 0.890227 0.455517i $$-0.150545\pi$$
0.890227 + 0.455517i $$0.150545\pi$$
$$548$$ 0 0
$$549$$ 24.0000i 1.02430i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 12.0000i 0.510292i
$$554$$ 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
$$559$$ 4.89898i 0.207205i
$$560$$ 0 0
$$561$$ 36.0000 36.0000i 1.51992 1.51992i
$$562$$ 0 0
$$563$$ 12.2474i 0.516168i −0.966122 0.258084i $$-0.916909\pi$$
0.966122 0.258084i $$-0.0830912\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −22.0454 −0.925820
$$568$$ 0 0
$$569$$ 30.0000i 1.25767i −0.777541 0.628833i $$-0.783533\pi$$
0.777541 0.628833i $$-0.216467\pi$$
$$570$$ 0 0
$$571$$ 9.79796i 0.410032i 0.978759 + 0.205016i $$0.0657246\pi$$
−0.978759 + 0.205016i $$0.934275\pi$$
$$572$$ 0 0
$$573$$ −18.0000 + 18.0000i −0.751961 + 0.751961i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 38.0000i 1.58196i 0.611842 + 0.790980i $$0.290429\pi$$
−0.611842 + 0.790980i $$0.709571\pi$$
$$578$$ 0 0
$$579$$ −12.2474 12.2474i −0.508987 0.508987i
$$580$$ 0 0
$$581$$ 18.0000i 0.746766i
$$582$$ 0 0
$$583$$ 29.3939 1.21737
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 36.7423i 1.51652i −0.651953 0.758259i $$-0.726050\pi$$
0.651953 0.758259i $$-0.273950\pi$$
$$588$$ 0 0
$$589$$ −48.0000 −1.97781
$$590$$ 0 0
$$591$$ −22.0454 + 22.0454i −0.906827 + 0.906827i
$$592$$ 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
$$597$$ 18.0000 + 18.0000i 0.736691 + 0.736691i
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 0 0
$$603$$ 22.0454i 0.897758i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 26.9444 1.09364 0.546819 0.837251i $$-0.315838\pi$$
0.546819 + 0.837251i $$0.315838\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.4949 −0.990957
$$612$$ 0 0
$$613$$ 38.0000i 1.53481i −0.641165 0.767403i $$-0.721549\pi$$
0.641165 0.767403i $$-0.278451\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ 4.89898i 0.196907i 0.995142 + 0.0984533i $$0.0313895\pi$$
−0.995142 + 0.0984533i $$0.968610\pi$$
$$620$$ 0 0
$$621$$ 9.00000 9.00000i 0.361158 0.361158i
$$622$$ 0 0
$$623$$ 29.3939i 1.17764i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 29.3939 + 29.3939i 1.17388 + 1.17388i
$$628$$ 0 0
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ 0 0
$$639$$ 14.6969i 0.581402i
$$640$$ 0 0
$$641$$ 18.0000i 0.710957i 0.934684 + 0.355479i $$0.115682\pi$$
−0.934684 + 0.355479i $$0.884318\pi$$
$$642$$ 0 0
$$643$$ −36.7423 −1.44898 −0.724488 0.689287i $$-0.757924\pi$$
−0.724488 + 0.689287i $$0.757924\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 26.9444i 1.05929i 0.848218 + 0.529647i $$0.177675\pi$$
−0.848218 + 0.529647i $$0.822325\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ −29.3939 29.3939i −1.15204 1.15204i
$$652$$ 0 0
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 42.0000 1.63858
$$658$$ 0 0
$$659$$ −29.3939 −1.14502 −0.572511 0.819897i $$-0.694031\pi$$
−0.572511 + 0.819897i $$0.694031\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 0 0
$$663$$ −14.6969 14.6969i −0.570782 0.570782i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −3.00000 + 3.00000i −0.115987 + 0.115987i
$$670$$ 0 0
$$671$$ −39.1918 −1.51298
$$672$$ 0 0
$$673$$ 26.0000i 1.00223i −0.865382 0.501113i $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 24.4949i 0.940028i
$$680$$ 0 0
$$681$$ −3.00000 3.00000i −0.114960 0.114960i
$$682$$ 0 0
$$683$$ 31.8434i 1.21845i 0.792996 + 0.609226i $$0.208520\pi$$
−0.792996 + 0.609226i $$0.791480\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 26.9444 26.9444i 1.02799 1.02799i
$$688$$ 0 0
$$689$$ 12.0000i 0.457164i
$$690$$ 0 0
$$691$$ 29.3939i 1.11820i 0.829102 + 0.559098i $$0.188852\pi$$
−0.829102 + 0.559098i $$0.811148\pi$$
$$692$$ 0 0
$$693$$ 36.0000i 1.36753i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ 0 0
$$699$$ 22.0454 22.0454i 0.833834 0.833834i
$$700$$ 0 0
$$701$$ 30.0000i 1.13308i 0.824033 + 0.566542i $$0.191719\pi$$
−0.824033 + 0.566542i $$0.808281\pi$$
$$702$$ 0 0
$$703$$ −9.79796 −0.369537
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 29.3939i 1.10547i
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −14.6969 −0.551178
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −12.0000 + 12.0000i −0.448148 + 0.448148i
$$718$$ 0 0
$$719$$ 9.79796 0.365402 0.182701 0.983169i $$-0.441516\pi$$
0.182701 + 0.983169i $$0.441516\pi$$
$$720$$ 0 0
$$721$$ −30.0000 −1.11726
$$722$$ 0 0
$$723$$ −4.89898 + 4.89898i −0.182195 + 0.182195i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 26.9444 0.999312 0.499656 0.866224i $$-0.333460\pi$$
0.499656 + 0.866224i $$0.333460\pi$$
$$728$$ 0 0
$$729$$ 27.0000i 1.00000i
$$730$$ 0 0
$$731$$ 14.6969 0.543586
$$732$$ 0 0
$$733$$ 34.0000i 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −36.0000 −1.32608
$$738$$ 0 0
$$739$$ 14.6969i 0.540636i 0.962771 + 0.270318i $$0.0871288\pi$$
−0.962771 + 0.270318i $$0.912871\pi$$
$$740$$ 0 0
$$741$$ 12.0000 12.0000i 0.440831 0.440831i
$$742$$ 0 0
$$743$$ 41.6413i 1.52767i 0.645410 + 0.763836i $$0.276686\pi$$
−0.645410 + 0.763836i $$0.723314\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 22.0454 0.806599
$$748$$ 0 0
$$749$$ 6.00000i 0.219235i
$$750$$ 0 0
$$751$$ 19.5959i 0.715065i 0.933901 + 0.357533i $$0.116382\pi$$
−0.933901 + 0.357533i $$0.883618\pi$$
$$752$$ 0 0
$$753$$ −30.0000 + 30.0000i −1.09326 + 1.09326i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 0 0
$$759$$ −14.6969 14.6969i −0.533465 0.533465i
$$760$$ 0 0
$$761$$ 48.0000i 1.74000i −0.493053 0.869999i $$-0.664119\pi$$
0.493053 0.869999i $$-0.335881\pi$$
$$762$$ 0 0
$$763$$ −9.79796 −0.354710
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 19.5959i 0.707568i
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −7.34847 + 7.34847i −0.264649 + 0.264649i
$$772$$ 0 0
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −6.00000 6.00000i −0.215249 0.215249i
$$778$$ 0 0
$$779$$ 29.3939 1.05314
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −46.5403 −1.65898 −0.829491 0.558520i $$-0.811370\pi$$
−0.829491 + 0.558520i $$0.811370\pi$$
$$788$$ 0 0
$$789$$ 3.00000 + 3.00000i 0.106803 + 0.106803i
$$790$$ 0 0
$$791$$ 14.6969 0.522563
$$792$$ 0 0
$$793$$ 16.0000i 0.568177i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ 73.4847i 2.59970i
$$800$$ 0 0
$$801$$ 36.0000 1.27200
$$802$$ 0 0
$$803$$ 68.5857i 2.42034i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0