# Properties

 Label 55.2.e.b.32.1 Level 55 Weight 2 Character 55.32 Analytic conductor 0.439 Analytic rank 0 Dimension 4 CM discriminant -11 Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$55 = 5 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 55.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## Embedding invariants

 Embedding label 32.1 Root $$-1.65831 + 0.500000i$$ Character $$\chi$$ = 55.32 Dual form 55.2.e.b.43.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-2.15831 + 2.15831i) q^{3} +2.00000i q^{4} +(1.65831 - 1.50000i) q^{5} -6.31662i q^{9} +O(q^{10})$$ $$q+(-2.15831 + 2.15831i) q^{3} +2.00000i q^{4} +(1.65831 - 1.50000i) q^{5} -6.31662i q^{9} +3.31662 q^{11} +(-4.31662 - 4.31662i) q^{12} +(-0.341688 + 6.81662i) q^{15} -4.00000 q^{16} +(3.00000 + 3.31662i) q^{20} +(2.84169 - 2.84169i) q^{23} +(0.500000 - 4.97494i) q^{25} +(7.15831 + 7.15831i) q^{27} -9.94987 q^{31} +(-7.15831 + 7.15831i) q^{33} +12.6332 q^{36} +(1.47494 + 1.47494i) q^{37} +6.63325i q^{44} +(-9.47494 - 10.4749i) q^{45} +(-9.31662 - 9.31662i) q^{47} +(8.63325 - 8.63325i) q^{48} +7.00000i q^{49} +(3.63325 - 3.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-13.6332 - 0.683375i) q^{60} -8.00000i q^{64} +(11.4749 + 11.4749i) q^{67} +12.2665i q^{69} -3.00000 q^{71} +(9.65831 + 11.8166i) q^{75} +(-6.63325 + 6.00000i) q^{80} -11.9499 q^{81} +9.00000i q^{89} +(5.68338 + 5.68338i) q^{92} +(21.4749 - 21.4749i) q^{93} +(-3.52506 - 3.52506i) q^{97} -20.9499i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + O(q^{10})$$ $$4q - 2q^{3} - 4q^{12} - 8q^{15} - 16q^{16} + 12q^{20} + 18q^{23} + 2q^{25} + 22q^{27} - 22q^{33} + 24q^{36} - 14q^{37} - 18q^{45} - 24q^{47} + 8q^{48} - 12q^{53} + 22q^{55} - 28q^{60} + 26q^{67} - 12q^{71} + 32q^{75} - 8q^{81} + 36q^{92} + 66q^{93} - 34q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$e\left(\frac{1}{4}\right)$$ $$-1$$

## 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 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ −2.15831 + 2.15831i −1.24610 + 1.24610i −0.288675 + 0.957427i $$0.593215\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 2.00000i 1.00000i
$$5$$ 1.65831 1.50000i 0.741620 0.670820i
$$6$$ 0 0
$$7$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$8$$ 0 0
$$9$$ 6.31662i 2.10554i
$$10$$ 0 0
$$11$$ 3.31662 1.00000
$$12$$ −4.31662 4.31662i −1.24610 1.24610i
$$13$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$14$$ 0 0
$$15$$ −0.341688 + 6.81662i −0.0882234 + 1.76004i
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 3.00000 + 3.31662i 0.670820 + 0.741620i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.84169 2.84169i 0.592533 0.592533i −0.345782 0.938315i $$-0.612386\pi$$
0.938315 + 0.345782i $$0.112386\pi$$
$$24$$ 0 0
$$25$$ 0.500000 4.97494i 0.100000 0.994987i
$$26$$ 0 0
$$27$$ 7.15831 + 7.15831i 1.37762 + 1.37762i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −9.94987 −1.78705 −0.893525 0.449013i $$-0.851776\pi$$
−0.893525 + 0.449013i $$0.851776\pi$$
$$32$$ 0 0
$$33$$ −7.15831 + 7.15831i −1.24610 + 1.24610i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 12.6332 2.10554
$$37$$ 1.47494 + 1.47494i 0.242478 + 0.242478i 0.817875 0.575396i $$-0.195152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$44$$ 6.63325i 1.00000i
$$45$$ −9.47494 10.4749i −1.41244 1.56151i
$$46$$ 0 0
$$47$$ −9.31662 9.31662i −1.35897 1.35897i −0.875190 0.483779i $$-0.839264\pi$$
−0.483779 0.875190i $$-0.660736\pi$$
$$48$$ 8.63325 8.63325i 1.24610 1.24610i
$$49$$ 7.00000i 1.00000i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 3.63325 3.63325i 0.499065 0.499065i −0.412082 0.911147i $$-0.635198\pi$$
0.911147 + 0.412082i $$0.135198\pi$$
$$54$$ 0 0
$$55$$ 5.50000 4.97494i 0.741620 0.670820i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.31662i 0.431788i −0.976417 0.215894i $$-0.930733\pi$$
0.976417 0.215894i $$-0.0692665\pi$$
$$60$$ −13.6332 0.683375i −1.76004 0.0882234i
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 8.00000i 1.00000i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.4749 + 11.4749i 1.40189 + 1.40189i 0.794101 + 0.607785i $$0.207942\pi$$
0.607785 + 0.794101i $$0.292058\pi$$
$$68$$ 0 0
$$69$$ 12.2665i 1.47671i
$$70$$ 0 0
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 0 0
$$73$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$74$$ 0 0
$$75$$ 9.65831 + 11.8166i 1.11525 + 1.36447i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −6.63325 + 6.00000i −0.741620 + 0.670820i
$$81$$ −11.9499 −1.32776
$$82$$ 0 0
$$83$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.00000i 0.953998i 0.878904 + 0.476999i $$0.158275\pi$$
−0.878904 + 0.476999i $$0.841725\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 5.68338 + 5.68338i 0.592533 + 0.592533i
$$93$$ 21.4749 21.4749i 2.22685 2.22685i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.52506 3.52506i −0.357916 0.357916i 0.505128 0.863044i $$-0.331445\pi$$
−0.863044 + 0.505128i $$0.831445\pi$$
$$98$$ 0 0
$$99$$ 20.9499i 2.10554i
$$100$$ 9.94987 + 1.00000i 0.994987 + 0.100000i
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ −7.94987 + 7.94987i −0.783324 + 0.783324i −0.980390 0.197066i $$-0.936859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$108$$ −14.3166 + 14.3166i −1.37762 + 1.37762i
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ −6.36675 −0.604305
$$112$$ 0 0
$$113$$ −12.1583 + 12.1583i −1.14376 + 1.14376i −0.156001 + 0.987757i $$0.549860\pi$$
−0.987757 + 0.156001i $$0.950140\pi$$
$$114$$ 0 0
$$115$$ 0.449874 8.97494i 0.0419510 0.836917i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 19.8997i 1.78705i
$$125$$ −6.63325 9.00000i −0.593296 0.804984i
$$126$$ 0 0
$$127$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ −14.3166 14.3166i −1.24610 1.24610i
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 22.6082 + 1.13325i 1.94580 + 0.0975346i
$$136$$ 0 0
$$137$$ −10.1082 10.1082i −0.863601 0.863601i 0.128154 0.991754i $$-0.459095\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 40.2164 3.38683
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 25.2665i 2.10554i
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −15.1082 15.1082i −1.24610 1.24610i
$$148$$ −2.94987 + 2.94987i −0.242478 + 0.242478i
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −16.5000 + 14.9248i −1.32531 + 1.19879i
$$156$$ 0 0
$$157$$ 16.4749 + 16.4749i 1.31484 + 1.31484i 0.917800 + 0.397043i $$0.129964\pi$$
0.397043 + 0.917800i $$0.370036\pi$$
$$158$$ 0 0
$$159$$ 15.6834i 1.24377i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −17.9499 + 17.9499i −1.40594 + 1.40594i −0.626608 + 0.779334i $$0.715557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 0 0
$$165$$ −1.13325 + 22.6082i −0.0882234 + 1.76004i
$$166$$ 0 0
$$167$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$168$$ 0 0
$$169$$ 13.0000i 1.00000i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −13.2665 −1.00000
$$177$$ 7.15831 + 7.15831i 0.538052 + 0.538052i
$$178$$ 0 0
$$179$$ 21.0000i 1.56961i −0.619740 0.784807i $$-0.712762\pi$$
0.619740 0.784807i $$-0.287238\pi$$
$$180$$ 20.9499 18.9499i 1.56151 1.41244i
$$181$$ −9.94987 −0.739568 −0.369784 0.929118i $$-0.620568\pi$$
−0.369784 + 0.929118i $$0.620568\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.65831 + 0.233501i 0.342486 + 0.0171673i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 18.6332 18.6332i 1.35897 1.35897i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 23.2164 1.67988 0.839939 0.542681i $$-0.182591\pi$$
0.839939 + 0.542681i $$0.182591\pi$$
$$192$$ 17.2665 + 17.2665i 1.24610 + 1.24610i
$$193$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −14.0000 −1.00000
$$197$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$198$$ 0 0
$$199$$ 19.8997i 1.41066i −0.708881 0.705328i $$-0.750800\pi$$
0.708881 0.705328i $$-0.249200\pi$$
$$200$$ 0 0
$$201$$ −49.5330 −3.49379
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −17.9499 17.9499i −1.24760 1.24760i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 7.26650 + 7.26650i 0.499065 + 0.499065i
$$213$$ 6.47494 6.47494i 0.443655 0.443655i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 9.94987 + 11.0000i 0.670820 + 0.741620i
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 14.4248 14.4248i 0.965957 0.965957i −0.0334825 0.999439i $$-0.510660\pi$$
0.999439 + 0.0334825i $$0.0106598\pi$$
$$224$$ 0 0
$$225$$ −31.4248 3.15831i −2.09499 0.210554i
$$226$$ 0 0
$$227$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$228$$ 0 0
$$229$$ 29.8496i 1.97252i 0.165205 + 0.986259i $$0.447172\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$234$$ 0 0
$$235$$ −29.4248 1.47494i −1.91946 0.0962143i
$$236$$ 6.63325 0.431788
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 1.36675 27.2665i 0.0882234 1.76004i
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 4.31662 4.31662i 0.276912 0.276912i
$$244$$ 0 0
$$245$$ 10.5000 + 11.6082i 0.670820 + 0.741620i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 27.0000 1.70422 0.852112 0.523359i $$-0.175321\pi$$
0.852112 + 0.523359i $$0.175321\pi$$
$$252$$ 0 0
$$253$$ 9.42481 9.42481i 0.592533 0.592533i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 22.2665 + 22.2665i 1.38895 + 1.38895i 0.827541 + 0.561405i $$0.189739\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$264$$ 0 0
$$265$$ 0.575188 11.4749i 0.0353335 0.704900i
$$266$$ 0 0
$$267$$ −19.4248 19.4248i −1.18878 1.18878i
$$268$$ −22.9499 + 22.9499i −1.40189 + 1.40189i
$$269$$ 13.2665i 0.808873i 0.914566 + 0.404436i $$0.132532\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.65831 16.5000i 0.100000 0.994987i
$$276$$ −24.5330 −1.47671
$$277$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$278$$ 0 0
$$279$$ 62.8496i 3.76271i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$284$$ 6.00000i 0.356034i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000i 1.00000i
$$290$$ 0 0
$$291$$ 15.2164 0.892000
$$292$$ 0 0
$$293$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$294$$ 0 0
$$295$$ −4.97494 5.50000i −0.289652 0.320222i
$$296$$ 0 0
$$297$$ 23.7414 + 23.7414i 1.37762 + 1.37762i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −23.6332 + 19.3166i −1.36447 + 1.11525i
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$308$$ 0 0
$$309$$ 34.3166i 1.95220i
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 24.4248 24.4248i 1.38057 1.38057i 0.536972 0.843600i $$-0.319568\pi$$
0.843600 0.536972i $$-0.180432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −25.1082 25.1082i −1.41022 1.41022i −0.758236 0.651981i $$-0.773938\pi$$
−0.651981 0.758236i $$-0.726062\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −12.0000 13.2665i −0.670820 0.741620i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 23.8997i 1.32776i
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −9.94987 −0.546895 −0.273447 0.961887i $$-0.588164\pi$$
−0.273447 + 0.961887i $$0.588164\pi$$
$$332$$ 0 0
$$333$$ 9.31662 9.31662i 0.510548 0.510548i
$$334$$ 0 0
$$335$$ 36.2414 + 1.81662i 1.98008 + 0.0992528i
$$336$$ 0 0
$$337$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$338$$ 0 0
$$339$$ 52.4829i 2.85048i
$$340$$ 0 0
$$341$$ −33.0000 −1.78705
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 18.3997 + 20.3417i 0.990609 + 1.09516i
$$346$$ 0 0
$$347$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −13.7414 + 13.7414i −0.731383 + 0.731383i −0.970894 0.239511i $$-0.923013\pi$$
0.239511 + 0.970894i $$0.423013\pi$$
$$354$$ 0 0
$$355$$ −4.97494 + 4.50000i −0.264042 + 0.238835i
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −23.7414 + 23.7414i −1.24610 + 1.24610i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −13.5251 13.5251i −0.706003 0.706003i 0.259690 0.965692i $$-0.416380\pi$$
−0.965692 + 0.259690i $$0.916380\pi$$
$$368$$ −11.3668 + 11.3668i −0.592533 + 0.592533i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 42.9499 + 42.9499i 2.22685 + 2.22685i
$$373$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$374$$ 0 0
$$375$$ 33.7414 + 5.10819i 1.74240 + 0.263786i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 29.8496i 1.53327i 0.642082 + 0.766636i $$0.278071\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 17.8417 17.8417i 0.911668 0.911668i −0.0847358 0.996403i $$-0.527005\pi$$
0.996403 + 0.0847358i $$0.0270046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 7.05013 7.05013i 0.357916 0.357916i
$$389$$ 36.4829i 1.84976i −0.380265 0.924878i $$-0.624167\pi$$
0.380265 0.924878i $$-0.375833\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 41.8997 2.10554
$$397$$ −20.8997 20.8997i −1.04893 1.04893i −0.998740 0.0501886i $$-0.984018\pi$$
−0.0501886 0.998740i $$-0.515982\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −2.00000 + 19.8997i −0.100000 + 0.994987i
$$401$$ −26.5330 −1.32499 −0.662497 0.749064i $$-0.730503\pi$$
−0.662497 + 0.749064i $$0.730503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −19.8166 + 17.9248i −0.984696 + 0.890691i
$$406$$ 0 0
$$407$$ 4.89181 + 4.89181i 0.242478 + 0.242478i
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 43.6332 2.15227
$$412$$ −15.8997 15.8997i −0.783324 0.783324i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 24.0000i 1.17248i 0.810139 + 0.586238i $$0.199392\pi$$
−0.810139 + 0.586238i $$0.800608\pi$$
$$420$$ 0 0
$$421$$ 39.7995 1.93971 0.969854 0.243685i $$-0.0783563\pi$$
0.969854 + 0.243685i $$0.0783563\pi$$
$$422$$ 0 0
$$423$$ −58.8496 + 58.8496i −2.86137 + 2.86137i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −28.6332 28.6332i −1.37762 1.37762i
$$433$$ 29.4248 29.4248i 1.41407 1.41407i 0.696826 0.717241i $$-0.254595\pi$$
0.717241 0.696826i $$-0.245405\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 44.2164 2.10554
$$442$$ 0 0
$$443$$ −28.7414 + 28.7414i −1.36555 + 1.36555i −0.498870 + 0.866677i $$0.666252\pi$$
−0.866677 + 0.498870i $$0.833748\pi$$
$$444$$ 12.7335i 0.604305i
$$445$$ 13.5000 + 14.9248i 0.639961 + 0.707504i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 39.0000i 1.84052i 0.391303 + 0.920262i $$0.372024\pi$$
−0.391303 + 0.920262i $$0.627976\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −24.3166 24.3166i −1.14376 1.14376i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 17.9499 + 0.899749i 0.836917 + 0.0419510i
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ −0.575188 + 0.575188i −0.0267313 + 0.0267313i −0.720346 0.693615i $$-0.756017\pi$$
0.693615 + 0.720346i $$0.256017\pi$$
$$464$$ 0 0
$$465$$ 3.39975 67.8246i 0.157660 3.14529i
$$466$$ 0 0
$$467$$ 23.0581 + 23.0581i 1.06700 + 1.06700i 0.997588 + 0.0694117i $$0.0221122\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −71.1161 −3.27686
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −22.9499 22.9499i −1.05080 1.05080i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 22.0000i 1.00000i
$$485$$ −11.1332 0.558061i −0.505535 0.0253403i
$$486$$ 0 0
$$487$$ 26.4749 + 26.4749i 1.19969 + 1.19969i 0.974258 + 0.225436i $$0.0723806\pi$$
0.225436 + 0.974258i $$0.427619\pi$$
$$488$$ 0 0
$$489$$ 77.4829i 3.50390i
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −31.4248 34.7414i −1.41244 1.56151i
$$496$$ 39.7995 1.78705
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 19.8997i 0.890835i −0.895323 0.445418i $$-0.853055\pi$$
0.895323 0.445418i $$-0.146945\pi$$
$$500$$ 18.0000 13.2665i 0.804984 0.593296i
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 28.0581 + 28.0581i 1.24610 + 1.24610i
$$508$$ 0 0
$$509$$ 3.31662i 0.147007i −0.997295 0.0735034i $$-0.976582\pi$$
0.997295 0.0735034i $$-0.0234180\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1.25856 + 25.1082i −0.0554589 + 1.10640i
$$516$$ 0 0
$$517$$ −30.8997 30.8997i −1.35897 1.35897i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −43.1161 −1.88895 −0.944476 0.328581i $$-0.893430\pi$$
−0.944476 + 0.328581i $$0.893430\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 28.6332 28.6332i 1.24610 1.24610i
$$529$$ 6.84962i 0.297810i
$$530$$ 0 0
$$531$$ −20.9499 −0.909147
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 45.3246 + 45.3246i 1.95590 + 1.95590i
$$538$$ 0 0
$$539$$ 23.2164i 1.00000i
$$540$$ −2.26650 + 45.2164i −0.0975346 + 1.94580i
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 21.4749 21.4749i 0.921578 0.921578i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$548$$ 20.2164 20.2164i 0.863601 0.863601i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −10.5581 + 9.55013i −0.448165 + 0.405380i
$$556$$ 0 0
$$557$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$564$$ 80.4327i 3.38683i
$$565$$ −1.92481 + 38.3997i −0.0809774 + 1.61549i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ −50.1082 + 50.1082i −2.09330 + 2.09330i
$$574$$ 0 0
$$575$$ −12.7164 15.5581i −0.530309 0.648816i
$$576$$ −50.5330 −2.10554
$$577$$ −18.5251 18.5251i −0.771208 0.771208i 0.207109 0.978318i $$-0.433594\pi$$
−0.978318 + 0.207109i $$0.933594\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 12.0501 12.0501i 0.499065 0.499065i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 20.6834 + 20.6834i 0.853694 + 0.853694i 0.990586 0.136892i $$-0.0437113\pi$$
−0.136892 + 0.990586i $$0.543711\pi$$
$$588$$ 30.2164 30.2164i 1.24610 1.24610i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −5.89975 5.89975i −0.242478 0.242478i
$$593$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 42.9499 + 42.9499i 1.75782 + 1.75782i
$$598$$ 0 0
$$599$$ 36.0000i 1.47092i −0.677568 0.735460i $$-0.736966\pi$$
0.677568 0.735460i $$-0.263034\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 72.4829 72.4829i 2.95173 2.95173i
$$604$$ 0 0
$$605$$ 18.2414 16.5000i 0.741620 0.670820i
$$606$$ 0 0
$$607$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −7.73350 7.73350i −0.311339 0.311339i 0.534089 0.845428i $$-0.320655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 1.00000i 0.0401934i −0.999798 0.0200967i $$-0.993603\pi$$
0.999798 0.0200967i $$-0.00639741\pi$$
$$620$$ −29.8496 33.0000i −1.19879 1.32531i
$$621$$ 40.6834 1.63257
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −24.5000 4.97494i −0.980000 0.198997i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −32.9499 + 32.9499i −1.31484 + 1.31484i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ −31.3668 −1.24377
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 18.9499i 0.749645i
$$640$$ 0 0
$$641$$ 23.2164 0.916992 0.458496 0.888697i $$-0.348388\pi$$
0.458496 + 0.888697i $$0.348388\pi$$
$$642$$ 0 0
$$643$$ −5.57519 + 5.57519i −0.219864 + 0.219864i −0.808441 0.588577i $$-0.799688\pi$$
0.588577 + 0.808441i $$0.299688\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.05806 + 8.05806i 0.316795 + 0.316795i 0.847535 0.530740i $$-0.178086\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 0 0
$$649$$ 11.0000i 0.431788i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −35.8997 35.8997i −1.40594 1.40594i
$$653$$ −27.1583 + 27.1583i −1.06279 + 1.06279i −0.0648948 + 0.997892i $$0.520671\pi$$
−0.997892 + 0.0648948i $$0.979329\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ −45.2164 2.26650i −1.76004 0.0882234i
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 62.2665i 2.40736i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$674$$ 0 0
$$675$$ 39.1913 32.0330i 1.50847 1.23295i
$$676$$ 26.0000 1.00000
$$677$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 35.2164 35.2164i 1.34752 1.34752i 0.459167 0.888350i $$-0.348148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 0 0
$$685$$ −31.9248 1.60025i −1.21978 0.0611425i
$$686$$ 0 0
$$687$$ −64.4248 64.4248i −2.45796 2.45796i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 26.5330i 1.00000i
$$705$$ 66.6913 60.3246i 2.51174 2.27195i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ −14.3166 + 14.3166i −0.538052 + 0.538052i
$$709$$ 19.0000i 0.713560i 0.934188 + 0.356780i $$0.116125\pi$$
−0.934188 + 0.356780i $$0.883875\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −28.2744 + 28.2744i −1.05889 + 1.05889i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 42.0000 1.56961
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 51.0000i 1.90198i −0.309223 0.950990i $$-0.600069\pi$$
0.309223 0.950990i $$-0.399931\pi$$
$$720$$ 37.8997 + 41.8997i 1.41244 + 1.56151i
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 19.8997i 0.739568i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 31.4749 + 31.4749i 1.16734 + 1.16734i 0.982831 + 0.184510i $$0.0590699\pi$$
0.184510 + 0.982831i $$0.440930\pi$$
$$728$$ 0 0
$$729$$ 17.2164i 0.637643i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$734$$ 0 0
$$735$$ −47.7164 2.39181i −1.76004 0.0882234i
$$736$$ 0 0
$$737$$ 38.0581 + 38.0581i 1.40189 + 1.40189i
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ −0.467002 + 9.31662i −0.0171673 + 0.342486i
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 37.2665 + 37.2665i 1.35897 + 1.35897i
$$753$$ −58.2744 + 58.2744i −2.12364 + 2.12364i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −0.899749 0.899749i −0.0327019 0.0327019i 0.690567 0.723269i $$-0.257361\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ 0 0
$$759$$ 40.6834i 1.47671i
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 46.4327i 1.67988i
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −34.5330 + 34.5330i −1.24610 + 1.24610i
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ −96.1161 −3.46154
$$772$$ 0 0
$$773$$ 33.6332 33.6332i 1.20970 1.20970i 0.238581 0.971123i $$-0.423318\pi$$
0.971123 0.238581i $$-0.0766824\pi$$
$$774$$ 0 0
$$775$$ −4.97494 + 49.5000i −0.178705 + 1.77809i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −9.94987 −0.356034
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 28.0000i 1.00000i
$$785$$ 52.0330 + 2.60819i 1.85714 + 0.0930902i
$$786$$ 0 0
$$787$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 23.5251 + 26.0079i 0.834348 + 0.922406i
$$796$$ 39.7995 1.41066
$$797$$ −26.6913 26.6913i −0.945455 0.945455i 0.0531327 0.998587i $$-0.483079\pi$$
−0.998587 + 0.0531327i $$0.983079\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 56.8496 2.00868
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 99.0660i 3.49379i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −28.6332 28.6332i −1.00794 1.00794i
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0