# Properties

 Label 3969.1.ca.a.3862.1 Level $3969$ Weight $1$ Character 3969.3862 Analytic conductor $1.981$ Analytic rank $0$ Dimension $12$ Projective image $D_{42}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3969.ca (of order $$42$$, degree $$12$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.98078903514$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1323) Projective image: $$D_{42}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{42} - \cdots)$$

## Embedding invariants

 Embedding label 3862.1 Root $$0.365341 + 0.930874i$$ of defining polynomial Character $$\chi$$ $$=$$ 3969.3862 Dual form 3969.1.ca.a.2782.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.826239 + 0.563320i) q^{4} +(0.733052 + 0.680173i) q^{7} +O(q^{10})$$ $$q+(-0.826239 + 0.563320i) q^{4} +(0.733052 + 0.680173i) q^{7} +(-0.548760 - 1.77904i) q^{13} +(0.365341 - 0.930874i) q^{16} +(-1.68862 - 0.974928i) q^{19} +(-0.222521 - 0.974928i) q^{25} +(-0.988831 - 0.149042i) q^{28} +(-1.72721 - 0.997204i) q^{31} +(-0.0111692 + 0.149042i) q^{37} +(1.88980 - 0.284841i) q^{43} +(0.0747301 + 0.997204i) q^{49} +(1.45557 + 1.16078i) q^{52} +(-0.634659 + 0.930874i) q^{61} +(0.222521 + 0.974928i) q^{64} +(0.988831 - 1.71271i) q^{67} +(0.590232 + 0.636119i) q^{73} +(1.94440 - 0.145713i) q^{76} +(-0.826239 - 1.43109i) q^{79} +(0.807782 - 1.67738i) q^{91} +(-0.510531 - 0.294755i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - q^{4} - q^{7} + O(q^{10})$$ $$12q - q^{4} - q^{7} + 3q^{13} + q^{16} - 2q^{25} + q^{28} - 3q^{31} - 13q^{37} + q^{43} + q^{49} + 7q^{52} - 11q^{61} + 2q^{64} - q^{67} - q^{79} + 3q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times$$.

 $$n$$ $$2108$$ $$3727$$ $$\chi(n)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{23}{42}\right)$$

## 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 −0.294755 0.955573i $$-0.595238\pi$$
0.294755 + 0.955573i $$0.404762\pi$$
$$3$$ 0 0
$$4$$ −0.826239 + 0.563320i −0.826239 + 0.563320i
$$5$$ 0 0 0.623490 0.781831i $$-0.285714\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$6$$ 0 0
$$7$$ 0.733052 + 0.680173i 0.733052 + 0.680173i
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 −0.974928 0.222521i $$-0.928571\pi$$
0.974928 + 0.222521i $$0.0714286\pi$$
$$12$$ 0 0
$$13$$ −0.548760 1.77904i −0.548760 1.77904i −0.623490 0.781831i $$-0.714286\pi$$
0.0747301 0.997204i $$-0.476190\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0.365341 0.930874i 0.365341 0.930874i
$$17$$ 0 0 −0.826239 0.563320i $$-0.809524\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$18$$ 0 0
$$19$$ −1.68862 0.974928i −1.68862 0.974928i −0.955573 0.294755i $$-0.904762\pi$$
−0.733052 0.680173i $$-0.761905\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 0.433884 0.900969i $$-0.357143\pi$$
−0.433884 + 0.900969i $$0.642857\pi$$
$$24$$ 0 0
$$25$$ −0.222521 0.974928i −0.222521 0.974928i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −0.988831 0.149042i −0.988831 0.149042i
$$29$$ 0 0 −0.997204 0.0747301i $$-0.976190\pi$$
0.997204 + 0.0747301i $$0.0238095\pi$$
$$30$$ 0 0
$$31$$ −1.72721 0.997204i −1.72721 0.997204i −0.900969 0.433884i $$-0.857143\pi$$
−0.826239 0.563320i $$-0.809524\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.0111692 + 0.149042i −0.0111692 + 0.149042i 0.988831 + 0.149042i $$0.0476190\pi$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 −0.988831 0.149042i $$-0.952381\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$42$$ 0 0
$$43$$ 1.88980 0.284841i 1.88980 0.284841i 0.900969 0.433884i $$-0.142857\pi$$
0.988831 + 0.149042i $$0.0476190\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$48$$ 0 0
$$49$$ 0.0747301 + 0.997204i 0.0747301 + 0.997204i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.45557 + 1.16078i 1.45557 + 1.16078i
$$53$$ 0 0 0.997204 0.0747301i $$-0.0238095\pi$$
−0.997204 + 0.0747301i $$0.976190\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 0.988831 0.149042i $$-0.0476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$60$$ 0 0
$$61$$ −0.634659 + 0.930874i −0.634659 + 0.930874i 0.365341 + 0.930874i $$0.380952\pi$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0.222521 + 0.974928i 0.222521 + 0.974928i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.988831 1.71271i 0.988831 1.71271i 0.365341 0.930874i $$-0.380952\pi$$
0.623490 0.781831i $$-0.285714\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 0.433884 0.900969i $$-0.357143\pi$$
−0.433884 + 0.900969i $$0.642857\pi$$
$$72$$ 0 0
$$73$$ 0.590232 + 0.636119i 0.590232 + 0.636119i 0.955573 0.294755i $$-0.0952381\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.94440 0.145713i 1.94440 0.145713i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −0.826239 1.43109i −0.826239 1.43109i −0.900969 0.433884i $$-0.857143\pi$$
0.0747301 0.997204i $$-0.476190\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$90$$ 0 0
$$91$$ 0.807782 1.67738i 0.807782 1.67738i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.510531 0.294755i −0.510531 0.294755i 0.222521 0.974928i $$-0.428571\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0.733052 + 0.680173i 0.733052 + 0.680173i
$$101$$ 0 0 0.623490 0.781831i $$-0.285714\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$102$$ 0 0
$$103$$ −1.06356 0.848162i −1.06356 0.848162i −0.0747301 0.997204i $$-0.523810\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 0.294755 0.955573i $$-0.404762\pi$$
−0.294755 + 0.955573i $$0.595238\pi$$
$$108$$ 0 0
$$109$$ −0.698220 + 0.215372i −0.698220 + 0.215372i −0.623490 0.781831i $$-0.714286\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0.900969 0.433884i 0.900969 0.433884i
$$113$$ 0 0 −0.680173 0.733052i $$-0.738095\pi$$
0.680173 + 0.733052i $$0.261905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0.900969 + 0.433884i 0.900969 + 0.433884i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 1.98883 0.149042i 1.98883 0.149042i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1.32091 0.636119i 1.32091 0.636119i 0.365341 0.930874i $$-0.380952\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 −0.623490 0.781831i $$-0.714286\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$132$$ 0 0
$$133$$ −0.574730 1.86323i −0.574730 1.86323i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.781831 0.623490i $$-0.214286\pi$$
−0.781831 + 0.623490i $$0.785714\pi$$
$$138$$ 0 0
$$139$$ 0.0444272 0.294755i 0.0444272 0.294755i −0.955573 0.294755i $$-0.904762\pi$$
1.00000 $$0$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −0.0747301 0.129436i −0.0747301 0.129436i
$$149$$ 0 0 −0.974928 0.222521i $$-0.928571\pi$$
0.974928 + 0.222521i $$0.0714286\pi$$
$$150$$ 0 0
$$151$$ −1.32091 0.636119i −1.32091 0.636119i −0.365341 0.930874i $$-0.619048\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.233052 + 1.54620i 0.233052 + 1.54620i 0.733052 + 0.680173i $$0.238095\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −0.722521 + 0.108903i −0.722521 + 0.108903i −0.500000 0.866025i $$-0.666667\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 −0.826239 0.563320i $$-0.809524\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$168$$ 0 0
$$169$$ −2.03759 + 1.38921i −2.03759 + 1.38921i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.40097 + 1.29991i −1.40097 + 1.29991i
$$173$$ 0 0 −0.0747301 0.997204i $$-0.523810\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$174$$ 0 0
$$175$$ 0.500000 0.866025i 0.500000 0.866025i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 0.563320 0.826239i $$-0.309524\pi$$
−0.563320 + 0.826239i $$0.690476\pi$$
$$180$$ 0 0
$$181$$ 1.52446 0.347948i 1.52446 0.347948i 0.623490 0.781831i $$-0.285714\pi$$
0.900969 + 0.433884i $$0.142857\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 0.149042 0.988831i $$-0.452381\pi$$
−0.149042 + 0.988831i $$0.547619\pi$$
$$192$$ 0 0
$$193$$ −0.266948 0.680173i −0.266948 0.680173i 0.733052 0.680173i $$-0.238095\pi$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −0.623490 0.781831i −0.623490 0.781831i
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −0.202749 + 1.34515i −0.202749 + 1.34515i 0.623490 + 0.781831i $$0.285714\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −1.85654 0.139129i −1.85654 0.139129i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.21135 1.12397i 1.21135 1.12397i 0.222521 0.974928i $$-0.428571\pi$$
0.988831 0.149042i $$-0.0476190\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.587862 1.90580i −0.587862 1.90580i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −0.488831 0.716983i −0.488831 0.716983i 0.500000 0.866025i $$-0.333333\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0.460898 0.367554i 0.460898 0.367554i −0.365341 0.930874i $$-0.619048\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 −0.997204 0.0747301i $$-0.976190\pi$$
0.997204 + 0.0747301i $$0.0238095\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 −0.930874 0.365341i $$-0.880952\pi$$
0.930874 + 0.365341i $$0.119048\pi$$
$$240$$ 0 0
$$241$$ −0.129334 + 0.268565i −0.129334 + 0.268565i −0.955573 0.294755i $$-0.904762\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 1.12664i 1.12664i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.807782 + 3.53912i −0.807782 + 3.53912i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 −0.623490 0.781831i $$-0.714286\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −0.733052 0.680173i −0.733052 0.680173i
$$257$$ 0 0 0.900969 0.433884i $$-0.142857\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$258$$ 0 0
$$259$$ −0.109562 + 0.101659i −0.109562 + 0.101659i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0.147791 + 1.97213i 0.147791 + 1.97213i
$$269$$ 0 0 0.733052 0.680173i $$-0.238095\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$270$$ 0 0
$$271$$ 0.167917 + 0.246289i 0.167917 + 0.246289i 0.900969 0.433884i $$-0.142857\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.78181 0.858075i 1.78181 0.858075i 0.826239 0.563320i $$-0.190476\pi$$
0.955573 0.294755i $$-0.0952381\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 0.680173 0.733052i $$-0.261905\pi$$
−0.680173 + 0.733052i $$0.738095\pi$$
$$282$$ 0 0
$$283$$ −0.766310 + 0.825886i −0.766310 + 0.825886i −0.988831 0.149042i $$-0.952381\pi$$
0.222521 + 0.974928i $$0.428571\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 0.365341 + 0.930874i 0.365341 + 0.930874i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −0.846011 0.193096i −0.846011 0.193096i
$$293$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.57906 + 1.07659i 1.57906 + 1.07659i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −1.52446 + 1.21572i −1.52446 + 1.21572i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0.574730 + 0.131178i 0.574730 + 0.131178i 0.500000 0.866025i $$-0.333333\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 0.0747301 0.997204i $$-0.476190\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$312$$ 0 0
$$313$$ −0.751509 + 0.433884i −0.751509 + 0.433884i −0.826239 0.563320i $$-0.809524\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 1.48883 + 0.716983i 1.48883 + 0.716983i
$$317$$ 0 0 0.997204 0.0747301i $$-0.0238095\pi$$
−0.997204 + 0.0747301i $$0.976190\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −1.61232 + 0.930874i −1.61232 + 0.930874i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0.367711 + 0.250701i 0.367711 + 0.250701i 0.733052 0.680173i $$-0.238095\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −0.455573 + 1.16078i −0.455573 + 1.16078i 0.500000 + 0.866025i $$0.333333\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −0.623490 + 0.781831i −0.623490 + 0.781831i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.563320 0.826239i $$-0.690476\pi$$
0.563320 + 0.826239i $$0.309524\pi$$
$$348$$ 0 0
$$349$$ −0.202749 1.34515i −0.202749 1.34515i −0.826239 0.563320i $$-0.809524\pi$$
0.623490 0.781831i $$-0.285714\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 −0.623490 0.781831i $$-0.714286\pi$$
0.623490 + 0.781831i $$0.285714\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 −0.149042 0.988831i $$-0.547619\pi$$
0.149042 + 0.988831i $$0.452381\pi$$
$$360$$ 0 0
$$361$$ 1.40097 + 2.42655i 1.40097 + 2.42655i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0.277479 + 1.84095i 0.277479 + 1.84095i
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.52446 0.347948i 1.52446 0.347948i 0.623490 0.781831i $$-0.285714\pi$$
0.900969 + 0.433884i $$0.142857\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −0.445042 −0.445042 −0.222521 0.974928i $$-0.571429\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0.425270 1.86323i 0.425270 1.86323i −0.0747301 0.997204i $$-0.523810\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 −0.222521 0.974928i $$-0.571429\pi$$
0.222521 + 0.974928i $$0.428571\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0.587862 0.0440542i 0.587862 0.0440542i
$$389$$ 0 0 −0.781831 0.623490i $$-0.785714\pi$$
0.781831 + 0.623490i $$0.214286\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.04876 0.411608i 1.04876 0.411608i 0.222521 0.974928i $$-0.428571\pi$$
0.826239 + 0.563320i $$0.190476\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −0.988831 0.149042i −0.988831 0.149042i
$$401$$ 0 0 0.974928 0.222521i $$-0.0714286\pi$$
−0.974928 + 0.222521i $$0.928571\pi$$
$$402$$ 0 0
$$403$$ −0.826239 + 3.61999i −0.826239 + 3.61999i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −1.12349 0.0841939i −1.12349 0.0841939i −0.500000 0.866025i $$-0.666667\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 1.35654 + 0.101659i 1.35654 + 0.101659i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 −0.0747301 0.997204i $$-0.523810\pi$$
0.0747301 + 0.997204i $$0.476190\pi$$
$$420$$ 0 0
$$421$$ 0.0931869 1.24349i 0.0931869 1.24349i −0.733052 0.680173i $$-0.761905\pi$$
0.826239 0.563320i $$-0.190476\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.09839 + 0.250701i −1.09839 + 0.250701i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.930874 0.365341i $$-0.880952\pi$$
0.930874 + 0.365341i $$0.119048\pi$$
$$432$$ 0 0
$$433$$ −1.06356 0.848162i −1.06356 0.848162i −0.0747301 0.997204i $$-0.523810\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0.455573 0.571270i 0.455573 0.571270i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0.460898 + 1.49419i 0.460898 + 1.49419i 0.826239 + 0.563320i $$0.190476\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 −0.294755 0.955573i $$-0.595238\pi$$
0.294755 + 0.955573i $$0.404762\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$449$$ 0 0 −0.781831 0.623490i $$-0.785714\pi$$
0.781831 + 0.623490i $$0.214286\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.0546039 + 0.139129i −0.0546039 + 0.139129i −0.955573 0.294755i $$-0.904762\pi$$
0.900969 + 0.433884i $$0.142857\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 0.0747301 0.997204i $$-0.476190\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$462$$ 0 0
$$463$$ 0.0931869 + 1.24349i 0.0931869 + 1.24349i 0.826239 + 0.563320i $$0.190476\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.826239 0.563320i $$-0.190476\pi$$
−0.826239 + 0.563320i $$0.809524\pi$$
$$468$$ 0 0
$$469$$ 1.88980 0.582926i 1.88980 0.582926i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −0.574730 + 1.86323i −0.574730 + 1.86323i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 0.222521 0.974928i $$-0.428571\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$480$$ 0 0
$$481$$ 0.271281 0.0619180i 0.271281 0.0619180i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −0.988831 + 0.149042i −0.988831 + 0.149042i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1.78181 0.268565i −1.78181 0.268565i −0.826239 0.563320i $$-0.809524\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −1.55929 + 1.24349i −1.55929 + 1.24349i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0.0332580 + 0.145713i 0.0332580 + 0.145713i 0.988831 0.149042i $$-0.0476190\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 0.222521 0.974928i $$-0.428571\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −0.733052 + 1.26968i −0.733052 + 1.26968i
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0.867767i 0.867767i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 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 $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$522$$ 0 0
$$523$$ −0.297251 1.97213i −0.297251 1.97213i −0.222521 0.974928i $$-0.571429\pi$$
−0.0747301 0.997204i $$-0.523810\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −0.623490 0.781831i −0.623490 0.781831i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.52446 + 1.21572i 1.52446 + 1.21572i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.326239 + 0.302705i 0.326239 + 0.302705i 0.826239 0.563320i $$-0.190476\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0.0546039 + 0.139129i 0.0546039 + 0.139129i 0.955573 0.294755i $$-0.0952381\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0.367711 1.61105i 0.367711 1.61105i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0.129334 + 0.268565i 0.129334 + 0.268565i
$$557$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$558$$ 0 0
$$559$$ −1.54379 3.20571i −1.54379 3.20571i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$570$$ 0 0
$$571$$ −0.134659 + 1.79690i −0.134659 + 1.79690i 0.365341 + 0.930874i $$0.380952\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0.0878620 + 0.284841i 0.0878620 + 0.284841i 0.988831 0.149042i $$-0.0476190\pi$$
−0.900969 + 0.433884i $$0.857143\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$588$$ 0 0
$$589$$ 1.94440 + 3.36781i 1.94440 + 3.36781i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0.134659 + 0.0648483i 0.134659 + 0.0648483i
$$593$$ 0 0 −0.365341 0.930874i $$-0.619048\pi$$
0.365341 + 0.930874i $$0.380952\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 0.680173 0.733052i $$-0.261905\pi$$
−0.680173 + 0.733052i $$0.738095\pi$$
$$600$$ 0 0
$$601$$ −1.17809 + 1.26968i −1.17809 + 1.26968i −0.222521 + 0.974928i $$0.571429\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1.44973 0.218511i 1.44973 0.218511i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1.21135 + 1.12397i −1.21135 + 1.12397i −0.222521 + 0.974928i $$0.571429\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 0.563320 0.826239i $$-0.309524\pi$$
−0.563320 + 0.826239i $$0.690476\pi$$
$$618$$ 0 0
$$619$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −0.900969 + 0.433884i −0.900969 + 0.433884i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −1.06356 1.14625i −1.06356 1.14625i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0.0931869 + 0.116853i 0.0931869 + 0.116853i 0.826239 0.563320i $$-0.190476\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.73305 0.680173i 1.73305 0.680173i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 0.433884 0.900969i $$-0.357143\pi$$
−0.433884 + 0.900969i $$0.642857\pi$$
$$642$$ 0 0
$$643$$ 0.277479 + 0.108903i 0.277479 + 0.108903i 0.500000 0.866025i $$-0.333333\pi$$
−0.222521 + 0.974928i $$0.571429\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.365341 0.930874i $$-0.380952\pi$$
−0.365341 + 0.930874i $$0.619048\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0.535628 0.496990i 0.535628 0.496990i
$$653$$ 0 0 0.781831 0.623490i $$-0.214286\pi$$
−0.781831 + 0.623490i $$0.785714\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 −0.563320 0.826239i $$-0.690476\pi$$
0.563320 + 0.826239i $$0.309524\pi$$
$$660$$ 0 0
$$661$$ 0 0 −0.955573 0.294755i $$-0.904762\pi$$
0.955573 + 0.294755i $$0.0952381\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0.914101 + 0.848162i 0.914101 + 0.848162i 0.988831 0.149042i $$-0.0476190\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0.900969 2.29563i 0.900969 2.29563i
$$677$$ 0 0 0.733052 0.680173i $$-0.238095\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$678$$ 0 0
$$679$$ −0.173761 0.563320i −0.173761 0.563320i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 0.149042 0.988831i $$-0.452381\pi$$
−0.149042 + 0.988831i $$0.547619\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0.425270 1.86323i 0.425270 1.86323i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0.258149 1.71271i 0.258149 1.71271i −0.365341 0.930874i $$-0.619048\pi$$
0.623490 0.781831i $$-0.285714\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0.0747301 + 0.997204i 0.0747301 + 0.997204i
$$701$$ 0 0 0.974928 0.222521i $$-0.0714286\pi$$
−0.974928 + 0.222521i $$0.928571\pi$$
$$702$$ 0 0
$$703$$ 0.164166 0.240787i 0.164166 0.240787i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −0.142820 1.90580i −0.142820 1.90580i −0.365341 0.930874i $$-0.619048\pi$$
0.222521 0.974928i $$-0.428571\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 0.988831 0.149042i $$-0.0476190\pi$$
−0.988831 + 0.149042i $$0.952381\pi$$
$$720$$ 0 0
$$721$$ −0.202749 1.34515i −0.202749 1.34515i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −1.06356 + 1.14625i −1.06356 + 1.14625i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −1.61232 + 0.632789i −1.61232 + 0.632789i −0.988831 0.149042i $$-0.952381\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 1.32624 + 0.302705i 1.32624 + 0.302705i 0.826239 0.563320i $$-0.190476\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −0.147791 1.97213i −0.147791 1.97213i −0.222521 0.974928i $$-0.571429\pi$$
0.0747301 0.997204i $$-0.476190\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 0.149042 0.988831i $$-0.452381\pi$$
−0.149042 + 0.988831i $$0.547619\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0.277479 + 0.347948i 0.277479 + 0.347948i 0.900969 0.433884i $$-0.142857\pi$$
−0.623490 + 0.781831i $$0.714286\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0.658322 + 0.317031i 0.658322 + 0.317031i 0.733052 0.680173i $$-0.238095\pi$$
−0.0747301 + 0.997204i $$0.523810\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 −0.900969 0.433884i $$-0.857143\pi$$
0.900969 + 0.433884i $$0.142857\pi$$
$$762$$ 0 0
$$763$$ −0.658322 0.317031i −0.658322 0.317031i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 0.733052 0.680173i $$-0.238095\pi$$
−0.733052 + 0.680173i $$0.761905\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0.603718 + 0.411608i 0.603718 + 0.411608i
$$773$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$774$$ 0 0
$$775$$ −0.587862 + 1.90580i −0.587862 + 1.90580i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0.955573 + 0.294755i 0.955573 + 0.294755i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.85654 0.728639i 1.85654 0.728639i 0.900969 0.433884i $$-0.142857\pi$$
0.955573 0.294755i $$-0.0952381\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2.00433 + 0.618255i 2.00433 + 0.618255i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −0.590232 1.22563i −0.590232 1.22563i
$$797$$ 0 0 0.955573 0.294755i $$-0.0952381\pi$$
−0.955573 + 0.294755i $$0.904762\pi$$
$$798$$ 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 0 −0.680173 0.733052i $$-0.738095\pi$$
0.680173 + 0.733052i $$0.261905\pi$$