# Properties

 Label 1200.2.h.k.1151.2 Level $1200$ Weight $2$ Character 1200.1151 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.h (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1151.2 Root $$-1.22474 + 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1151 Dual form 1200.2.h.k.1151.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.22474 + 1.22474i) q^{3} +2.44949i q^{7} -3.00000i q^{9} +O(q^{10})$$ $$q+(-1.22474 + 1.22474i) q^{3} +2.44949i q^{7} -3.00000i q^{9} +4.89898 q^{11} +2.00000 q^{13} -6.00000i q^{17} -4.89898i q^{19} +(-3.00000 - 3.00000i) q^{21} +2.44949 q^{23} +(3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(-6.00000 + 6.00000i) q^{33} -2.00000 q^{37} +(-2.44949 + 2.44949i) q^{39} +6.00000i q^{41} -2.44949i q^{43} +12.2474 q^{47} +1.00000 q^{49} +(7.34847 + 7.34847i) q^{51} -6.00000i q^{53} +(6.00000 + 6.00000i) q^{57} -9.79796 q^{59} +8.00000 q^{61} +7.34847 q^{63} +7.34847i q^{67} +(-3.00000 + 3.00000i) q^{69} +4.89898 q^{71} +14.0000 q^{73} +12.0000i q^{77} +4.89898i q^{79} -9.00000 q^{81} -7.34847 q^{83} +12.0000i q^{89} +4.89898i q^{91} +(-12.0000 - 12.0000i) q^{93} -10.0000 q^{97} -14.6969i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 8q^{13} - 12q^{21} - 24q^{33} - 8q^{37} + 4q^{49} + 24q^{57} + 32q^{61} - 12q^{69} + 56q^{73} - 36q^{81} - 48q^{93} - 40q^{97} + O(q^{100})$$

## 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
$$7$$ 2.44949i 0.925820i 0.886405 + 0.462910i $$0.153195\pi$$
−0.886405 + 0.462910i $$0.846805\pi$$
$$8$$ 0 0
$$9$$ 3.00000i 1.00000i
$$10$$ 0 0
$$11$$ 4.89898 1.47710 0.738549 0.674200i $$-0.235511\pi$$
0.738549 + 0.674200i $$0.235511\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000i 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ 4.89898i 1.12390i −0.827170 0.561951i $$-0.810051\pi$$
0.827170 0.561951i $$-0.189949\pi$$
$$20$$ 0 0
$$21$$ −3.00000 3.00000i −0.654654 0.654654i
$$22$$ 0 0
$$23$$ 2.44949 0.510754 0.255377 0.966842i $$-0.417800\pi$$
0.255377 + 0.966842i $$0.417800\pi$$
$$24$$ 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
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −2.44949 + 2.44949i −0.392232 + 0.392232i
$$40$$ 0 0
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 0 0
$$43$$ 2.44949i 0.373544i −0.982403 0.186772i $$-0.940197\pi$$
0.982403 0.186772i $$-0.0598025\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.2474 1.78647 0.893237 0.449586i $$-0.148429\pi$$
0.893237 + 0.449586i $$0.148429\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 7.34847 + 7.34847i 1.02899 + 1.02899i
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 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
$$63$$ 7.34847 0.925820
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.34847i 0.897758i 0.893592 + 0.448879i $$0.148177\pi$$
−0.893592 + 0.448879i $$0.851823\pi$$
$$68$$ 0 0
$$69$$ −3.00000 + 3.00000i −0.361158 + 0.361158i
$$70$$ 0 0
$$71$$ 4.89898 0.581402 0.290701 0.956814i $$-0.406112\pi$$
0.290701 + 0.956814i $$0.406112\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 12.0000i 1.36753i
$$78$$ 0 0
$$79$$ 4.89898i 0.551178i 0.961276 + 0.275589i $$0.0888729\pi$$
−0.961276 + 0.275589i $$0.911127\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ −7.34847 −0.806599 −0.403300 0.915068i $$-0.632137\pi$$
−0.403300 + 0.915068i $$0.632137\pi$$
$$84$$ 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
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ 14.6969i 1.47710i
$$100$$ 0 0
$$101$$ 12.0000i 1.19404i −0.802225 0.597022i $$-0.796350\pi$$
0.802225 0.597022i $$-0.203650\pi$$
$$102$$ 0 0
$$103$$ 12.2474i 1.20678i 0.797447 + 0.603388i $$0.206183\pi$$
−0.797447 + 0.603388i $$0.793817\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.44949 −0.236801 −0.118401 0.992966i $$-0.537777\pi$$
−0.118401 + 0.992966i $$0.537777\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 2.44949 2.44949i 0.232495 0.232495i
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 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
$$127$$ 7.34847i 0.652071i 0.945357 + 0.326036i $$0.105713\pi$$
−0.945357 + 0.326036i $$0.894287\pi$$
$$128$$ 0 0
$$129$$ 3.00000 + 3.00000i 0.264135 + 0.264135i
$$130$$ 0 0
$$131$$ 4.89898 0.428026 0.214013 0.976831i $$-0.431347\pi$$
0.214013 + 0.976831i $$0.431347\pi$$
$$132$$ 0 0
$$133$$ 12.0000 1.04053
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 4.89898i 0.415526i −0.978179 0.207763i $$-0.933382\pi$$
0.978179 0.207763i $$-0.0666183\pi$$
$$140$$ 0 0
$$141$$ −15.0000 + 15.0000i −1.26323 + 1.26323i
$$142$$ 0 0
$$143$$ 9.79796 0.819346
$$144$$ 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
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ 7.34847 + 7.34847i 0.582772 + 0.582772i
$$160$$ 0 0
$$161$$ 6.00000i 0.472866i
$$162$$ 0 0
$$163$$ 7.34847i 0.575577i −0.957694 0.287788i $$-0.907080\pi$$
0.957694 0.287788i $$-0.0929199\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −17.1464 −1.32683 −0.663415 0.748251i $$-0.730894\pi$$
−0.663415 + 0.748251i $$0.730894\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −14.6969 −1.12390
$$172$$ 0 0
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 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
$$187$$ 29.3939i 2.14949i
$$188$$ 0 0
$$189$$ −9.00000 + 9.00000i −0.654654 + 0.654654i
$$190$$ 0 0
$$191$$ −14.6969 −1.06343 −0.531717 0.846922i $$-0.678453\pi$$
−0.531717 + 0.846922i $$0.678453\pi$$
$$192$$ 0 0
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 14.6969i 1.04184i 0.853606 + 0.520919i $$0.174411\pi$$
−0.853606 + 0.520919i $$0.825589\pi$$
$$200$$ 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
$$207$$ 7.34847i 0.510754i
$$208$$ 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
$$217$$ −24.0000 −1.62923
$$218$$ 0 0
$$219$$ −17.1464 + 17.1464i −1.15865 + 1.15865i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 2.44949i 0.164030i −0.996631 0.0820150i $$-0.973864\pi$$
0.996631 0.0820150i $$-0.0261355\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.44949 0.162578 0.0812892 0.996691i $$-0.474096\pi$$
0.0812892 + 0.996691i $$0.474096\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ −14.6969 14.6969i −0.966988 0.966988i
$$232$$ 0 0
$$233$$ 18.0000i 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 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
$$247$$ 9.79796i 0.623429i
$$248$$ 0 0
$$249$$ 9.00000 9.00000i 0.570352 0.570352i
$$250$$ 0 0
$$251$$ −24.4949 −1.54610 −0.773052 0.634343i $$-0.781271\pi$$
−0.773052 + 0.634343i $$0.781271\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.00000i 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ 0 0
$$259$$ 4.89898i 0.304408i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2.44949 0.151042 0.0755210 0.997144i $$-0.475938\pi$$
0.0755210 + 0.997144i $$0.475938\pi$$
$$264$$ 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
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ 29.3939 1.75977
$$280$$ 0 0
$$281$$ 6.00000i 0.357930i −0.983855 0.178965i $$-0.942725\pi$$
0.983855 0.178965i $$-0.0572749\pi$$
$$282$$ 0 0
$$283$$ 26.9444i 1.60168i −0.598880 0.800839i $$-0.704387\pi$$
0.598880 0.800839i $$-0.295613\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −14.6969 −0.867533
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 12.2474 12.2474i 0.717958 0.717958i
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 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
$$307$$ 2.44949i 0.139800i 0.997554 + 0.0698999i $$0.0222680\pi$$
−0.997554 + 0.0698999i $$0.977732\pi$$
$$308$$ 0 0
$$309$$ −15.0000 15.0000i −0.853320 0.853320i
$$310$$ 0 0
$$311$$ −24.4949 −1.38898 −0.694489 0.719503i $$-0.744370\pi$$
−0.694489 + 0.719503i $$0.744370\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 3.00000 3.00000i 0.167444 0.167444i
$$322$$ 0 0
$$323$$ −29.3939 −1.63552
$$324$$ 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
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ −7.34847 7.34847i −0.399114 0.399114i
$$340$$ 0 0
$$341$$ 48.0000i 2.59935i
$$342$$ 0 0
$$343$$ 19.5959i 1.05808i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7.34847 −0.394486 −0.197243 0.980355i $$-0.563199\pi$$
−0.197243 + 0.980355i $$0.563199\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 7.34847 + 7.34847i 0.392232 + 0.392232i
$$352$$ 0 0
$$353$$ 6.00000i 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 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
$$367$$ 7.34847i 0.383587i −0.981435 0.191793i $$-0.938570\pi$$
0.981435 0.191793i $$-0.0614304\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ 14.6969 0.763027
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 14.6969i 0.754931i −0.926024 0.377466i $$-0.876796\pi$$
0.926024 0.377466i $$-0.123204\pi$$
$$380$$ 0 0
$$381$$ −9.00000 9.00000i −0.461084 0.461084i
$$382$$ 0 0
$$383$$ −2.44949 −0.125163 −0.0625815 0.998040i $$-0.519933\pi$$
−0.0625815 + 0.998040i $$0.519933\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −7.34847 −0.373544
$$388$$ 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
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ −14.6969 + 14.6969i −0.735767 + 0.735767i
$$400$$ 0 0
$$401$$ 24.0000i 1.19850i −0.800561 0.599251i $$-0.795465\pi$$
0.800561 0.599251i $$-0.204535\pi$$
$$402$$ 0 0
$$403$$ 19.5959i 0.976142i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.79796 −0.485667
$$408$$ 0 0
$$409$$ 16.0000 0.791149 0.395575 0.918434i $$-0.370545\pi$$
0.395575 + 0.918434i $$0.370545\pi$$
$$410$$ 0 0
$$411$$ −7.34847 7.34847i −0.362473 0.362473i
$$412$$ 0 0
$$413$$ 24.0000i 1.18096i
$$414$$ 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
$$423$$ 36.7423i 1.78647i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 19.5959i 0.948313i
$$428$$ 0 0
$$429$$ −12.0000 + 12.0000i −0.579365 + 0.579365i
$$430$$ 0 0
$$431$$ 24.4949 1.17988 0.589939 0.807448i $$-0.299152\pi$$
0.589939 + 0.807448i $$0.299152\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 12.0000i 0.574038i
$$438$$ 0 0
$$439$$ 24.4949i 1.16908i 0.811366 + 0.584539i $$0.198725\pi$$
−0.811366 + 0.584539i $$0.801275\pi$$
$$440$$ 0 0
$$441$$ 3.00000i 0.142857i
$$442$$ 0 0
$$443$$ 17.1464 0.814651 0.407326 0.913283i $$-0.366461\pi$$
0.407326 + 0.913283i $$0.366461\pi$$
$$444$$ 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
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ 22.0454 22.0454i 1.02899 1.02899i
$$460$$ 0 0
$$461$$ 12.0000i 0.558896i 0.960161 + 0.279448i $$0.0901514\pi$$
−0.960161 + 0.279448i $$0.909849\pi$$
$$462$$ 0 0
$$463$$ 2.44949i 0.113837i −0.998379 0.0569187i $$-0.981872\pi$$
0.998379 0.0569187i $$-0.0181276\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 17.1464 0.793442 0.396721 0.917939i $$-0.370148\pi$$
0.396721 + 0.917939i $$0.370148\pi$$
$$468$$ 0 0
$$469$$ −18.0000 −0.831163
$$470$$ 0 0
$$471$$ −12.2474 + 12.2474i −0.564333 + 0.564333i
$$472$$ 0 0
$$473$$ 12.0000i 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −18.0000 −0.824163
$$478$$ 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
$$487$$ 22.0454i 0.998973i −0.866322 0.499486i $$-0.833522\pi$$
0.866322 0.499486i $$-0.166478\pi$$
$$488$$ 0 0
$$489$$ 9.00000 + 9.00000i 0.406994 + 0.406994i
$$490$$ 0 0
$$491$$ 14.6969 0.663264 0.331632 0.943409i $$-0.392401\pi$$
0.331632 + 0.943409i $$0.392401\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ 4.89898i 0.219308i 0.993970 + 0.109654i $$0.0349744\pi$$
−0.993970 + 0.109654i $$0.965026\pi$$
$$500$$ 0 0
$$501$$ 21.0000 21.0000i 0.938211 0.938211i
$$502$$ 0 0
$$503$$ 36.7423 1.63826 0.819130 0.573608i $$-0.194457\pi$$
0.819130 + 0.573608i $$0.194457\pi$$
$$504$$ 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
$$517$$ 60.0000 2.63880
$$518$$ 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
$$523$$ 2.44949i 0.107109i −0.998565 0.0535544i $$-0.982945\pi$$
0.998565 0.0535544i $$-0.0170550\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 58.7878 2.56083
$$528$$ 0 0
$$529$$ −17.0000 −0.739130
$$530$$ 0 0
$$531$$ 29.3939i 1.27559i
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 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
$$547$$ 41.6413i 1.78045i 0.455517 + 0.890227i $$0.349455\pi$$
−0.455517 + 0.890227i $$0.650545\pi$$
$$548$$ 0 0
$$549$$ 24.0000i 1.02430i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −12.0000 −0.510292
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000i 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ 4.89898i 0.207205i
$$560$$ 0 0
$$561$$ 36.0000 + 36.0000i 1.51992 + 1.51992i
$$562$$ 0 0
$$563$$ 12.2474 0.516168 0.258084 0.966122i $$-0.416909\pi$$
0.258084 + 0.966122i $$0.416909\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 22.0454i 0.925820i
$$568$$ 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
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 0 0
$$579$$ −12.2474 + 12.2474i −0.508987 + 0.508987i
$$580$$ 0 0
$$581$$ 18.0000i 0.746766i
$$582$$ 0 0
$$583$$ 29.3939i 1.21737i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −36.7423 −1.51652 −0.758259 0.651953i $$-0.773950\pi$$
−0.758259 + 0.651953i $$0.773950\pi$$
$$588$$ 0 0
$$589$$ 48.0000 1.97781
$$590$$ 0 0
$$591$$ 22.0454 + 22.0454i 0.906827 + 0.906827i
$$592$$ 0 0
$$593$$ 42.0000i 1.72473i 0.506284 + 0.862367i $$0.331019\pi$$
−0.506284 + 0.862367i $$0.668981\pi$$
$$594$$ 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
$$603$$ 22.0454 0.897758
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 26.9444i 1.09364i 0.837251 + 0.546819i $$0.184162\pi$$
−0.837251 + 0.546819i $$0.815838\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.4949 0.990957
$$612$$ 0 0
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 0 0
$$619$$ 4.89898i 0.196907i −0.995142 0.0984533i $$-0.968610\pi$$
0.995142 0.0984533i $$-0.0313895\pi$$
$$620$$ 0 0
$$621$$ 9.00000 + 9.00000i 0.361158 + 0.361158i
$$622$$ 0 0
$$623$$ −29.3939 −1.17764
$$624$$ 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
$$637$$ 2.00000 0.0792429
$$638$$ 0 0
$$639$$ 14.6969i 0.581402i
$$640$$ 0 0
$$641$$ 18.0000i 0.710957i −0.934684 0.355479i $$-0.884318\pi$$
0.934684 0.355479i $$-0.115682\pi$$
$$642$$ 0 0
$$643$$ 36.7423i 1.44898i 0.689287 + 0.724488i $$0.257924\pi$$
−0.689287 + 0.724488i $$0.742076\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 26.9444 1.05929 0.529647 0.848218i $$-0.322325\pi$$
0.529647 + 0.848218i $$0.322325\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 29.3939 29.3939i 1.15204 1.15204i
$$652$$ 0 0
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 42.0000i 1.63858i
$$658$$ 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
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ 24.4949i 0.940028i
$$680$$ 0 0
$$681$$ −3.00000 + 3.00000i −0.114960 + 0.114960i
$$682$$ 0 0
$$683$$ −31.8434 −1.21845 −0.609226 0.792996i $$-0.708520\pi$$
−0.609226 + 0.792996i $$0.708520\pi$$
$$684$$ 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
$$693$$ 36.0000 1.36753
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 0 0
$$699$$ 22.0454 + 22.0454i 0.833834 + 0.833834i
$$700$$ 0 0
$$701$$ 30.0000i 1.13308i −0.824033 0.566542i $$-0.808281\pi$$
0.824033 0.566542i $$-0.191719\pi$$
$$702$$ 0 0
$$703$$ 9.79796i 0.369537i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 29.3939 1.10547
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 14.6969 0.551178
$$712$$ 0 0
$$713$$ 24.0000i 0.898807i
$$714$$ 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
$$727$$ 26.9444i 0.999312i 0.866224 + 0.499656i $$0.166540\pi$$
−0.866224 + 0.499656i $$0.833460\pi$$
$$728$$ 0 0
$$729$$ 27.0000i 1.00000i
$$730$$ 0 0
$$731$$ −14.6969 −0.543586
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 36.0000i 1.32608i
$$738$$ 0 0
$$739$$ 14.6969i 0.540636i −0.962771 0.270318i $$-0.912871\pi$$
0.962771 0.270318i $$-0.0871288\pi$$
$$740$$ 0 0
$$741$$ 12.0000 + 12.0000i 0.440831 + 0.440831i
$$742$$ 0 0
$$743$$ −41.6413 −1.52767 −0.763836 0.645410i $$-0.776686\pi$$
−0.763836 + 0.645410i $$0.776686\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 22.0454i 0.806599i
$$748$$ 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
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ −14.6969 + 14.6969i −0.533465 + 0.533465i
$$760$$ 0 0
$$761$$ 48.0000i 1.74000i 0.493053 + 0.869999i $$0.335881\pi$$
−0.493053 + 0.869999i $$0.664119\pi$$
$$762$$ 0 0
$$763$$ 9.79796i 0.354710i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −19.5959 −0.707568
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 7.34847 + 7.34847i 0.264649 + 0.264649i
$$772$$ 0 0
$$773$$ 42.0000i 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 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
$$787$$ 46.5403i 1.65898i −0.558520 0.829491i $$-0.688630\pi$$
0.558520 0.829491i $$-0.311370\pi$$
$$788$$ 0 0
$$789$$ −3.00000 + 3.00000i −0.106803 + 0.106803i
$$790$$ 0 0
$$791$$ −14.6969 −0.522563
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 42.0000i 1.48772i 0.668338 + 0.743858i $$0.267006\pi$$
−0.668338 + 0.743858i $$0.732994\pi$$
$$798$$ 0 0
$$799$$ 73.4847i 2.59970i
$$800$$ 0 0
$$801$$ 36.0000 1.27200
$$802$$ 0 0
$$803$$ 68.5857 2.42034
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$