# Properties

 Label 588.1.z.a.389.1 Level $588$ Weight $1$ Character 588.389 Analytic conductor $0.293$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 588.z (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.293450227428$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{21}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{21} - \cdots)$$

## Embedding invariants

 Embedding label 389.1 Root $$-0.988831 + 0.149042i$$ of defining polynomial Character $$\chi$$ $$=$$ 588.389 Dual form 588.1.z.a.65.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(0.0747301 - 0.997204i) q^{3} +(0.365341 - 0.930874i) q^{7} +(-0.988831 - 0.149042i) q^{9} +O(q^{10})$$ $$q+(0.0747301 - 0.997204i) q^{3} +(0.365341 - 0.930874i) q^{7} +(-0.988831 - 0.149042i) q^{9} +(0.0931869 + 0.116853i) q^{13} +(-0.365341 - 0.632789i) q^{19} +(-0.900969 - 0.433884i) q^{21} +(0.365341 + 0.930874i) q^{25} +(-0.222521 + 0.974928i) q^{27} +(0.733052 - 1.26968i) q^{31} +(-1.40097 + 1.29991i) q^{37} +(0.123490 - 0.0841939i) q^{39} +(1.78181 + 0.858075i) q^{43} +(-0.733052 - 0.680173i) q^{49} +(-0.658322 + 0.317031i) q^{57} +(0.326239 - 0.302705i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.826239 + 1.43109i) q^{67} +(0.698220 + 1.77904i) q^{73} +(0.955573 - 0.294755i) q^{75} +(0.733052 + 1.26968i) q^{79} +(0.955573 + 0.294755i) q^{81} +(0.142820 - 0.0440542i) q^{91} +(-1.21135 - 0.825886i) q^{93} +1.24698 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$12 q + q^{3} + q^{7} + q^{9} + 2 q^{13} - q^{19} - 2 q^{21} + q^{25} - 2 q^{27} - q^{31} - 8 q^{37} - 8 q^{39} + 2 q^{43} + q^{49} + 2 q^{57} - 5 q^{61} - 6 q^{63} - q^{67} - q^{73} + q^{75} - q^{79} + q^{81} - q^{91} - q^{93} - 4 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0.0747301 0.997204i 0.0747301 0.997204i
$$4$$ 0 0
$$5$$ 0 0 −0.826239 0.563320i $$-0.809524\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$6$$ 0 0
$$7$$ 0.365341 0.930874i 0.365341 0.930874i
$$8$$ 0 0
$$9$$ −0.988831 0.149042i −0.988831 0.149042i
$$10$$ 0 0
$$11$$ 0 0 0.988831 0.149042i $$-0.0476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$12$$ 0 0
$$13$$ 0.0931869 + 0.116853i 0.0931869 + 0.116853i 0.826239 0.563320i $$-0.190476\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$18$$ 0 0
$$19$$ −0.365341 0.632789i −0.365341 0.632789i 0.623490 0.781831i $$-0.285714\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$20$$ 0 0
$$21$$ −0.900969 0.433884i −0.900969 0.433884i
$$22$$ 0 0
$$23$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$24$$ 0 0
$$25$$ 0.365341 + 0.930874i 0.365341 + 0.930874i
$$26$$ 0 0
$$27$$ −0.222521 + 0.974928i −0.222521 + 0.974928i
$$28$$ 0 0
$$29$$ 0 0 −0.222521 0.974928i $$-0.571429\pi$$
0.222521 + 0.974928i $$0.428571\pi$$
$$30$$ 0 0
$$31$$ 0.733052 1.26968i 0.733052 1.26968i −0.222521 0.974928i $$-0.571429\pi$$
0.955573 0.294755i $$-0.0952381\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.40097 + 1.29991i −1.40097 + 1.29991i −0.500000 + 0.866025i $$0.666667\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$38$$ 0 0
$$39$$ 0.123490 0.0841939i 0.123490 0.0841939i
$$40$$ 0 0
$$41$$ 0 0 0.900969 0.433884i $$-0.142857\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$42$$ 0 0
$$43$$ 1.78181 + 0.858075i 1.78181 + 0.858075i 0.955573 + 0.294755i $$0.0952381\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.365341 0.930874i $$-0.380952\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$48$$ 0 0
$$49$$ −0.733052 0.680173i −0.733052 0.680173i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 −0.733052 0.680173i $$-0.761905\pi$$
0.733052 + 0.680173i $$0.238095\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.658322 + 0.317031i −0.658322 + 0.317031i
$$58$$ 0 0
$$59$$ 0 0 0.826239 0.563320i $$-0.190476\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$60$$ 0 0
$$61$$ 0.326239 0.302705i 0.326239 0.302705i −0.500000 0.866025i $$-0.666667\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$62$$ 0 0
$$63$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −0.826239 + 1.43109i −0.826239 + 1.43109i 0.0747301 + 0.997204i $$0.476190\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 0.222521 0.974928i $$-0.428571\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$72$$ 0 0
$$73$$ 0.698220 + 1.77904i 0.698220 + 1.77904i 0.623490 + 0.781831i $$0.285714\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$74$$ 0 0
$$75$$ 0.955573 0.294755i 0.955573 0.294755i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0.733052 + 1.26968i 0.733052 + 1.26968i 0.955573 + 0.294755i $$0.0952381\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$80$$ 0 0
$$81$$ 0.955573 + 0.294755i 0.955573 + 0.294755i
$$82$$ 0 0
$$83$$ 0 0 0.623490 0.781831i $$-0.285714\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 −0.988831 0.149042i $$-0.952381\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$90$$ 0 0
$$91$$ 0.142820 0.0440542i 0.142820 0.0440542i
$$92$$ 0 0
$$93$$ −1.21135 0.825886i −1.21135 0.825886i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.24698 1.24698 0.623490 0.781831i $$-0.285714\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 0.0747301 0.997204i $$-0.476190\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$102$$ 0 0
$$103$$ −1.63402 1.11406i −1.63402 1.11406i −0.900969 0.433884i $$-0.857143\pi$$
−0.733052 0.680173i $$-0.761905\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 −0.988831 0.149042i $$-0.952381\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$108$$ 0 0
$$109$$ −1.63402 + 0.246289i −1.63402 + 0.246289i −0.900969 0.433884i $$-0.857143\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$110$$ 0 0
$$111$$ 1.19158 + 1.49419i 1.19158 + 1.49419i
$$112$$ 0 0
$$113$$ 0 0 0.623490 0.781831i $$-0.285714\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −0.0747301 0.129436i −0.0747301 0.129436i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0.955573 0.294755i 0.955573 0.294755i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −0.162592 0.712362i −0.162592 0.712362i −0.988831 0.149042i $$-0.952381\pi$$
0.826239 0.563320i $$-0.190476\pi$$
$$128$$ 0 0
$$129$$ 0.988831 1.71271i 0.988831 1.71271i
$$130$$ 0 0
$$131$$ 0 0 −0.0747301 0.997204i $$-0.523810\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$132$$ 0 0
$$133$$ −0.722521 + 0.108903i −0.722521 + 0.108903i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.826239 0.563320i $$-0.190476\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$138$$ 0 0
$$139$$ −1.48883 + 0.716983i −1.48883 + 0.716983i −0.988831 0.149042i $$-0.952381\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −0.733052 + 0.680173i −0.733052 + 0.680173i
$$148$$ 0 0
$$149$$ 0 0 0.365341 0.930874i $$-0.380952\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$150$$ 0 0
$$151$$ −0.914101 0.848162i −0.914101 0.848162i 0.0747301 0.997204i $$-0.476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1.48883 + 1.01507i −1.48883 + 1.01507i −0.500000 + 0.866025i $$0.666667\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −0.134659 1.79690i −0.134659 1.79690i −0.500000 0.866025i $$-0.666667\pi$$
0.365341 0.930874i $$-0.380952\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 −0.222521 0.974928i $$-0.571429\pi$$
0.222521 + 0.974928i $$0.428571\pi$$
$$168$$ 0 0
$$169$$ 0.217550 0.953150i 0.217550 0.953150i
$$170$$ 0 0
$$171$$ 0.266948 + 0.680173i 0.266948 + 0.680173i
$$172$$ 0 0
$$173$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$174$$ 0 0
$$175$$ 1.00000 1.00000
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$180$$ 0 0
$$181$$ 1.03030 1.29196i 1.03030 1.29196i 0.0747301 0.997204i $$-0.476190\pi$$
0.955573 0.294755i $$-0.0952381\pi$$
$$182$$ 0 0
$$183$$ −0.277479 0.347948i −0.277479 0.347948i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0.826239 + 0.563320i 0.826239 + 0.563320i
$$190$$ 0 0
$$191$$ 0 0 −0.826239 0.563320i $$-0.809524\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$192$$ 0 0
$$193$$ 0.0111692 0.149042i 0.0111692 0.149042i −0.988831 0.149042i $$-0.952381\pi$$
1.00000 $$0$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 0.0931869 1.24349i 0.0931869 1.24349i −0.733052 0.680173i $$-0.761905\pi$$
0.826239 0.563320i $$-0.190476\pi$$
$$200$$ 0 0
$$201$$ 1.36534 + 0.930874i 1.36534 + 0.930874i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i $$-0.857143\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.914101 1.14625i −0.914101 1.14625i
$$218$$ 0 0
$$219$$ 1.82624 0.563320i 1.82624 0.563320i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0.0990311 0.433884i 0.0990311 0.433884i −0.900969 0.433884i $$-0.857143\pi$$
1.00000 $$0$$
$$224$$ 0 0
$$225$$ −0.222521 0.974928i −0.222521 0.974928i
$$226$$ 0 0
$$227$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$228$$ 0 0
$$229$$ −0.147791 1.97213i −0.147791 1.97213i −0.222521 0.974928i $$-0.571429\pi$$
0.0747301 0.997204i $$-0.476190\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 0.733052 0.680173i $$-0.238095\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1.32091 0.636119i 1.32091 0.636119i
$$238$$ 0 0
$$239$$ 0 0 −0.900969 0.433884i $$-0.857143\pi$$
0.900969 + 0.433884i $$0.142857\pi$$
$$240$$ 0 0
$$241$$ 1.32091 + 1.22563i 1.32091 + 1.22563i 0.955573 + 0.294755i $$0.0952381\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$242$$ 0 0
$$243$$ 0.365341 0.930874i 0.365341 0.930874i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.0398981 0.101659i 0.0398981 0.101659i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 −0.900969 0.433884i $$-0.857143\pi$$
0.900969 + 0.433884i $$0.142857\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 0.733052 0.680173i $$-0.238095\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$258$$ 0 0
$$259$$ 0.698220 + 1.77904i 0.698220 + 1.77904i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 −0.365341 0.930874i $$-0.619048\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$270$$ 0 0
$$271$$ −1.72188 + 0.531130i −1.72188 + 0.531130i −0.988831 0.149042i $$-0.952381\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$272$$ 0 0
$$273$$ −0.0332580 0.145713i −0.0332580 0.145713i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.142820 + 0.0440542i 0.142820 + 0.0440542i 0.365341 0.930874i $$-0.380952\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$278$$ 0 0
$$279$$ −0.914101 + 1.14625i −0.914101 + 1.14625i
$$280$$ 0 0
$$281$$ 0 0 −0.623490 0.781831i $$-0.714286\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$282$$ 0 0
$$283$$ −1.88980 + 0.284841i −1.88980 + 0.284841i −0.988831 0.149042i $$-0.952381\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 0.826239 + 0.563320i 0.826239 + 0.563320i
$$290$$ 0 0
$$291$$ 0.0931869 1.24349i 0.0931869 1.24349i
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.44973 1.34515i 1.44973 1.34515i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −1.23305 1.54620i −1.23305 1.54620i −0.733052 0.680173i $$-0.761905\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$308$$ 0 0
$$309$$ −1.23305 + 1.54620i −1.23305 + 1.54620i
$$310$$ 0 0
$$311$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$312$$ 0 0
$$313$$ 0.733052 + 1.26968i 0.733052 + 1.26968i 0.955573 + 0.294755i $$0.0952381\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −0.0747301 + 0.129436i −0.0747301 + 0.129436i
$$326$$ 0 0
$$327$$ 0.123490 + 1.64786i 0.123490 + 1.64786i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −0.535628 + 0.496990i −0.535628 + 0.496990i −0.900969 0.433884i $$-0.857143\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$332$$ 0 0
$$333$$ 1.57906 1.07659i 1.57906 1.07659i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1.48883 0.716983i −1.48883 0.716983i −0.500000 0.866025i $$-0.666667\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −0.900969 + 0.433884i −0.900969 + 0.433884i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.733052 0.680173i $$-0.761905\pi$$
0.733052 + 0.680173i $$0.238095\pi$$
$$348$$ 0 0
$$349$$ −1.12349 0.541044i −1.12349 0.541044i −0.222521 0.974928i $$-0.571429\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$350$$ 0 0
$$351$$ −0.134659 + 0.0648483i −0.134659 + 0.0648483i
$$352$$ 0 0
$$353$$ 0 0 0.826239 0.563320i $$-0.190476\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 −0.0747301 0.997204i $$-0.523810\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$360$$ 0 0
$$361$$ 0.233052 0.403658i 0.233052 0.403658i
$$362$$ 0 0
$$363$$ −0.222521 0.974928i −0.222521 0.974928i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0.603718 + 1.53825i 0.603718 + 1.53825i 0.826239 + 0.563320i $$0.190476\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0.988831 + 1.71271i 0.988831 + 1.71271i 0.623490 + 0.781831i $$0.285714\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0.455573 + 0.571270i 0.455573 + 0.571270i 0.955573 0.294755i $$-0.0952381\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$380$$ 0 0
$$381$$ −0.722521 + 0.108903i −0.722521 + 0.108903i
$$382$$ 0 0
$$383$$ 0 0 −0.988831 0.149042i $$-0.952381\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.63402 1.11406i −1.63402 1.11406i
$$388$$ 0 0
$$389$$ 0 0 0.0747301 0.997204i $$-0.476190\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.57906 + 1.07659i 1.57906 + 1.07659i 0.955573 + 0.294755i $$0.0952381\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$398$$ 0 0
$$399$$ 0.0546039 + 0.728639i 0.0546039 + 0.728639i
$$400$$ 0 0
$$401$$ 0 0 −0.988831 0.149042i $$-0.952381\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$402$$ 0 0
$$403$$ 0.216677 0.0326588i 0.216677 0.0326588i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −1.40097 0.432142i −1.40097 0.432142i −0.500000 0.866025i $$-0.666667\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0.603718 + 1.53825i 0.603718 + 1.53825i
$$418$$ 0 0
$$419$$ 0 0 0.222521 0.974928i $$-0.428571\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$420$$ 0 0
$$421$$ −0.0332580 0.145713i −0.0332580 0.145713i 0.955573 0.294755i $$-0.0952381\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −0.162592 0.414278i −0.162592 0.414278i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 0.826239 0.563320i $$-0.190476\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$432$$ 0 0
$$433$$ −0.658322 + 0.317031i −0.658322 + 0.317031i −0.733052 0.680173i $$-0.761905\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −0.658322 + 1.67738i −0.658322 + 1.67738i 0.0747301 + 0.997204i $$0.476190\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$440$$ 0 0
$$441$$ 0.623490 + 0.781831i 0.623490 + 0.781831i
$$442$$ 0 0
$$443$$ 0 0 0.365341 0.930874i $$-0.380952\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 0.900969 0.433884i $$-0.142857\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −0.914101 + 0.848162i −0.914101 + 0.848162i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.109562 1.46200i −0.109562 1.46200i −0.733052 0.680173i $$-0.761905\pi$$
0.623490 0.781831i $$-0.285714\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 −0.222521 0.974928i $$-0.571429\pi$$
0.222521 + 0.974928i $$0.428571\pi$$
$$462$$ 0 0
$$463$$ −0.367711 + 1.61105i −0.367711 + 1.61105i 0.365341 + 0.930874i $$0.380952\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$468$$ 0 0
$$469$$ 1.03030 + 1.29196i 1.03030 + 1.29196i
$$470$$ 0 0
$$471$$ 0.900969 + 1.56052i 0.900969 + 1.56052i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0.455573 0.571270i 0.455573 0.571270i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 0.988831 0.149042i $$-0.0476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$480$$ 0 0
$$481$$ −0.282450 0.0425725i −0.282450 0.0425725i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0.142820 1.90580i 0.142820 1.90580i −0.222521 0.974928i $$-0.571429\pi$$
0.365341 0.930874i $$-0.380952\pi$$
$$488$$ 0 0
$$489$$ −1.80194 −1.80194
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.44973 + 0.218511i 1.44973 + 0.218511i 0.826239 0.563320i $$-0.190476\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 −0.623490 0.781831i $$-0.714286\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −0.934227 0.288171i −0.934227 0.288171i
$$508$$ 0 0
$$509$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$510$$ 0 0
$$511$$ 1.91115 1.91115
$$512$$ 0 0
$$513$$ 0.698220 0.215372i 0.698220 0.215372i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$522$$ 0 0
$$523$$ 0.142820 + 1.90580i 0.142820 + 1.90580i 0.365341 + 0.930874i $$0.380952\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$524$$ 0 0
$$525$$ 0.0747301 0.997204i 0.0747301 0.997204i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 0.826239 0.563320i 0.826239 0.563320i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.266948 0.680173i 0.266948 0.680173i −0.733052 0.680173i $$-0.761905\pi$$
1.00000 $$0$$
$$542$$ 0 0
$$543$$ −1.21135 1.12397i −1.21135 1.12397i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0.400969 0.193096i 0.400969 0.193096i −0.222521 0.974928i $$-0.571429\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$548$$ 0 0
$$549$$ −0.367711 + 0.250701i −0.367711 + 0.250701i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 1.44973 0.218511i 1.44973 0.218511i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$558$$ 0 0
$$559$$ 0.0657731 + 0.288171i 0.0657731 + 0.288171i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 −0.365341 0.930874i $$-0.619048\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0.623490 0.781831i 0.623490 0.781831i
$$568$$ 0 0
$$569$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$570$$ 0 0
$$571$$ 1.82624 + 0.563320i 1.82624 + 0.563320i 1.00000 $$0$$
0.826239 + 0.563320i $$0.190476\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1.63402 + 0.246289i −1.63402 + 0.246289i −0.900969 0.433884i $$-0.857143\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$578$$ 0 0
$$579$$ −0.147791 0.0222759i −0.147791 0.0222759i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ −1.07126 −1.07126
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 −0.826239 0.563320i $$-0.809524\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.23305 0.185853i −1.23305 0.185853i
$$598$$ 0 0
$$599$$ 0 0 0.988831 0.149042i $$-0.0476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$600$$ 0 0
$$601$$ −0.623490 0.781831i −0.623490 0.781831i 0.365341 0.930874i $$-0.380952\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$602$$ 0 0
$$603$$ 1.03030 1.29196i 1.03030 1.29196i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −0.162592 0.414278i −0.162592 0.414278i 0.826239 0.563320i $$-0.190476\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 −0.222521 0.974928i $$-0.571429\pi$$
0.222521 + 0.974928i $$0.428571\pi$$
$$618$$ 0 0
$$619$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −0.733052 + 0.680173i −0.733052 + 0.680173i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0.400969 + 0.193096i 0.400969 + 0.193096i 0.623490 0.781831i $$-0.285714\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$632$$ 0 0
$$633$$ 0.326239 + 0.302705i 0.326239 + 0.302705i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.0111692 0.149042i 0.0111692 0.149042i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 −0.733052 0.680173i $$-0.761905\pi$$
0.733052 + 0.680173i $$0.238095\pi$$
$$642$$ 0 0
$$643$$ −0.134659 0.0648483i −0.134659 0.0648483i 0.365341 0.930874i $$-0.380952\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.826239 0.563320i $$-0.190476\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −1.21135 + 0.825886i −1.21135 + 0.825886i
$$652$$ 0 0
$$653$$ 0 0 −0.0747301 0.997204i $$-0.523810\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −0.425270 1.86323i −0.425270 1.86323i
$$658$$ 0 0
$$659$$ 0 0 0.222521 0.974928i $$-0.428571\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$660$$ 0 0
$$661$$ −0.365341 0.930874i −0.365341 0.930874i −0.988831 0.149042i $$-0.952381\pi$$
0.623490 0.781831i $$-0.285714\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −0.425270 0.131178i −0.425270 0.131178i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0.0931869 + 0.116853i 0.0931869 + 0.116853i 0.826239 0.563320i $$-0.190476\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$674$$ 0 0
$$675$$ −0.988831 + 0.149042i −0.988831 + 0.149042i
$$676$$ 0 0
$$677$$ 0 0 −0.988831 0.149042i $$-0.952381\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$678$$ 0 0
$$679$$ 0.455573 1.16078i 0.455573 1.16078i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 0.0747301 0.997204i $$-0.476190\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −1.97766 −1.97766
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −0.826239 0.563320i −0.826239 0.563320i 0.0747301 0.997204i $$-0.476190\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 0.623490 0.781831i $$-0.285714\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$702$$ 0 0
$$703$$ 1.33440 + 0.411608i 1.33440 + 0.411608i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.19158 0.367554i 1.19158 0.367554i 0.365341 0.930874i $$-0.380952\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$710$$ 0 0
$$711$$ −0.535628 1.36476i −0.535628 1.36476i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 −0.0747301 0.997204i $$-0.523810\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$720$$ 0 0
$$721$$ −1.63402 + 1.11406i −1.63402 + 1.11406i
$$722$$ 0 0
$$723$$ 1.32091 1.22563i 1.32091 1.22563i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0.900969 0.433884i 0.900969 0.433884i 0.0747301 0.997204i $$-0.476190\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$728$$ 0 0
$$729$$ −0.900969 0.433884i −0.900969 0.433884i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0.266948 0.680173i 0.266948 0.680173i −0.733052 0.680173i $$-0.761905\pi$$
1.00000 $$0$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −0.109562 0.101659i −0.109562 0.101659i 0.623490 0.781831i $$-0.285714\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$740$$ 0 0
$$741$$ −0.0983929 0.0473835i −0.0983929 0.0473835i
$$742$$ 0 0
$$743$$ 0 0 0.900969 0.433884i $$-0.142857\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −0.147791 1.97213i −0.147791 1.97213i −0.222521 0.974928i $$-0.571429\pi$$
0.0747301 0.997204i $$-0.476190\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i $$-0.571429\pi$$
0.623490 0.781831i $$-0.285714\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$762$$ 0 0
$$763$$ −0.367711 + 1.61105i −0.367711 + 1.61105i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −0.623490 + 0.781831i −0.623490 + 0.781831i −0.988831 0.149042i $$-0.952381\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 0.988831 0.149042i $$-0.0476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$774$$ 0 0
$$775$$ 1.44973 + 0.218511i 1.44973 + 0.218511i
$$776$$ 0 0
$$777$$ 1.82624 0.563320i 1.82624 0.563320i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −0.0332580 + 0.443797i −0.0332580 + 0.443797i 0.955573 + 0.294755i $$0.0952381\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0.0657731 + 0.00991370i 0.0657731 + 0.00991370i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 −0.623490 0.781831i $$-0.714286\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0