# Properties

 Label 4560.2.d.e.2431.1 Level $4560$ Weight $2$ Character 4560.2431 Analytic conductor $36.412$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.9821011968.3 Defining polynomial: $$x^{6} + 20 x^{4} + 118 x^{2} + 192$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2431.1 Root $$-3.24237i$$ of defining polynomial Character $$\chi$$ $$=$$ 4560.2431 Dual form 4560.2.d.e.2431.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -3.24237i q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -3.24237i q^{7} +1.00000 q^{9} -5.82288i q^{11} -3.24237i q^{13} +1.00000 q^{15} -4.51298 q^{17} +(-3.51298 - 2.58050i) q^{19} +3.24237i q^{21} -5.56750i q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.82288i q^{29} -7.85398 q^{31} +5.82288i q^{33} +3.24237i q^{35} +0.255376i q^{37} +3.24237i q^{39} -0.661868i q^{41} +0.255376i q^{43} -1.00000 q^{45} +0.406492i q^{47} -3.51298 q^{49} +4.51298 q^{51} +5.56750i q^{53} +5.82288i q^{55} +(3.51298 + 2.58050i) q^{57} +9.02595 q^{59} -5.85398 q^{61} -3.24237i q^{63} +3.24237i q^{65} +10.3670 q^{67} +5.56750i q^{69} +9.02595 q^{71} -5.68495 q^{73} -1.00000 q^{75} -18.8799 q^{77} +4.65900 q^{79} +1.00000 q^{81} -6.07825i q^{83} +4.51298 q^{85} +5.82288i q^{87} -3.64886i q^{89} -10.5130 q^{91} +7.85398 q^{93} +(3.51298 + 2.58050i) q^{95} +13.0536i q^{97} -5.82288i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{3} - 6q^{5} + 6q^{9} + O(q^{10})$$ $$6q - 6q^{3} - 6q^{5} + 6q^{9} + 6q^{15} - 4q^{17} + 2q^{19} + 6q^{25} - 6q^{27} - 12q^{31} - 6q^{45} + 2q^{49} + 4q^{51} - 2q^{57} + 8q^{59} + 4q^{67} + 8q^{71} - 6q^{75} - 32q^{77} + 40q^{79} + 6q^{81} + 4q^{85} - 40q^{91} + 12q^{93} - 2q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$$.

 $$n$$ $$1141$$ $$1711$$ $$1921$$ $$2737$$ $$3041$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 3.24237i 1.22550i −0.790276 0.612751i $$-0.790063\pi$$
0.790276 0.612751i $$-0.209937\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.82288i 1.75566i −0.478970 0.877832i $$-0.658990\pi$$
0.478970 0.877832i $$-0.341010\pi$$
$$12$$ 0 0
$$13$$ 3.24237i 0.899272i −0.893212 0.449636i $$-0.851554\pi$$
0.893212 0.449636i $$-0.148446\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −4.51298 −1.09456 −0.547279 0.836950i $$-0.684336\pi$$
−0.547279 + 0.836950i $$0.684336\pi$$
$$18$$ 0 0
$$19$$ −3.51298 2.58050i −0.805932 0.592008i
$$20$$ 0 0
$$21$$ 3.24237i 0.707544i
$$22$$ 0 0
$$23$$ 5.56750i 1.16090i −0.814295 0.580452i $$-0.802876\pi$$
0.814295 0.580452i $$-0.197124\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.82288i 1.08128i −0.841254 0.540640i $$-0.818182\pi$$
0.841254 0.540640i $$-0.181818\pi$$
$$30$$ 0 0
$$31$$ −7.85398 −1.41062 −0.705308 0.708901i $$-0.749191\pi$$
−0.705308 + 0.708901i $$0.749191\pi$$
$$32$$ 0 0
$$33$$ 5.82288i 1.01363i
$$34$$ 0 0
$$35$$ 3.24237i 0.548061i
$$36$$ 0 0
$$37$$ 0.255376i 0.0419836i 0.999780 + 0.0209918i $$0.00668239\pi$$
−0.999780 + 0.0209918i $$0.993318\pi$$
$$38$$ 0 0
$$39$$ 3.24237i 0.519195i
$$40$$ 0 0
$$41$$ 0.661868i 0.103366i −0.998664 0.0516832i $$-0.983541\pi$$
0.998664 0.0516832i $$-0.0164586\pi$$
$$42$$ 0 0
$$43$$ 0.255376i 0.0389445i 0.999810 + 0.0194723i $$0.00619860\pi$$
−0.999810 + 0.0194723i $$0.993801\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 0.406492i 0.0592930i 0.999560 + 0.0296465i $$0.00943815\pi$$
−0.999560 + 0.0296465i $$0.990562\pi$$
$$48$$ 0 0
$$49$$ −3.51298 −0.501854
$$50$$ 0 0
$$51$$ 4.51298 0.631943
$$52$$ 0 0
$$53$$ 5.56750i 0.764755i 0.924006 + 0.382377i $$0.124895\pi$$
−0.924006 + 0.382377i $$0.875105\pi$$
$$54$$ 0 0
$$55$$ 5.82288i 0.785156i
$$56$$ 0 0
$$57$$ 3.51298 + 2.58050i 0.465305 + 0.341796i
$$58$$ 0 0
$$59$$ 9.02595 1.17508 0.587539 0.809196i $$-0.300097\pi$$
0.587539 + 0.809196i $$0.300097\pi$$
$$60$$ 0 0
$$61$$ −5.85398 −0.749525 −0.374762 0.927121i $$-0.622276\pi$$
−0.374762 + 0.927121i $$0.622276\pi$$
$$62$$ 0 0
$$63$$ 3.24237i 0.408500i
$$64$$ 0 0
$$65$$ 3.24237i 0.402167i
$$66$$ 0 0
$$67$$ 10.3670 1.26652 0.633262 0.773937i $$-0.281715\pi$$
0.633262 + 0.773937i $$0.281715\pi$$
$$68$$ 0 0
$$69$$ 5.56750i 0.670248i
$$70$$ 0 0
$$71$$ 9.02595 1.07118 0.535592 0.844477i $$-0.320089\pi$$
0.535592 + 0.844477i $$0.320089\pi$$
$$72$$ 0 0
$$73$$ −5.68495 −0.665373 −0.332687 0.943037i $$-0.607955\pi$$
−0.332687 + 0.943037i $$0.607955\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −18.8799 −2.15157
$$78$$ 0 0
$$79$$ 4.65900 0.524178 0.262089 0.965044i $$-0.415589\pi$$
0.262089 + 0.965044i $$0.415589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.07825i 0.667175i −0.942719 0.333587i $$-0.891741\pi$$
0.942719 0.333587i $$-0.108259\pi$$
$$84$$ 0 0
$$85$$ 4.51298 0.489501
$$86$$ 0 0
$$87$$ 5.82288i 0.624278i
$$88$$ 0 0
$$89$$ 3.64886i 0.386779i −0.981122 0.193389i $$-0.938052\pi$$
0.981122 0.193389i $$-0.0619481\pi$$
$$90$$ 0 0
$$91$$ −10.5130 −1.10206
$$92$$ 0 0
$$93$$ 7.85398 0.814419
$$94$$ 0 0
$$95$$ 3.51298 + 2.58050i 0.360424 + 0.264754i
$$96$$ 0 0
$$97$$ 13.0536i 1.32540i 0.748887 + 0.662698i $$0.230589\pi$$
−0.748887 + 0.662698i $$0.769411\pi$$
$$98$$ 0 0
$$99$$ 5.82288i 0.585221i
$$100$$ 0 0
$$101$$ 13.7080 1.36399 0.681996 0.731356i $$-0.261112\pi$$
0.681996 + 0.731356i $$0.261112\pi$$
$$102$$ 0 0
$$103$$ −5.02595 −0.495222 −0.247611 0.968860i $$-0.579645\pi$$
−0.247611 + 0.968860i $$0.579645\pi$$
$$104$$ 0 0
$$105$$ 3.24237i 0.316423i
$$106$$ 0 0
$$107$$ 17.3929 1.68144 0.840718 0.541474i $$-0.182133\pi$$
0.840718 + 0.541474i $$0.182133\pi$$
$$108$$ 0 0
$$109$$ 4.65025i 0.445414i −0.974885 0.222707i $$-0.928511\pi$$
0.974885 0.222707i $$-0.0714893\pi$$
$$110$$ 0 0
$$111$$ 0.255376i 0.0242392i
$$112$$ 0 0
$$113$$ 11.2393i 1.05730i 0.848840 + 0.528650i $$0.177302\pi$$
−0.848840 + 0.528650i $$0.822698\pi$$
$$114$$ 0 0
$$115$$ 5.56750i 0.519172i
$$116$$ 0 0
$$117$$ 3.24237i 0.299757i
$$118$$ 0 0
$$119$$ 14.6327i 1.34138i
$$120$$ 0 0
$$121$$ −22.9059 −2.08235
$$122$$ 0 0
$$123$$ 0.661868i 0.0596787i
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −7.68495 −0.681929 −0.340965 0.940076i $$-0.610754\pi$$
−0.340965 + 0.940076i $$0.610754\pi$$
$$128$$ 0 0
$$129$$ 0.255376i 0.0224846i
$$130$$ 0 0
$$131$$ 1.00139i 0.0874919i −0.999043 0.0437460i $$-0.986071\pi$$
0.999043 0.0437460i $$-0.0139292\pi$$
$$132$$ 0 0
$$133$$ −8.36695 + 11.3904i −0.725507 + 0.987671i
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −3.68495 −0.314827 −0.157413 0.987533i $$-0.550315\pi$$
−0.157413 + 0.987533i $$0.550315\pi$$
$$138$$ 0 0
$$139$$ 1.32374i 0.112278i −0.998423 0.0561389i $$-0.982121\pi$$
0.998423 0.0561389i $$-0.0178790\pi$$
$$140$$ 0 0
$$141$$ 0.406492i 0.0342328i
$$142$$ 0 0
$$143$$ −18.8799 −1.57882
$$144$$ 0 0
$$145$$ 5.82288i 0.483564i
$$146$$ 0 0
$$147$$ 3.51298 0.289745
$$148$$ 0 0
$$149$$ 15.0260 1.23097 0.615487 0.788147i $$-0.288959\pi$$
0.615487 + 0.788147i $$0.288959\pi$$
$$150$$ 0 0
$$151$$ 22.2209 1.80831 0.904157 0.427201i $$-0.140500\pi$$
0.904157 + 0.427201i $$0.140500\pi$$
$$152$$ 0 0
$$153$$ −4.51298 −0.364852
$$154$$ 0 0
$$155$$ 7.85398 0.630847
$$156$$ 0 0
$$157$$ −4.36695 −0.348521 −0.174260 0.984700i $$-0.555753\pi$$
−0.174260 + 0.984700i $$0.555753\pi$$
$$158$$ 0 0
$$159$$ 5.56750i 0.441531i
$$160$$ 0 0
$$161$$ −18.0519 −1.42269
$$162$$ 0 0
$$163$$ 6.56889i 0.514515i 0.966343 + 0.257258i $$0.0828189\pi$$
−0.966343 + 0.257258i $$0.917181\pi$$
$$164$$ 0 0
$$165$$ 5.82288i 0.453310i
$$166$$ 0 0
$$167$$ −1.48702 −0.115069 −0.0575347 0.998344i $$-0.518324\pi$$
−0.0575347 + 0.998344i $$0.518324\pi$$
$$168$$ 0 0
$$169$$ 2.48702 0.191310
$$170$$ 0 0
$$171$$ −3.51298 2.58050i −0.268644 0.197336i
$$172$$ 0 0
$$173$$ 16.4003i 1.24689i 0.781868 + 0.623445i $$0.214267\pi$$
−0.781868 + 0.623445i $$0.785733\pi$$
$$174$$ 0 0
$$175$$ 3.24237i 0.245100i
$$176$$ 0 0
$$177$$ −9.02595 −0.678432
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 24.6152i 1.82964i −0.403866 0.914818i $$-0.632334\pi$$
0.403866 0.914818i $$-0.367666\pi$$
$$182$$ 0 0
$$183$$ 5.85398 0.432738
$$184$$ 0 0
$$185$$ 0.255376i 0.0187756i
$$186$$ 0 0
$$187$$ 26.2785i 1.92167i
$$188$$ 0 0
$$189$$ 3.24237i 0.235848i
$$190$$ 0 0
$$191$$ 1.17262i 0.0848479i 0.999100 + 0.0424239i $$0.0135080\pi$$
−0.999100 + 0.0424239i $$0.986492\pi$$
$$192$$ 0 0
$$193$$ 17.5356i 1.26224i −0.775685 0.631120i $$-0.782596\pi$$
0.775685 0.631120i $$-0.217404\pi$$
$$194$$ 0 0
$$195$$ 3.24237i 0.232191i
$$196$$ 0 0
$$197$$ 7.48702 0.533428 0.266714 0.963776i $$-0.414062\pi$$
0.266714 + 0.963776i $$0.414062\pi$$
$$198$$ 0 0
$$199$$ 0.510752i 0.0362063i 0.999836 + 0.0181031i $$0.00576272\pi$$
−0.999836 + 0.0181031i $$0.994237\pi$$
$$200$$ 0 0
$$201$$ −10.3670 −0.731228
$$202$$ 0 0
$$203$$ −18.8799 −1.32511
$$204$$ 0 0
$$205$$ 0.661868i 0.0462269i
$$206$$ 0 0
$$207$$ 5.56750i 0.386968i
$$208$$ 0 0
$$209$$ −15.0260 + 20.4556i −1.03937 + 1.41495i
$$210$$ 0 0
$$211$$ −18.3670 −1.26443 −0.632217 0.774792i $$-0.717855\pi$$
−0.632217 + 0.774792i $$0.717855\pi$$
$$212$$ 0 0
$$213$$ −9.02595 −0.618448
$$214$$ 0 0
$$215$$ 0.255376i 0.0174165i
$$216$$ 0 0
$$217$$ 25.4655i 1.72871i
$$218$$ 0 0
$$219$$ 5.68495 0.384153
$$220$$ 0 0
$$221$$ 14.6327i 0.984305i
$$222$$ 0 0
$$223$$ −23.0779 −1.54541 −0.772704 0.634767i $$-0.781096\pi$$
−0.772704 + 0.634767i $$0.781096\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 8.85693 0.587855 0.293927 0.955828i $$-0.405038\pi$$
0.293927 + 0.955828i $$0.405038\pi$$
$$228$$ 0 0
$$229$$ −20.5130 −1.35554 −0.677768 0.735276i $$-0.737053\pi$$
−0.677768 + 0.735276i $$0.737053\pi$$
$$230$$ 0 0
$$231$$ 18.8799 1.24221
$$232$$ 0 0
$$233$$ −28.7858 −1.88582 −0.942911 0.333046i $$-0.891924\pi$$
−0.942911 + 0.333046i $$0.891924\pi$$
$$234$$ 0 0
$$235$$ 0.406492i 0.0265166i
$$236$$ 0 0
$$237$$ −4.65900 −0.302635
$$238$$ 0 0
$$239$$ 25.6166i 1.65700i −0.559988 0.828501i $$-0.689194\pi$$
0.559988 0.828501i $$-0.310806\pi$$
$$240$$ 0 0
$$241$$ 22.4785i 1.44797i 0.689816 + 0.723984i $$0.257691\pi$$
−0.689816 + 0.723984i $$0.742309\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 3.51298 0.224436
$$246$$ 0 0
$$247$$ −8.36695 + 11.3904i −0.532376 + 0.724752i
$$248$$ 0 0
$$249$$ 6.07825i 0.385194i
$$250$$ 0 0
$$251$$ 15.8054i 0.997626i −0.866710 0.498813i $$-0.833769\pi$$
0.866710 0.498813i $$-0.166231\pi$$
$$252$$ 0 0
$$253$$ −32.4189 −2.03816
$$254$$ 0 0
$$255$$ −4.51298 −0.282614
$$256$$ 0 0
$$257$$ 18.0262i 1.12445i −0.826986 0.562223i $$-0.809946\pi$$
0.826986 0.562223i $$-0.190054\pi$$
$$258$$ 0 0
$$259$$ 0.828025 0.0514510
$$260$$ 0 0
$$261$$ 5.82288i 0.360427i
$$262$$ 0 0
$$263$$ 18.0262i 1.11155i −0.831334 0.555773i $$-0.812423\pi$$
0.831334 0.555773i $$-0.187577\pi$$
$$264$$ 0 0
$$265$$ 5.56750i 0.342009i
$$266$$ 0 0
$$267$$ 3.64886i 0.223307i
$$268$$ 0 0
$$269$$ 3.34663i 0.204048i −0.994782 0.102024i $$-0.967468\pi$$
0.994782 0.102024i $$-0.0325318\pi$$
$$270$$ 0 0
$$271$$ 3.83727i 0.233098i 0.993185 + 0.116549i $$0.0371831\pi$$
−0.993185 + 0.116549i $$0.962817\pi$$
$$272$$ 0 0
$$273$$ 10.5130 0.636274
$$274$$ 0 0
$$275$$ 5.82288i 0.351133i
$$276$$ 0 0
$$277$$ 21.7080 1.30430 0.652152 0.758088i $$-0.273866\pi$$
0.652152 + 0.758088i $$0.273866\pi$$
$$278$$ 0 0
$$279$$ −7.85398 −0.470205
$$280$$ 0 0
$$281$$ 17.8081i 1.06235i 0.847264 + 0.531173i $$0.178248\pi$$
−0.847264 + 0.531173i $$0.821752\pi$$
$$282$$ 0 0
$$283$$ 14.8881i 0.885007i 0.896767 + 0.442504i $$0.145910\pi$$
−0.896767 + 0.442504i $$0.854090\pi$$
$$284$$ 0 0
$$285$$ −3.51298 2.58050i −0.208091 0.152856i
$$286$$ 0 0
$$287$$ −2.14602 −0.126676
$$288$$ 0 0
$$289$$ 3.36695 0.198056
$$290$$ 0 0
$$291$$ 13.0536i 0.765218i
$$292$$ 0 0
$$293$$ 17.2133i 1.00561i 0.864400 + 0.502804i $$0.167698\pi$$
−0.864400 + 0.502804i $$0.832302\pi$$
$$294$$ 0 0
$$295$$ −9.02595 −0.525511
$$296$$ 0 0
$$297$$ 5.82288i 0.337878i
$$298$$ 0 0
$$299$$ −18.0519 −1.04397
$$300$$ 0 0
$$301$$ 0.828025 0.0477265
$$302$$ 0 0
$$303$$ −13.7080 −0.787501
$$304$$ 0 0
$$305$$ 5.85398 0.335198
$$306$$ 0 0
$$307$$ 6.97405 0.398030 0.199015 0.979996i $$-0.436226\pi$$
0.199015 + 0.979996i $$0.436226\pi$$
$$308$$ 0 0
$$309$$ 5.02595 0.285916
$$310$$ 0 0
$$311$$ 9.83138i 0.557486i −0.960366 0.278743i $$-0.910082\pi$$
0.960366 0.278743i $$-0.0899178\pi$$
$$312$$ 0 0
$$313$$ −14.7109 −0.831509 −0.415755 0.909477i $$-0.636483\pi$$
−0.415755 + 0.909477i $$0.636483\pi$$
$$314$$ 0 0
$$315$$ 3.24237i 0.182687i
$$316$$ 0 0
$$317$$ 25.5325i 1.43405i −0.697049 0.717024i $$-0.745504\pi$$
0.697049 0.717024i $$-0.254496\pi$$
$$318$$ 0 0
$$319$$ −33.9059 −1.89836
$$320$$ 0 0
$$321$$ −17.3929 −0.970777
$$322$$ 0 0
$$323$$ 15.8540 + 11.6458i 0.882139 + 0.647987i
$$324$$ 0 0
$$325$$ 3.24237i 0.179854i
$$326$$ 0 0
$$327$$ 4.65025i 0.257160i
$$328$$ 0 0
$$329$$ 1.31800 0.0726636
$$330$$ 0 0
$$331$$ −3.53893 −0.194517 −0.0972585 0.995259i $$-0.531007\pi$$
−0.0972585 + 0.995259i $$0.531007\pi$$
$$332$$ 0 0
$$333$$ 0.255376i 0.0139945i
$$334$$ 0 0
$$335$$ −10.3670 −0.566407
$$336$$ 0 0
$$337$$ 29.8231i 1.62457i 0.583262 + 0.812284i $$0.301776\pi$$
−0.583262 + 0.812284i $$0.698224\pi$$
$$338$$ 0 0
$$339$$ 11.2393i 0.610433i
$$340$$ 0 0
$$341$$ 45.7327i 2.47657i
$$342$$ 0 0
$$343$$ 11.3062i 0.610479i
$$344$$ 0 0
$$345$$ 5.56750i 0.299744i
$$346$$ 0 0
$$347$$ 3.56472i 0.191364i 0.995412 + 0.0956820i $$0.0305032\pi$$
−0.995412 + 0.0956820i $$0.969497\pi$$
$$348$$ 0 0
$$349$$ 28.4189 1.52123 0.760613 0.649205i $$-0.224898\pi$$
0.760613 + 0.649205i $$0.224898\pi$$
$$350$$ 0 0
$$351$$ 3.24237i 0.173065i
$$352$$ 0 0
$$353$$ −12.0519 −0.641458 −0.320729 0.947171i $$-0.603928\pi$$
−0.320729 + 0.947171i $$0.603928\pi$$
$$354$$ 0 0
$$355$$ −9.02595 −0.479048
$$356$$ 0 0
$$357$$ 14.6327i 0.774447i
$$358$$ 0 0
$$359$$ 35.0884i 1.85189i 0.377654 + 0.925947i $$0.376731\pi$$
−0.377654 + 0.925947i $$0.623269\pi$$
$$360$$ 0 0
$$361$$ 5.68200 + 18.1305i 0.299053 + 0.954237i
$$362$$ 0 0
$$363$$ 22.9059 1.20225
$$364$$ 0 0
$$365$$ 5.68495 0.297564
$$366$$ 0 0
$$367$$ 32.1683i 1.67917i −0.543225 0.839587i $$-0.682797\pi$$
0.543225 0.839587i $$-0.317203\pi$$
$$368$$ 0 0
$$369$$ 0.661868i 0.0344555i
$$370$$ 0 0
$$371$$ 18.0519 0.937208
$$372$$ 0 0
$$373$$ 2.42939i 0.125789i 0.998020 + 0.0628945i $$0.0200332\pi$$
−0.998020 + 0.0628945i $$0.979967\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −18.8799 −0.972366
$$378$$ 0 0
$$379$$ −3.53893 −0.181783 −0.0908913 0.995861i $$-0.528972\pi$$
−0.0908913 + 0.995861i $$0.528972\pi$$
$$380$$ 0 0
$$381$$ 7.68495 0.393712
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 18.8799 0.962210
$$386$$ 0 0
$$387$$ 0.255376i 0.0129815i
$$388$$ 0 0
$$389$$ 21.0779 1.06869 0.534345 0.845267i $$-0.320558\pi$$
0.534345 + 0.845267i $$0.320558\pi$$
$$390$$ 0 0
$$391$$ 25.1260i 1.27068i
$$392$$ 0 0
$$393$$ 1.00139i 0.0505135i
$$394$$ 0 0
$$395$$ −4.65900 −0.234420
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 8.36695 11.3904i 0.418872 0.570232i
$$400$$ 0 0
$$401$$ 14.1421i 0.706223i −0.935581 0.353112i $$-0.885124\pi$$
0.935581 0.353112i $$-0.114876\pi$$
$$402$$ 0 0
$$403$$ 25.4655i 1.26853i
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 1.48702 0.0737090
$$408$$ 0 0
$$409$$ 18.6412i 0.921750i −0.887465 0.460875i $$-0.847536\pi$$
0.887465 0.460875i $$-0.152464\pi$$
$$410$$ 0 0
$$411$$ 3.68495 0.181765
$$412$$ 0 0
$$413$$ 29.2655i 1.44006i
$$414$$ 0 0
$$415$$ 6.07825i 0.298370i
$$416$$ 0 0
$$417$$ 1.32374i 0.0648237i
$$418$$ 0 0
$$419$$ 6.63586i 0.324183i −0.986776 0.162091i $$-0.948176\pi$$
0.986776 0.162091i $$-0.0518240\pi$$
$$420$$ 0 0
$$421$$ 12.1565i 0.592472i −0.955115 0.296236i $$-0.904269\pi$$
0.955115 0.296236i $$-0.0957314\pi$$
$$422$$ 0 0
$$423$$ 0.406492i 0.0197643i
$$424$$ 0 0
$$425$$ −4.51298 −0.218911
$$426$$ 0 0
$$427$$ 18.9808i 0.918544i
$$428$$ 0 0
$$429$$ 18.8799 0.911532
$$430$$ 0 0
$$431$$ 9.02595 0.434765 0.217382 0.976087i $$-0.430248\pi$$
0.217382 + 0.976087i $$0.430248\pi$$
$$432$$ 0 0
$$433$$ 14.8881i 0.715478i 0.933822 + 0.357739i $$0.116452\pi$$
−0.933822 + 0.357739i $$0.883548\pi$$
$$434$$ 0 0
$$435$$ 5.82288i 0.279186i
$$436$$ 0 0
$$437$$ −14.3670 + 19.5585i −0.687265 + 0.935610i
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −3.51298 −0.167285
$$442$$ 0 0
$$443$$ 34.0200i 1.61634i 0.588950 + 0.808170i $$0.299542\pi$$
−0.588950 + 0.808170i $$0.700458\pi$$
$$444$$ 0 0
$$445$$ 3.64886i 0.172973i
$$446$$ 0 0
$$447$$ −15.0260 −0.710703
$$448$$ 0 0
$$449$$ 30.6064i 1.44441i −0.691681 0.722203i $$-0.743130\pi$$
0.691681 0.722203i $$-0.256870\pi$$
$$450$$ 0 0
$$451$$ −3.85398 −0.181477
$$452$$ 0 0
$$453$$ −22.2209 −1.04403
$$454$$ 0 0
$$455$$ 10.5130 0.492856
$$456$$ 0 0
$$457$$ 12.3670 0.578502 0.289251 0.957253i $$-0.406594\pi$$
0.289251 + 0.957253i $$0.406594\pi$$
$$458$$ 0 0
$$459$$ 4.51298 0.210648
$$460$$ 0 0
$$461$$ 33.4159 1.55633 0.778167 0.628057i $$-0.216150\pi$$
0.778167 + 0.628057i $$0.216150\pi$$
$$462$$ 0 0
$$463$$ 17.0621i 0.792945i 0.918047 + 0.396472i $$0.129766\pi$$
−0.918047 + 0.396472i $$0.870234\pi$$
$$464$$ 0 0
$$465$$ −7.85398 −0.364219
$$466$$ 0 0
$$467$$ 27.0245i 1.25055i −0.780406 0.625273i $$-0.784988\pi$$
0.780406 0.625273i $$-0.215012\pi$$
$$468$$ 0 0
$$469$$ 33.6135i 1.55213i
$$470$$ 0 0
$$471$$ 4.36695 0.201219
$$472$$ 0 0
$$473$$ 1.48702 0.0683734
$$474$$ 0 0
$$475$$ −3.51298 2.58050i −0.161186 0.118402i
$$476$$ 0 0
$$477$$ 5.56750i 0.254918i
$$478$$ 0 0
$$479$$ 21.7794i 0.995124i −0.867428 0.497562i $$-0.834229\pi$$
0.867428 0.497562i $$-0.165771\pi$$
$$480$$ 0 0
$$481$$ 0.828025 0.0377547
$$482$$ 0 0
$$483$$ 18.0519 0.821390
$$484$$ 0 0
$$485$$ 13.0536i 0.592735i
$$486$$ 0 0
$$487$$ −15.3929 −0.697519 −0.348760 0.937212i $$-0.613397\pi$$
−0.348760 + 0.937212i $$0.613397\pi$$
$$488$$ 0 0
$$489$$ 6.56889i 0.297055i
$$490$$ 0 0
$$491$$ 17.4686i 0.788348i −0.919036 0.394174i $$-0.871031\pi$$
0.919036 0.394174i $$-0.128969\pi$$
$$492$$ 0 0
$$493$$ 26.2785i 1.18352i
$$494$$ 0 0
$$495$$ 5.82288i 0.261719i
$$496$$ 0 0
$$497$$ 29.2655i 1.31274i
$$498$$ 0 0
$$499$$ 5.67176i 0.253903i −0.991909 0.126951i $$-0.959481\pi$$
0.991909 0.126951i $$-0.0405192\pi$$
$$500$$ 0 0
$$501$$ 1.48702 0.0664353
$$502$$ 0 0
$$503$$ 18.0262i 0.803750i 0.915695 + 0.401875i $$0.131641\pi$$
−0.915695 + 0.401875i $$0.868359\pi$$
$$504$$ 0 0
$$505$$ −13.7080 −0.609996
$$506$$ 0 0
$$507$$ −2.48702 −0.110453
$$508$$ 0 0
$$509$$ 14.3104i 0.634297i −0.948376 0.317149i $$-0.897275\pi$$
0.948376 0.317149i $$-0.102725\pi$$
$$510$$ 0 0
$$511$$ 18.4327i 0.815416i
$$512$$ 0 0
$$513$$ 3.51298 + 2.58050i 0.155102 + 0.113932i
$$514$$ 0 0
$$515$$ 5.02595 0.221470
$$516$$ 0 0
$$517$$ 2.36695 0.104098
$$518$$ 0 0
$$519$$ 16.4003i 0.719892i
$$520$$ 0 0
$$521$$ 6.29634i 0.275848i −0.990443 0.137924i $$-0.955957\pi$$
0.990443 0.137924i $$-0.0440429\pi$$
$$522$$ 0 0
$$523$$ −34.7628 −1.52007 −0.760036 0.649881i $$-0.774819\pi$$
−0.760036 + 0.649881i $$0.774819\pi$$
$$524$$ 0 0
$$525$$ 3.24237i 0.141509i
$$526$$ 0 0
$$527$$ 35.4448 1.54400
$$528$$ 0 0
$$529$$ −7.99705 −0.347698
$$530$$ 0 0
$$531$$ 9.02595 0.391693
$$532$$ 0 0
$$533$$ −2.14602 −0.0929546
$$534$$ 0 0
$$535$$ −17.3929 −0.751961
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 20.4556i 0.881086i
$$540$$ 0 0
$$541$$ −35.4159 −1.52265 −0.761324 0.648371i $$-0.775451\pi$$
−0.761324 + 0.648371i $$0.775451\pi$$
$$542$$ 0 0
$$543$$ 24.6152i 1.05634i
$$544$$ 0 0
$$545$$ 4.65025i 0.199195i
$$546$$ 0 0
$$547$$ −15.7080 −0.671624 −0.335812 0.941929i $$-0.609011\pi$$
−0.335812 + 0.941929i $$0.609011\pi$$
$$548$$ 0 0
$$549$$ −5.85398 −0.249842
$$550$$ 0 0
$$551$$ −15.0260 + 20.4556i −0.640127 + 0.871439i
$$552$$ 0 0
$$553$$ 15.1062i 0.642381i
$$554$$ 0 0
$$555$$ 0.255376i 0.0108401i
$$556$$ 0 0
$$557$$ −6.99705 −0.296475 −0.148237 0.988952i $$-0.547360\pi$$
−0.148237 + 0.988952i $$0.547360\pi$$
$$558$$ 0 0
$$559$$ 0.828025 0.0350217
$$560$$ 0 0
$$561$$ 26.2785i 1.10948i
$$562$$ 0 0
$$563$$ −9.19498 −0.387522 −0.193761 0.981049i $$-0.562069\pi$$
−0.193761 + 0.981049i $$0.562069\pi$$
$$564$$ 0 0
$$565$$ 11.2393i 0.472839i
$$566$$ 0 0
$$567$$ 3.24237i 0.136167i
$$568$$ 0 0
$$569$$ 12.6099i 0.528632i 0.964436 + 0.264316i $$0.0851463\pi$$
−0.964436 + 0.264316i $$0.914854\pi$$
$$570$$ 0 0
$$571$$ 32.4237i 1.35689i −0.734651 0.678445i $$-0.762654\pi$$
0.734651 0.678445i $$-0.237346\pi$$
$$572$$ 0 0
$$573$$ 1.17262i 0.0489869i
$$574$$ 0 0
$$575$$ 5.56750i 0.232181i
$$576$$ 0 0
$$577$$ 21.3929 0.890598 0.445299 0.895382i $$-0.353097\pi$$
0.445299 + 0.895382i $$0.353097\pi$$
$$578$$ 0 0
$$579$$ 17.5356i 0.728755i
$$580$$ 0 0
$$581$$ −19.7080 −0.817624
$$582$$ 0 0
$$583$$ 32.4189 1.34265
$$584$$ 0 0
$$585$$ 3.24237i 0.134056i
$$586$$ 0 0
$$587$$ 37.6890i 1.55559i −0.628517 0.777795i $$-0.716338\pi$$
0.628517 0.777795i $$-0.283662\pi$$
$$588$$ 0 0
$$589$$ 27.5908 + 20.2672i 1.13686 + 0.835096i
$$590$$ 0 0
$$591$$ −7.48702 −0.307975
$$592$$ 0 0
$$593$$ −32.4189 −1.33128 −0.665641 0.746272i $$-0.731842\pi$$
−0.665641 + 0.746272i $$0.731842\pi$$
$$594$$ 0 0
$$595$$ 14.6327i 0.599884i
$$596$$ 0 0
$$597$$ 0.510752i 0.0209037i
$$598$$ 0 0
$$599$$ −4.29205 −0.175368 −0.0876841 0.996148i $$-0.527947\pi$$
−0.0876841 + 0.996148i $$0.527947\pi$$
$$600$$ 0 0
$$601$$ 40.0983i 1.63564i 0.575473 + 0.817821i $$0.304818\pi$$
−0.575473 + 0.817821i $$0.695182\pi$$
$$602$$ 0 0
$$603$$ 10.3670 0.422175
$$604$$ 0 0
$$605$$ 22.9059 0.931256
$$606$$ 0 0
$$607$$ −25.7369 −1.04463 −0.522313 0.852754i $$-0.674931\pi$$
−0.522313 + 0.852754i $$0.674931\pi$$
$$608$$ 0 0
$$609$$ 18.8799 0.765053
$$610$$ 0 0
$$611$$ 1.31800 0.0533205
$$612$$ 0 0
$$613$$ 32.0749 1.29549 0.647747 0.761856i $$-0.275712\pi$$
0.647747 + 0.761856i $$0.275712\pi$$
$$614$$ 0 0
$$615$$ 0.661868i 0.0266891i
$$616$$ 0 0
$$617$$ 13.5389 0.545057 0.272528 0.962148i $$-0.412140\pi$$
0.272528 + 0.962148i $$0.412140\pi$$
$$618$$ 0 0
$$619$$ 21.2887i 0.855666i −0.903858 0.427833i $$-0.859277\pi$$
0.903858 0.427833i $$-0.140723\pi$$
$$620$$ 0 0
$$621$$ 5.56750i 0.223416i
$$622$$ 0 0
$$623$$ −11.8310 −0.473998
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 15.0260 20.4556i 0.600079 0.816919i
$$628$$ 0 0
$$629$$ 1.15251i 0.0459535i
$$630$$ 0 0
$$631$$ 40.4005i 1.60832i 0.594414 + 0.804159i $$0.297384\pi$$
−0.594414 + 0.804159i $$0.702616\pi$$
$$632$$ 0 0
$$633$$ 18.3670 0.730021
$$634$$ 0 0
$$635$$ 7.68495 0.304968
$$636$$ 0 0
$$637$$ 11.3904i 0.451303i
$$638$$ 0 0
$$639$$ 9.02595 0.357061
$$640$$ 0 0
$$641$$ 18.4528i 0.728843i 0.931234 + 0.364422i $$0.118733\pi$$
−0.931234 + 0.364422i $$0.881267\pi$$
$$642$$ 0 0
$$643$$ 12.7141i 0.501396i 0.968065 + 0.250698i $$0.0806600\pi$$
−0.968065 + 0.250698i $$0.919340\pi$$
$$644$$ 0 0
$$645$$ 0.255376i 0.0100554i
$$646$$ 0 0
$$647$$ 8.72573i 0.343044i 0.985180 + 0.171522i $$0.0548684\pi$$
−0.985180 + 0.171522i $$0.945132\pi$$
$$648$$ 0 0
$$649$$ 52.5570i 2.06304i
$$650$$ 0 0
$$651$$ 25.4655i 0.998072i
$$652$$ 0 0
$$653$$ −11.8829 −0.465013 −0.232506 0.972595i $$-0.574693\pi$$
−0.232506 + 0.972595i $$0.574693\pi$$
$$654$$ 0 0
$$655$$ 1.00139i 0.0391276i
$$656$$ 0 0
$$657$$ −5.68495 −0.221791
$$658$$ 0 0
$$659$$ −13.3180 −0.518796 −0.259398 0.965771i $$-0.583524\pi$$
−0.259398 + 0.965771i $$0.583524\pi$$
$$660$$ 0 0
$$661$$ 15.7853i 0.613975i −0.951714 0.306988i $$-0.900679\pi$$
0.951714 0.306988i $$-0.0993210\pi$$
$$662$$ 0 0
$$663$$ 14.6327i 0.568289i
$$664$$ 0 0
$$665$$ 8.36695 11.3904i 0.324457 0.441700i
$$666$$ 0 0
$$667$$ −32.4189 −1.25526
$$668$$ 0 0
$$669$$ 23.0779 0.892241
$$670$$ 0 0
$$671$$ 34.0870i 1.31591i
$$672$$ 0 0
$$673$$ 45.1378i 1.73994i 0.493109 + 0.869968i $$0.335861\pi$$
−0.493109 + 0.869968i $$0.664139\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 4.54599i 0.174717i 0.996177 + 0.0873584i $$0.0278425\pi$$
−0.996177 + 0.0873584i $$0.972157\pi$$
$$678$$ 0 0
$$679$$ 42.3247 1.62427
$$680$$ 0 0
$$681$$ −8.85693 −0.339398
$$682$$ 0 0
$$683$$ 26.7569 1.02382 0.511912 0.859038i $$-0.328937\pi$$
0.511912 + 0.859038i $$0.328937\pi$$
$$684$$ 0 0
$$685$$ 3.68495 0.140795
$$686$$ 0 0
$$687$$ 20.5130 0.782619
$$688$$ 0 0
$$689$$ 18.0519 0.687723
$$690$$ 0 0
$$691$$ 11.4372i 0.435093i −0.976050 0.217546i $$-0.930195\pi$$
0.976050 0.217546i $$-0.0698053\pi$$
$$692$$ 0 0
$$693$$ −18.8799 −0.717189
$$694$$ 0 0
$$695$$ 1.32374i 0.0502122i
$$696$$ 0 0
$$697$$ 2.98700i 0.113141i
$$698$$ 0 0
$$699$$ 28.7858 1.08878
$$700$$ 0 0
$$701$$ 21.0779 0.796100 0.398050 0.917364i $$-0.369687\pi$$
0.398050 + 0.917364i $$0.369687\pi$$
$$702$$ 0 0
$$703$$ 0.658999 0.897131i 0.0248546 0.0338359i
$$704$$ 0 0
$$705$$ 0.406492i 0.0153094i
$$706$$ 0 0
$$707$$ 44.4463i 1.67157i
$$708$$ 0 0
$$709$$ 37.9578 1.42553 0.712767 0.701401i $$-0.247442\pi$$
0.712767 + 0.701401i $$0.247442\pi$$
$$710$$ 0 0
$$711$$ 4.65900 0.174726
$$712$$ 0 0
$$713$$ 43.7270i 1.63759i
$$714$$ 0 0
$$715$$ 18.8799 0.706069
$$716$$ 0 0
$$717$$ 25.6166i 0.956671i
$$718$$ 0 0
$$719$$ 17.9421i 0.669127i −0.942373 0.334563i $$-0.891411\pi$$
0.942373 0.334563i $$-0.108589\pi$$
$$720$$ 0 0
$$721$$ 16.2960i 0.606895i
$$722$$ 0 0
$$723$$ 22.4785i 0.835985i
$$724$$ 0 0
$$725$$ 5.82288i 0.216256i
$$726$$ 0 0
$$727$$ 7.89263i 0.292721i −0.989231 0.146361i $$-0.953244\pi$$
0.989231 0.146361i $$-0.0467560\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.15251i 0.0426270i
$$732$$ 0 0
$$733$$ 25.6850 0.948695 0.474348 0.880338i $$-0.342684\pi$$
0.474348 + 0.880338i $$0.342684\pi$$
$$734$$ 0 0
$$735$$ −3.51298 −0.129578
$$736$$ 0 0
$$737$$ 60.3655i 2.22359i
$$738$$ 0 0
$$739$$ 16.5045i 0.607129i 0.952811 + 0.303564i $$0.0981768\pi$$
−0.952811 + 0.303564i $$0.901823\pi$$
$$740$$ 0 0
$$741$$ 8.36695 11.3904i 0.307368 0.418436i
$$742$$ 0 0
$$743$$ −34.6820 −1.27236 −0.636180 0.771541i $$-0.719486\pi$$
−0.636180 + 0.771541i $$0.719486\pi$$
$$744$$ 0 0
$$745$$ −15.0260 −0.550508
$$746$$ 0 0
$$747$$ 6.07825i 0.222392i
$$748$$ 0 0
$$749$$ 56.3943i 2.06060i
$$750$$ 0 0
$$751$$ 40.2728 1.46958 0.734788 0.678297i $$-0.237282\pi$$
0.734788 + 0.678297i $$0.237282\pi$$
$$752$$ 0 0
$$753$$ 15.8054i 0.575980i
$$754$$ 0 0
$$755$$ −22.2209 −0.808702
$$756$$ 0 0
$$757$$ −29.7080 −1.07975 −0.539877 0.841744i $$-0.681529\pi$$
−0.539877 + 0.841744i $$0.681529\pi$$
$$758$$ 0 0
$$759$$ 32.4189 1.17673
$$760$$ 0 0
$$761$$ −52.4478 −1.90123 −0.950615 0.310373i $$-0.899546\pi$$
−0.950615 + 0.310373i $$0.899546\pi$$
$$762$$ 0 0
$$763$$ −15.0779 −0.545855
$$764$$ 0 0
$$765$$ 4.51298 0.163167
$$766$$ 0 0
$$767$$ 29.2655i 1.05672i
$$768$$ 0 0
$$769$$ 38.7858 1.39865 0.699326 0.714803i $$-0.253484\pi$$
0.699326 + 0.714803i $$0.253484\pi$$
$$770$$ 0 0
$$771$$ 18.0262i 0.649199i
$$772$$ 0 0
$$773$$ 44.4760i 1.59969i −0.600207 0.799845i $$-0.704915\pi$$
0.600207 0.799845i $$-0.295085\pi$$
$$774$$ 0 0
$$775$$ −7.85398 −0.282123
$$776$$ 0 0
$$777$$ −0.828025 −0.0297052
$$778$$ 0 0
$$779$$ −1.70795 + 2.32513i −0.0611938 + 0.0833063i
$$780$$ 0 0
$$781$$ 52.5570i 1.88064i
$$782$$ 0 0
$$783$$ 5.82288i 0.208093i
$$784$$ 0 0
$$785$$ 4.36695 0.155863
$$786$$ 0 0
$$787$$ 13.3410 0.475555 0.237778 0.971320i $$-0.423581\pi$$
0.237778 + 0.971320i $$0.423581\pi$$
$$788$$ 0 0
$$789$$ 18.0262i 0.641751i
$$790$$ 0 0
$$791$$ 36.4419 1.29572
$$792$$ 0 0
$$793$$ 18.9808i 0.674027i
$$794$$ 0 0
$$795$$ 5.56750i 0.197459i
$$796$$ 0 0
$$797$$ 10.5602i 0.374062i 0.982354 + 0.187031i $$0.0598865\pi$$
−0.982354 + 0.187031i $$0.940114\pi$$
$$798$$ 0 0
$$799$$ 1.83449i 0.0648996i
$$800$$ 0 0
$$801$$ 3.64886i 0.128926i
$$802$$ 0 0
$$803$$ 33.1028i 1.16817i
$$804$$