# Properties

 Label 1014.2.b.e.337.2 Level $1014$ Weight $2$ Character 1014.337 Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.2 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.2.b.e.337.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -0.267949i q^{5} -1.00000i q^{6} +0.732051i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -0.267949i q^{5} -1.00000i q^{6} +0.732051i q^{7} +1.00000i q^{8} +1.00000 q^{9} -0.267949 q^{10} +4.73205i q^{11} -1.00000 q^{12} +0.732051 q^{14} -0.267949i q^{15} +1.00000 q^{16} +2.26795 q^{17} -1.00000i q^{18} -1.26795i q^{19} +0.267949i q^{20} +0.732051i q^{21} +4.73205 q^{22} +6.19615 q^{23} +1.00000i q^{24} +4.92820 q^{25} +1.00000 q^{27} -0.732051i q^{28} +2.46410 q^{29} -0.267949 q^{30} +5.46410i q^{31} -1.00000i q^{32} +4.73205i q^{33} -2.26795i q^{34} +0.196152 q^{35} -1.00000 q^{36} +10.4641i q^{37} -1.26795 q^{38} +0.267949 q^{40} -11.3923i q^{41} +0.732051 q^{42} -7.66025 q^{43} -4.73205i q^{44} -0.267949i q^{45} -6.19615i q^{46} -8.19615i q^{47} +1.00000 q^{48} +6.46410 q^{49} -4.92820i q^{50} +2.26795 q^{51} +0.464102 q^{53} -1.00000i q^{54} +1.26795 q^{55} -0.732051 q^{56} -1.26795i q^{57} -2.46410i q^{58} +8.00000i q^{59} +0.267949i q^{60} +1.19615 q^{61} +5.46410 q^{62} +0.732051i q^{63} -1.00000 q^{64} +4.73205 q^{66} +11.1244i q^{67} -2.26795 q^{68} +6.19615 q^{69} -0.196152i q^{70} -1.26795i q^{71} +1.00000i q^{72} -9.73205i q^{73} +10.4641 q^{74} +4.92820 q^{75} +1.26795i q^{76} -3.46410 q^{77} -9.46410 q^{79} -0.267949i q^{80} +1.00000 q^{81} -11.3923 q^{82} -10.1962i q^{83} -0.732051i q^{84} -0.607695i q^{85} +7.66025i q^{86} +2.46410 q^{87} -4.73205 q^{88} -2.53590i q^{89} -0.267949 q^{90} -6.19615 q^{92} +5.46410i q^{93} -8.19615 q^{94} -0.339746 q^{95} -1.00000i q^{96} +6.00000i q^{97} -6.46410i q^{98} +4.73205i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 8 q^{10} - 4 q^{12} - 4 q^{14} + 4 q^{16} + 16 q^{17} + 12 q^{22} + 4 q^{23} - 8 q^{25} + 4 q^{27} - 4 q^{29} - 8 q^{30} - 20 q^{35} - 4 q^{36} - 12 q^{38} + 8 q^{40} - 4 q^{42} + 4 q^{43} + 4 q^{48} + 12 q^{49} + 16 q^{51} - 12 q^{53} + 12 q^{55} + 4 q^{56} - 16 q^{61} + 8 q^{62} - 4 q^{64} + 12 q^{66} - 16 q^{68} + 4 q^{69} + 28 q^{74} - 8 q^{75} - 24 q^{79} + 4 q^{81} - 4 q^{82} - 4 q^{87} - 12 q^{88} - 8 q^{90} - 4 q^{92} - 12 q^{94} - 36 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 8 * q^10 - 4 * q^12 - 4 * q^14 + 4 * q^16 + 16 * q^17 + 12 * q^22 + 4 * q^23 - 8 * q^25 + 4 * q^27 - 4 * q^29 - 8 * q^30 - 20 * q^35 - 4 * q^36 - 12 * q^38 + 8 * q^40 - 4 * q^42 + 4 * q^43 + 4 * q^48 + 12 * q^49 + 16 * q^51 - 12 * q^53 + 12 * q^55 + 4 * q^56 - 16 * q^61 + 8 * q^62 - 4 * q^64 + 12 * q^66 - 16 * q^68 + 4 * q^69 + 28 * q^74 - 8 * q^75 - 24 * q^79 + 4 * q^81 - 4 * q^82 - 4 * q^87 - 12 * q^88 - 8 * q^90 - 4 * q^92 - 12 * q^94 - 36 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 0.267949i − 0.119831i −0.998203 0.0599153i $$-0.980917\pi$$
0.998203 0.0599153i $$-0.0190830\pi$$
$$6$$ − 1.00000i − 0.408248i
$$7$$ 0.732051i 0.276689i 0.990384 + 0.138345i $$0.0441781\pi$$
−0.990384 + 0.138345i $$0.955822\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −0.267949 −0.0847330
$$11$$ 4.73205i 1.42677i 0.700774 + 0.713384i $$0.252838\pi$$
−0.700774 + 0.713384i $$0.747162\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 0.732051 0.195649
$$15$$ − 0.267949i − 0.0691842i
$$16$$ 1.00000 0.250000
$$17$$ 2.26795 0.550058 0.275029 0.961436i $$-0.411312\pi$$
0.275029 + 0.961436i $$0.411312\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ − 1.26795i − 0.290887i −0.989367 0.145444i $$-0.953539\pi$$
0.989367 0.145444i $$-0.0464610\pi$$
$$20$$ 0.267949i 0.0599153i
$$21$$ 0.732051i 0.159747i
$$22$$ 4.73205 1.00888
$$23$$ 6.19615 1.29199 0.645994 0.763343i $$-0.276443\pi$$
0.645994 + 0.763343i $$0.276443\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ 4.92820 0.985641
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 0.732051i − 0.138345i
$$29$$ 2.46410 0.457572 0.228786 0.973477i $$-0.426524\pi$$
0.228786 + 0.973477i $$0.426524\pi$$
$$30$$ −0.267949 −0.0489206
$$31$$ 5.46410i 0.981382i 0.871334 + 0.490691i $$0.163256\pi$$
−0.871334 + 0.490691i $$0.836744\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.73205i 0.823744i
$$34$$ − 2.26795i − 0.388950i
$$35$$ 0.196152 0.0331558
$$36$$ −1.00000 −0.166667
$$37$$ 10.4641i 1.72029i 0.510052 + 0.860144i $$0.329626\pi$$
−0.510052 + 0.860144i $$0.670374\pi$$
$$38$$ −1.26795 −0.205689
$$39$$ 0 0
$$40$$ 0.267949 0.0423665
$$41$$ − 11.3923i − 1.77918i −0.456761 0.889590i $$-0.650990\pi$$
0.456761 0.889590i $$-0.349010\pi$$
$$42$$ 0.732051 0.112958
$$43$$ −7.66025 −1.16818 −0.584089 0.811690i $$-0.698548\pi$$
−0.584089 + 0.811690i $$0.698548\pi$$
$$44$$ − 4.73205i − 0.713384i
$$45$$ − 0.267949i − 0.0399435i
$$46$$ − 6.19615i − 0.913573i
$$47$$ − 8.19615i − 1.19553i −0.801671 0.597766i $$-0.796055\pi$$
0.801671 0.597766i $$-0.203945\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 6.46410 0.923443
$$50$$ − 4.92820i − 0.696953i
$$51$$ 2.26795 0.317576
$$52$$ 0 0
$$53$$ 0.464102 0.0637493 0.0318746 0.999492i $$-0.489852\pi$$
0.0318746 + 0.999492i $$0.489852\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 1.26795 0.170970
$$56$$ −0.732051 −0.0978244
$$57$$ − 1.26795i − 0.167944i
$$58$$ − 2.46410i − 0.323552i
$$59$$ 8.00000i 1.04151i 0.853706 + 0.520756i $$0.174350\pi$$
−0.853706 + 0.520756i $$0.825650\pi$$
$$60$$ 0.267949i 0.0345921i
$$61$$ 1.19615 0.153152 0.0765758 0.997064i $$-0.475601\pi$$
0.0765758 + 0.997064i $$0.475601\pi$$
$$62$$ 5.46410 0.693942
$$63$$ 0.732051i 0.0922297i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.73205 0.582475
$$67$$ 11.1244i 1.35906i 0.733649 + 0.679528i $$0.237815\pi$$
−0.733649 + 0.679528i $$0.762185\pi$$
$$68$$ −2.26795 −0.275029
$$69$$ 6.19615 0.745929
$$70$$ − 0.196152i − 0.0234447i
$$71$$ − 1.26795i − 0.150478i −0.997166 0.0752389i $$-0.976028\pi$$
0.997166 0.0752389i $$-0.0239720\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 9.73205i − 1.13905i −0.821974 0.569525i $$-0.807127\pi$$
0.821974 0.569525i $$-0.192873\pi$$
$$74$$ 10.4641 1.21643
$$75$$ 4.92820 0.569060
$$76$$ 1.26795i 0.145444i
$$77$$ −3.46410 −0.394771
$$78$$ 0 0
$$79$$ −9.46410 −1.06479 −0.532397 0.846495i $$-0.678709\pi$$
−0.532397 + 0.846495i $$0.678709\pi$$
$$80$$ − 0.267949i − 0.0299576i
$$81$$ 1.00000 0.111111
$$82$$ −11.3923 −1.25807
$$83$$ − 10.1962i − 1.11917i −0.828772 0.559587i $$-0.810960\pi$$
0.828772 0.559587i $$-0.189040\pi$$
$$84$$ − 0.732051i − 0.0798733i
$$85$$ − 0.607695i − 0.0659138i
$$86$$ 7.66025i 0.826026i
$$87$$ 2.46410 0.264179
$$88$$ −4.73205 −0.504438
$$89$$ − 2.53590i − 0.268805i −0.990927 0.134402i $$-0.957089\pi$$
0.990927 0.134402i $$-0.0429115\pi$$
$$90$$ −0.267949 −0.0282443
$$91$$ 0 0
$$92$$ −6.19615 −0.645994
$$93$$ 5.46410i 0.566601i
$$94$$ −8.19615 −0.845369
$$95$$ −0.339746 −0.0348572
$$96$$ − 1.00000i − 0.102062i
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ − 6.46410i − 0.652973i
$$99$$ 4.73205i 0.475589i
$$100$$ −4.92820 −0.492820
$$101$$ 11.9282 1.18690 0.593450 0.804871i $$-0.297765\pi$$
0.593450 + 0.804871i $$0.297765\pi$$
$$102$$ − 2.26795i − 0.224560i
$$103$$ 18.7321 1.84572 0.922862 0.385131i $$-0.125844\pi$$
0.922862 + 0.385131i $$0.125844\pi$$
$$104$$ 0 0
$$105$$ 0.196152 0.0191425
$$106$$ − 0.464102i − 0.0450775i
$$107$$ −0.196152 −0.0189628 −0.00948139 0.999955i $$-0.503018\pi$$
−0.00948139 + 0.999955i $$0.503018\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 5.46410i 0.523366i 0.965154 + 0.261683i $$0.0842775\pi$$
−0.965154 + 0.261683i $$0.915723\pi$$
$$110$$ − 1.26795i − 0.120894i
$$111$$ 10.4641i 0.993209i
$$112$$ 0.732051i 0.0691723i
$$113$$ −18.6603 −1.75541 −0.877705 0.479202i $$-0.840926\pi$$
−0.877705 + 0.479202i $$0.840926\pi$$
$$114$$ −1.26795 −0.118754
$$115$$ − 1.66025i − 0.154819i
$$116$$ −2.46410 −0.228786
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 1.66025i 0.152195i
$$120$$ 0.267949 0.0244603
$$121$$ −11.3923 −1.03566
$$122$$ − 1.19615i − 0.108295i
$$123$$ − 11.3923i − 1.02721i
$$124$$ − 5.46410i − 0.490691i
$$125$$ − 2.66025i − 0.237940i
$$126$$ 0.732051 0.0652163
$$127$$ −17.8564 −1.58450 −0.792250 0.610197i $$-0.791090\pi$$
−0.792250 + 0.610197i $$0.791090\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −7.66025 −0.674448
$$130$$ 0 0
$$131$$ 13.4641 1.17636 0.588182 0.808729i $$-0.299844\pi$$
0.588182 + 0.808729i $$0.299844\pi$$
$$132$$ − 4.73205i − 0.411872i
$$133$$ 0.928203 0.0804854
$$134$$ 11.1244 0.960998
$$135$$ − 0.267949i − 0.0230614i
$$136$$ 2.26795i 0.194475i
$$137$$ − 1.92820i − 0.164738i −0.996602 0.0823688i $$-0.973751\pi$$
0.996602 0.0823688i $$-0.0262485\pi$$
$$138$$ − 6.19615i − 0.527452i
$$139$$ −9.85641 −0.836009 −0.418005 0.908445i $$-0.637270\pi$$
−0.418005 + 0.908445i $$0.637270\pi$$
$$140$$ −0.196152 −0.0165779
$$141$$ − 8.19615i − 0.690241i
$$142$$ −1.26795 −0.106404
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 0.660254i − 0.0548311i
$$146$$ −9.73205 −0.805430
$$147$$ 6.46410 0.533150
$$148$$ − 10.4641i − 0.860144i
$$149$$ − 2.80385i − 0.229700i −0.993383 0.114850i $$-0.963361\pi$$
0.993383 0.114850i $$-0.0366388\pi$$
$$150$$ − 4.92820i − 0.402386i
$$151$$ 3.26795i 0.265942i 0.991120 + 0.132971i $$0.0424517\pi$$
−0.991120 + 0.132971i $$0.957548\pi$$
$$152$$ 1.26795 0.102844
$$153$$ 2.26795 0.183353
$$154$$ 3.46410i 0.279145i
$$155$$ 1.46410 0.117599
$$156$$ 0 0
$$157$$ −23.5885 −1.88256 −0.941282 0.337622i $$-0.890378\pi$$
−0.941282 + 0.337622i $$0.890378\pi$$
$$158$$ 9.46410i 0.752923i
$$159$$ 0.464102 0.0368057
$$160$$ −0.267949 −0.0211832
$$161$$ 4.53590i 0.357479i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 6.53590i − 0.511931i −0.966686 0.255966i $$-0.917607\pi$$
0.966686 0.255966i $$-0.0823934\pi$$
$$164$$ 11.3923i 0.889590i
$$165$$ 1.26795 0.0987097
$$166$$ −10.1962 −0.791375
$$167$$ − 2.53590i − 0.196234i −0.995175 0.0981169i $$-0.968718\pi$$
0.995175 0.0981169i $$-0.0312819\pi$$
$$168$$ −0.732051 −0.0564789
$$169$$ 0 0
$$170$$ −0.607695 −0.0466081
$$171$$ − 1.26795i − 0.0969625i
$$172$$ 7.66025 0.584089
$$173$$ 16.3923 1.24628 0.623142 0.782109i $$-0.285856\pi$$
0.623142 + 0.782109i $$0.285856\pi$$
$$174$$ − 2.46410i − 0.186803i
$$175$$ 3.60770i 0.272716i
$$176$$ 4.73205i 0.356692i
$$177$$ 8.00000i 0.601317i
$$178$$ −2.53590 −0.190074
$$179$$ −22.0526 −1.64829 −0.824143 0.566382i $$-0.808343\pi$$
−0.824143 + 0.566382i $$0.808343\pi$$
$$180$$ 0.267949i 0.0199718i
$$181$$ −8.80385 −0.654385 −0.327192 0.944958i $$-0.606103\pi$$
−0.327192 + 0.944958i $$0.606103\pi$$
$$182$$ 0 0
$$183$$ 1.19615 0.0884221
$$184$$ 6.19615i 0.456786i
$$185$$ 2.80385 0.206143
$$186$$ 5.46410 0.400647
$$187$$ 10.7321i 0.784805i
$$188$$ 8.19615i 0.597766i
$$189$$ 0.732051i 0.0532489i
$$190$$ 0.339746i 0.0246478i
$$191$$ 6.92820 0.501307 0.250654 0.968077i $$-0.419354\pi$$
0.250654 + 0.968077i $$0.419354\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 8.26795i − 0.595140i −0.954700 0.297570i $$-0.903824\pi$$
0.954700 0.297570i $$-0.0961762\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −6.46410 −0.461722
$$197$$ − 9.85641i − 0.702240i −0.936330 0.351120i $$-0.885801\pi$$
0.936330 0.351120i $$-0.114199\pi$$
$$198$$ 4.73205 0.336292
$$199$$ 3.80385 0.269648 0.134824 0.990870i $$-0.456953\pi$$
0.134824 + 0.990870i $$0.456953\pi$$
$$200$$ 4.92820i 0.348477i
$$201$$ 11.1244i 0.784652i
$$202$$ − 11.9282i − 0.839265i
$$203$$ 1.80385i 0.126605i
$$204$$ −2.26795 −0.158788
$$205$$ −3.05256 −0.213200
$$206$$ − 18.7321i − 1.30512i
$$207$$ 6.19615 0.430662
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ − 0.196152i − 0.0135358i
$$211$$ −4.39230 −0.302379 −0.151189 0.988505i $$-0.548310\pi$$
−0.151189 + 0.988505i $$0.548310\pi$$
$$212$$ −0.464102 −0.0318746
$$213$$ − 1.26795i − 0.0868784i
$$214$$ 0.196152i 0.0134087i
$$215$$ 2.05256i 0.139983i
$$216$$ 1.00000i 0.0680414i
$$217$$ −4.00000 −0.271538
$$218$$ 5.46410 0.370076
$$219$$ − 9.73205i − 0.657631i
$$220$$ −1.26795 −0.0854851
$$221$$ 0 0
$$222$$ 10.4641 0.702305
$$223$$ 13.0718i 0.875352i 0.899133 + 0.437676i $$0.144198\pi$$
−0.899133 + 0.437676i $$0.855802\pi$$
$$224$$ 0.732051 0.0489122
$$225$$ 4.92820 0.328547
$$226$$ 18.6603i 1.24126i
$$227$$ 1.80385i 0.119726i 0.998207 + 0.0598628i $$0.0190663\pi$$
−0.998207 + 0.0598628i $$0.980934\pi$$
$$228$$ 1.26795i 0.0839720i
$$229$$ 15.8564i 1.04782i 0.851773 + 0.523910i $$0.175527\pi$$
−0.851773 + 0.523910i $$0.824473\pi$$
$$230$$ −1.66025 −0.109474
$$231$$ −3.46410 −0.227921
$$232$$ 2.46410i 0.161776i
$$233$$ −19.8564 −1.30084 −0.650418 0.759576i $$-0.725406\pi$$
−0.650418 + 0.759576i $$0.725406\pi$$
$$234$$ 0 0
$$235$$ −2.19615 −0.143261
$$236$$ − 8.00000i − 0.520756i
$$237$$ −9.46410 −0.614759
$$238$$ 1.66025 0.107618
$$239$$ 9.66025i 0.624870i 0.949939 + 0.312435i $$0.101145\pi$$
−0.949939 + 0.312435i $$0.898855\pi$$
$$240$$ − 0.267949i − 0.0172960i
$$241$$ − 17.5885i − 1.13297i −0.824071 0.566486i $$-0.808302\pi$$
0.824071 0.566486i $$-0.191698\pi$$
$$242$$ 11.3923i 0.732325i
$$243$$ 1.00000 0.0641500
$$244$$ −1.19615 −0.0765758
$$245$$ − 1.73205i − 0.110657i
$$246$$ −11.3923 −0.726347
$$247$$ 0 0
$$248$$ −5.46410 −0.346971
$$249$$ − 10.1962i − 0.646155i
$$250$$ −2.66025 −0.168249
$$251$$ −6.53590 −0.412542 −0.206271 0.978495i $$-0.566133\pi$$
−0.206271 + 0.978495i $$0.566133\pi$$
$$252$$ − 0.732051i − 0.0461149i
$$253$$ 29.3205i 1.84336i
$$254$$ 17.8564i 1.12041i
$$255$$ − 0.607695i − 0.0380553i
$$256$$ 1.00000 0.0625000
$$257$$ −26.6603 −1.66302 −0.831510 0.555509i $$-0.812523\pi$$
−0.831510 + 0.555509i $$0.812523\pi$$
$$258$$ 7.66025i 0.476907i
$$259$$ −7.66025 −0.475985
$$260$$ 0 0
$$261$$ 2.46410 0.152524
$$262$$ − 13.4641i − 0.831815i
$$263$$ 28.0526 1.72979 0.864897 0.501949i $$-0.167383\pi$$
0.864897 + 0.501949i $$0.167383\pi$$
$$264$$ −4.73205 −0.291238
$$265$$ − 0.124356i − 0.00763911i
$$266$$ − 0.928203i − 0.0569118i
$$267$$ − 2.53590i − 0.155194i
$$268$$ − 11.1244i − 0.679528i
$$269$$ −1.46410 −0.0892679 −0.0446339 0.999003i $$-0.514212\pi$$
−0.0446339 + 0.999003i $$0.514212\pi$$
$$270$$ −0.267949 −0.0163069
$$271$$ 5.85641i 0.355751i 0.984053 + 0.177876i $$0.0569225\pi$$
−0.984053 + 0.177876i $$0.943078\pi$$
$$272$$ 2.26795 0.137515
$$273$$ 0 0
$$274$$ −1.92820 −0.116487
$$275$$ 23.3205i 1.40628i
$$276$$ −6.19615 −0.372965
$$277$$ 2.26795 0.136268 0.0681339 0.997676i $$-0.478295\pi$$
0.0681339 + 0.997676i $$0.478295\pi$$
$$278$$ 9.85641i 0.591148i
$$279$$ 5.46410i 0.327127i
$$280$$ 0.196152i 0.0117223i
$$281$$ − 22.3205i − 1.33153i −0.746162 0.665765i $$-0.768105\pi$$
0.746162 0.665765i $$-0.231895\pi$$
$$282$$ −8.19615 −0.488074
$$283$$ −8.33975 −0.495746 −0.247873 0.968792i $$-0.579732\pi$$
−0.247873 + 0.968792i $$0.579732\pi$$
$$284$$ 1.26795i 0.0752389i
$$285$$ −0.339746 −0.0201248
$$286$$ 0 0
$$287$$ 8.33975 0.492280
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −11.8564 −0.697436
$$290$$ −0.660254 −0.0387715
$$291$$ 6.00000i 0.351726i
$$292$$ 9.73205i 0.569525i
$$293$$ − 14.5167i − 0.848072i −0.905645 0.424036i $$-0.860613\pi$$
0.905645 0.424036i $$-0.139387\pi$$
$$294$$ − 6.46410i − 0.376994i
$$295$$ 2.14359 0.124805
$$296$$ −10.4641 −0.608214
$$297$$ 4.73205i 0.274581i
$$298$$ −2.80385 −0.162423
$$299$$ 0 0
$$300$$ −4.92820 −0.284530
$$301$$ − 5.60770i − 0.323222i
$$302$$ 3.26795 0.188049
$$303$$ 11.9282 0.685257
$$304$$ − 1.26795i − 0.0727219i
$$305$$ − 0.320508i − 0.0183522i
$$306$$ − 2.26795i − 0.129650i
$$307$$ 8.58846i 0.490169i 0.969502 + 0.245085i $$0.0788157\pi$$
−0.969502 + 0.245085i $$0.921184\pi$$
$$308$$ 3.46410 0.197386
$$309$$ 18.7321 1.06563
$$310$$ − 1.46410i − 0.0831554i
$$311$$ −15.6603 −0.888012 −0.444006 0.896024i $$-0.646443\pi$$
−0.444006 + 0.896024i $$0.646443\pi$$
$$312$$ 0 0
$$313$$ 13.4641 0.761036 0.380518 0.924774i $$-0.375746\pi$$
0.380518 + 0.924774i $$0.375746\pi$$
$$314$$ 23.5885i 1.33117i
$$315$$ 0.196152 0.0110519
$$316$$ 9.46410 0.532397
$$317$$ − 3.33975i − 0.187579i −0.995592 0.0937894i $$-0.970102\pi$$
0.995592 0.0937894i $$-0.0298980\pi$$
$$318$$ − 0.464102i − 0.0260255i
$$319$$ 11.6603i 0.652849i
$$320$$ 0.267949i 0.0149788i
$$321$$ −0.196152 −0.0109482
$$322$$ 4.53590 0.252776
$$323$$ − 2.87564i − 0.160005i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −6.53590 −0.361990
$$327$$ 5.46410i 0.302166i
$$328$$ 11.3923 0.629035
$$329$$ 6.00000 0.330791
$$330$$ − 1.26795i − 0.0697983i
$$331$$ − 20.0000i − 1.09930i −0.835395 0.549650i $$-0.814761\pi$$
0.835395 0.549650i $$-0.185239\pi$$
$$332$$ 10.1962i 0.559587i
$$333$$ 10.4641i 0.573429i
$$334$$ −2.53590 −0.138758
$$335$$ 2.98076 0.162856
$$336$$ 0.732051i 0.0399366i
$$337$$ −6.85641 −0.373492 −0.186746 0.982408i $$-0.559794\pi$$
−0.186746 + 0.982408i $$0.559794\pi$$
$$338$$ 0 0
$$339$$ −18.6603 −1.01349
$$340$$ 0.607695i 0.0329569i
$$341$$ −25.8564 −1.40020
$$342$$ −1.26795 −0.0685628
$$343$$ 9.85641i 0.532196i
$$344$$ − 7.66025i − 0.413013i
$$345$$ − 1.66025i − 0.0893851i
$$346$$ − 16.3923i − 0.881256i
$$347$$ 8.87564 0.476470 0.238235 0.971208i $$-0.423431\pi$$
0.238235 + 0.971208i $$0.423431\pi$$
$$348$$ −2.46410 −0.132090
$$349$$ − 19.3205i − 1.03420i −0.855924 0.517102i $$-0.827011\pi$$
0.855924 0.517102i $$-0.172989\pi$$
$$350$$ 3.60770 0.192839
$$351$$ 0 0
$$352$$ 4.73205 0.252219
$$353$$ − 19.7846i − 1.05303i −0.850166 0.526514i $$-0.823499\pi$$
0.850166 0.526514i $$-0.176501\pi$$
$$354$$ 8.00000 0.425195
$$355$$ −0.339746 −0.0180318
$$356$$ 2.53590i 0.134402i
$$357$$ 1.66025i 0.0878700i
$$358$$ 22.0526i 1.16551i
$$359$$ 23.1244i 1.22046i 0.792226 + 0.610228i $$0.208922\pi$$
−0.792226 + 0.610228i $$0.791078\pi$$
$$360$$ 0.267949 0.0141222
$$361$$ 17.3923 0.915384
$$362$$ 8.80385i 0.462720i
$$363$$ −11.3923 −0.597941
$$364$$ 0 0
$$365$$ −2.60770 −0.136493
$$366$$ − 1.19615i − 0.0625239i
$$367$$ −14.7321 −0.769007 −0.384503 0.923124i $$-0.625627\pi$$
−0.384503 + 0.923124i $$0.625627\pi$$
$$368$$ 6.19615 0.322997
$$369$$ − 11.3923i − 0.593060i
$$370$$ − 2.80385i − 0.145765i
$$371$$ 0.339746i 0.0176387i
$$372$$ − 5.46410i − 0.283300i
$$373$$ −10.2679 −0.531654 −0.265827 0.964021i $$-0.585645\pi$$
−0.265827 + 0.964021i $$0.585645\pi$$
$$374$$ 10.7321 0.554941
$$375$$ − 2.66025i − 0.137375i
$$376$$ 8.19615 0.422684
$$377$$ 0 0
$$378$$ 0.732051 0.0376526
$$379$$ − 1.46410i − 0.0752058i −0.999293 0.0376029i $$-0.988028\pi$$
0.999293 0.0376029i $$-0.0119722\pi$$
$$380$$ 0.339746 0.0174286
$$381$$ −17.8564 −0.914811
$$382$$ − 6.92820i − 0.354478i
$$383$$ − 5.46410i − 0.279203i −0.990208 0.139601i $$-0.955418\pi$$
0.990208 0.139601i $$-0.0445821\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0.928203i 0.0473056i
$$386$$ −8.26795 −0.420828
$$387$$ −7.66025 −0.389393
$$388$$ − 6.00000i − 0.304604i
$$389$$ 29.7846 1.51014 0.755070 0.655644i $$-0.227603\pi$$
0.755070 + 0.655644i $$0.227603\pi$$
$$390$$ 0 0
$$391$$ 14.0526 0.710668
$$392$$ 6.46410i 0.326486i
$$393$$ 13.4641 0.679174
$$394$$ −9.85641 −0.496559
$$395$$ 2.53590i 0.127595i
$$396$$ − 4.73205i − 0.237795i
$$397$$ − 0.392305i − 0.0196892i −0.999952 0.00984461i $$-0.996866\pi$$
0.999952 0.00984461i $$-0.00313369\pi$$
$$398$$ − 3.80385i − 0.190670i
$$399$$ 0.928203 0.0464683
$$400$$ 4.92820 0.246410
$$401$$ 21.9282i 1.09504i 0.836792 + 0.547521i $$0.184428\pi$$
−0.836792 + 0.547521i $$0.815572\pi$$
$$402$$ 11.1244 0.554832
$$403$$ 0 0
$$404$$ −11.9282 −0.593450
$$405$$ − 0.267949i − 0.0133145i
$$406$$ 1.80385 0.0895235
$$407$$ −49.5167 −2.45445
$$408$$ 2.26795i 0.112280i
$$409$$ − 14.2679i − 0.705505i −0.935717 0.352752i $$-0.885246\pi$$
0.935717 0.352752i $$-0.114754\pi$$
$$410$$ 3.05256i 0.150755i
$$411$$ − 1.92820i − 0.0951113i
$$412$$ −18.7321 −0.922862
$$413$$ −5.85641 −0.288175
$$414$$ − 6.19615i − 0.304524i
$$415$$ −2.73205 −0.134111
$$416$$ 0 0
$$417$$ −9.85641 −0.482670
$$418$$ − 6.00000i − 0.293470i
$$419$$ −10.5359 −0.514712 −0.257356 0.966317i $$-0.582851\pi$$
−0.257356 + 0.966317i $$0.582851\pi$$
$$420$$ −0.196152 −0.00957126
$$421$$ − 32.7128i − 1.59432i −0.603765 0.797162i $$-0.706333\pi$$
0.603765 0.797162i $$-0.293667\pi$$
$$422$$ 4.39230i 0.213814i
$$423$$ − 8.19615i − 0.398511i
$$424$$ 0.464102i 0.0225388i
$$425$$ 11.1769 0.542160
$$426$$ −1.26795 −0.0614323
$$427$$ 0.875644i 0.0423754i
$$428$$ 0.196152 0.00948139
$$429$$ 0 0
$$430$$ 2.05256 0.0989832
$$431$$ 11.1244i 0.535841i 0.963441 + 0.267921i $$0.0863365\pi$$
−0.963441 + 0.267921i $$0.913663\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 14.8564 0.713953 0.356977 0.934113i $$-0.383808\pi$$
0.356977 + 0.934113i $$0.383808\pi$$
$$434$$ 4.00000i 0.192006i
$$435$$ − 0.660254i − 0.0316568i
$$436$$ − 5.46410i − 0.261683i
$$437$$ − 7.85641i − 0.375823i
$$438$$ −9.73205 −0.465015
$$439$$ 17.6603 0.842878 0.421439 0.906857i $$-0.361525\pi$$
0.421439 + 0.906857i $$0.361525\pi$$
$$440$$ 1.26795i 0.0604471i
$$441$$ 6.46410 0.307814
$$442$$ 0 0
$$443$$ 36.3923 1.72905 0.864525 0.502589i $$-0.167619\pi$$
0.864525 + 0.502589i $$0.167619\pi$$
$$444$$ − 10.4641i − 0.496604i
$$445$$ −0.679492 −0.0322110
$$446$$ 13.0718 0.618968
$$447$$ − 2.80385i − 0.132617i
$$448$$ − 0.732051i − 0.0345861i
$$449$$ − 23.3205i − 1.10056i −0.834979 0.550281i $$-0.814520\pi$$
0.834979 0.550281i $$-0.185480\pi$$
$$450$$ − 4.92820i − 0.232318i
$$451$$ 53.9090 2.53847
$$452$$ 18.6603 0.877705
$$453$$ 3.26795i 0.153542i
$$454$$ 1.80385 0.0846588
$$455$$ 0 0
$$456$$ 1.26795 0.0593772
$$457$$ − 18.6603i − 0.872890i −0.899731 0.436445i $$-0.856237\pi$$
0.899731 0.436445i $$-0.143763\pi$$
$$458$$ 15.8564 0.740921
$$459$$ 2.26795 0.105859
$$460$$ 1.66025i 0.0774097i
$$461$$ − 25.7321i − 1.19846i −0.800577 0.599231i $$-0.795473\pi$$
0.800577 0.599231i $$-0.204527\pi$$
$$462$$ 3.46410i 0.161165i
$$463$$ − 28.0526i − 1.30371i −0.758342 0.651856i $$-0.773990\pi$$
0.758342 0.651856i $$-0.226010\pi$$
$$464$$ 2.46410 0.114393
$$465$$ 1.46410 0.0678961
$$466$$ 19.8564i 0.919830i
$$467$$ 12.5885 0.582524 0.291262 0.956643i $$-0.405925\pi$$
0.291262 + 0.956643i $$0.405925\pi$$
$$468$$ 0 0
$$469$$ −8.14359 −0.376036
$$470$$ 2.19615i 0.101301i
$$471$$ −23.5885 −1.08690
$$472$$ −8.00000 −0.368230
$$473$$ − 36.2487i − 1.66672i
$$474$$ 9.46410i 0.434701i
$$475$$ − 6.24871i − 0.286711i
$$476$$ − 1.66025i − 0.0760976i
$$477$$ 0.464102 0.0212498
$$478$$ 9.66025 0.441850
$$479$$ 26.5359i 1.21246i 0.795291 + 0.606228i $$0.207318\pi$$
−0.795291 + 0.606228i $$0.792682\pi$$
$$480$$ −0.267949 −0.0122302
$$481$$ 0 0
$$482$$ −17.5885 −0.801132
$$483$$ 4.53590i 0.206391i
$$484$$ 11.3923 0.517832
$$485$$ 1.60770 0.0730017
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 21.1244i − 0.957236i −0.878023 0.478618i $$-0.841138\pi$$
0.878023 0.478618i $$-0.158862\pi$$
$$488$$ 1.19615i 0.0541473i
$$489$$ − 6.53590i − 0.295564i
$$490$$ −1.73205 −0.0782461
$$491$$ −5.26795 −0.237739 −0.118870 0.992910i $$-0.537927\pi$$
−0.118870 + 0.992910i $$0.537927\pi$$
$$492$$ 11.3923i 0.513605i
$$493$$ 5.58846 0.251691
$$494$$ 0 0
$$495$$ 1.26795 0.0569901
$$496$$ 5.46410i 0.245345i
$$497$$ 0.928203 0.0416356
$$498$$ −10.1962 −0.456901
$$499$$ − 32.0000i − 1.43252i −0.697835 0.716258i $$-0.745853\pi$$
0.697835 0.716258i $$-0.254147\pi$$
$$500$$ 2.66025i 0.118970i
$$501$$ − 2.53590i − 0.113296i
$$502$$ 6.53590i 0.291711i
$$503$$ −10.9808 −0.489608 −0.244804 0.969573i $$-0.578724\pi$$
−0.244804 + 0.969573i $$0.578724\pi$$
$$504$$ −0.732051 −0.0326081
$$505$$ − 3.19615i − 0.142227i
$$506$$ 29.3205 1.30346
$$507$$ 0 0
$$508$$ 17.8564 0.792250
$$509$$ − 10.2679i − 0.455119i −0.973764 0.227559i $$-0.926925\pi$$
0.973764 0.227559i $$-0.0730746\pi$$
$$510$$ −0.607695 −0.0269092
$$511$$ 7.12436 0.315163
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.26795i − 0.0559813i
$$514$$ 26.6603i 1.17593i
$$515$$ − 5.01924i − 0.221174i
$$516$$ 7.66025 0.337224
$$517$$ 38.7846 1.70575
$$518$$ 7.66025i 0.336572i
$$519$$ 16.3923 0.719542
$$520$$ 0 0
$$521$$ −17.4449 −0.764273 −0.382137 0.924106i $$-0.624812\pi$$
−0.382137 + 0.924106i $$0.624812\pi$$
$$522$$ − 2.46410i − 0.107851i
$$523$$ 36.4449 1.59362 0.796811 0.604228i $$-0.206518\pi$$
0.796811 + 0.604228i $$0.206518\pi$$
$$524$$ −13.4641 −0.588182
$$525$$ 3.60770i 0.157453i
$$526$$ − 28.0526i − 1.22315i
$$527$$ 12.3923i 0.539817i
$$528$$ 4.73205i 0.205936i
$$529$$ 15.3923 0.669231
$$530$$ −0.124356 −0.00540166
$$531$$ 8.00000i 0.347170i
$$532$$ −0.928203 −0.0402427
$$533$$ 0 0
$$534$$ −2.53590 −0.109739
$$535$$ 0.0525589i 0.00227232i
$$536$$ −11.1244 −0.480499
$$537$$ −22.0526 −0.951638
$$538$$ 1.46410i 0.0631219i
$$539$$ 30.5885i 1.31754i
$$540$$ 0.267949i 0.0115307i
$$541$$ 40.3205i 1.73351i 0.498731 + 0.866757i $$0.333800\pi$$
−0.498731 + 0.866757i $$0.666200\pi$$
$$542$$ 5.85641 0.251554
$$543$$ −8.80385 −0.377809
$$544$$ − 2.26795i − 0.0972375i
$$545$$ 1.46410 0.0627152
$$546$$ 0 0
$$547$$ 6.19615 0.264928 0.132464 0.991188i $$-0.457711\pi$$
0.132464 + 0.991188i $$0.457711\pi$$
$$548$$ 1.92820i 0.0823688i
$$549$$ 1.19615 0.0510505
$$550$$ 23.3205 0.994390
$$551$$ − 3.12436i − 0.133102i
$$552$$ 6.19615i 0.263726i
$$553$$ − 6.92820i − 0.294617i
$$554$$ − 2.26795i − 0.0963559i
$$555$$ 2.80385 0.119017
$$556$$ 9.85641 0.418005
$$557$$ − 30.3731i − 1.28695i −0.765468 0.643474i $$-0.777492\pi$$
0.765468 0.643474i $$-0.222508\pi$$
$$558$$ 5.46410 0.231314
$$559$$ 0 0
$$560$$ 0.196152 0.00828895
$$561$$ 10.7321i 0.453108i
$$562$$ −22.3205 −0.941534
$$563$$ −21.0718 −0.888070 −0.444035 0.896009i $$-0.646453\pi$$
−0.444035 + 0.896009i $$0.646453\pi$$
$$564$$ 8.19615i 0.345120i
$$565$$ 5.00000i 0.210352i
$$566$$ 8.33975i 0.350546i
$$567$$ 0.732051i 0.0307432i
$$568$$ 1.26795 0.0532020
$$569$$ 38.6410 1.61992 0.809958 0.586488i $$-0.199490\pi$$
0.809958 + 0.586488i $$0.199490\pi$$
$$570$$ 0.339746i 0.0142304i
$$571$$ 24.0526 1.00657 0.503284 0.864121i $$-0.332125\pi$$
0.503284 + 0.864121i $$0.332125\pi$$
$$572$$ 0 0
$$573$$ 6.92820 0.289430
$$574$$ − 8.33975i − 0.348094i
$$575$$ 30.5359 1.27343
$$576$$ −1.00000 −0.0416667
$$577$$ − 0.267949i − 0.0111549i −0.999984 0.00557744i $$-0.998225\pi$$
0.999984 0.00557744i $$-0.00177536\pi$$
$$578$$ 11.8564i 0.493161i
$$579$$ − 8.26795i − 0.343604i
$$580$$ 0.660254i 0.0274156i
$$581$$ 7.46410 0.309663
$$582$$ 6.00000 0.248708
$$583$$ 2.19615i 0.0909553i
$$584$$ 9.73205 0.402715
$$585$$ 0 0
$$586$$ −14.5167 −0.599678
$$587$$ − 16.0000i − 0.660391i −0.943913 0.330195i $$-0.892885\pi$$
0.943913 0.330195i $$-0.107115\pi$$
$$588$$ −6.46410 −0.266575
$$589$$ 6.92820 0.285472
$$590$$ − 2.14359i − 0.0882503i
$$591$$ − 9.85641i − 0.405438i
$$592$$ 10.4641i 0.430072i
$$593$$ − 36.8564i − 1.51351i −0.653698 0.756756i $$-0.726783\pi$$
0.653698 0.756756i $$-0.273217\pi$$
$$594$$ 4.73205 0.194158
$$595$$ 0.444864 0.0182376
$$596$$ 2.80385i 0.114850i
$$597$$ 3.80385 0.155681
$$598$$ 0 0
$$599$$ −9.46410 −0.386693 −0.193346 0.981131i $$-0.561934\pi$$
−0.193346 + 0.981131i $$0.561934\pi$$
$$600$$ 4.92820i 0.201193i
$$601$$ −5.92820 −0.241816 −0.120908 0.992664i $$-0.538581\pi$$
−0.120908 + 0.992664i $$0.538581\pi$$
$$602$$ −5.60770 −0.228553
$$603$$ 11.1244i 0.453019i
$$604$$ − 3.26795i − 0.132971i
$$605$$ 3.05256i 0.124104i
$$606$$ − 11.9282i − 0.484550i
$$607$$ 0.784610 0.0318463 0.0159232 0.999873i $$-0.494931\pi$$
0.0159232 + 0.999873i $$0.494931\pi$$
$$608$$ −1.26795 −0.0514221
$$609$$ 1.80385i 0.0730956i
$$610$$ −0.320508 −0.0129770
$$611$$ 0 0
$$612$$ −2.26795 −0.0916764
$$613$$ − 11.3923i − 0.460131i −0.973175 0.230065i $$-0.926106\pi$$
0.973175 0.230065i $$-0.0738940\pi$$
$$614$$ 8.58846 0.346602
$$615$$ −3.05256 −0.123091
$$616$$ − 3.46410i − 0.139573i
$$617$$ − 35.2487i − 1.41906i −0.704675 0.709530i $$-0.748907\pi$$
0.704675 0.709530i $$-0.251093\pi$$
$$618$$ − 18.7321i − 0.753514i
$$619$$ 10.5359i 0.423474i 0.977327 + 0.211737i $$0.0679119\pi$$
−0.977327 + 0.211737i $$0.932088\pi$$
$$620$$ −1.46410 −0.0587997
$$621$$ 6.19615 0.248643
$$622$$ 15.6603i 0.627919i
$$623$$ 1.85641 0.0743754
$$624$$ 0 0
$$625$$ 23.9282 0.957128
$$626$$ − 13.4641i − 0.538134i
$$627$$ 6.00000 0.239617
$$628$$ 23.5885 0.941282
$$629$$ 23.7321i 0.946259i
$$630$$ − 0.196152i − 0.00781490i
$$631$$ 47.7128i 1.89942i 0.313135 + 0.949709i $$0.398621\pi$$
−0.313135 + 0.949709i $$0.601379\pi$$
$$632$$ − 9.46410i − 0.376462i
$$633$$ −4.39230 −0.174578
$$634$$ −3.33975 −0.132638
$$635$$ 4.78461i 0.189871i
$$636$$ −0.464102 −0.0184028
$$637$$ 0 0
$$638$$ 11.6603 0.461634
$$639$$ − 1.26795i − 0.0501593i
$$640$$ 0.267949 0.0105916
$$641$$ −25.9808 −1.02618 −0.513089 0.858335i $$-0.671499\pi$$
−0.513089 + 0.858335i $$0.671499\pi$$
$$642$$ 0.196152i 0.00774152i
$$643$$ − 13.8564i − 0.546443i −0.961951 0.273222i $$-0.911911\pi$$
0.961951 0.273222i $$-0.0880892\pi$$
$$644$$ − 4.53590i − 0.178739i
$$645$$ 2.05256i 0.0808194i
$$646$$ −2.87564 −0.113141
$$647$$ −26.2487 −1.03194 −0.515972 0.856606i $$-0.672569\pi$$
−0.515972 + 0.856606i $$0.672569\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −37.8564 −1.48599
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ 6.53590i 0.255966i
$$653$$ −10.5359 −0.412302 −0.206151 0.978520i $$-0.566094\pi$$
−0.206151 + 0.978520i $$0.566094\pi$$
$$654$$ 5.46410 0.213663
$$655$$ − 3.60770i − 0.140964i
$$656$$ − 11.3923i − 0.444795i
$$657$$ − 9.73205i − 0.379683i
$$658$$ − 6.00000i − 0.233904i
$$659$$ 38.2487 1.48996 0.744979 0.667088i $$-0.232459\pi$$
0.744979 + 0.667088i $$0.232459\pi$$
$$660$$ −1.26795 −0.0493549
$$661$$ − 9.39230i − 0.365318i −0.983176 0.182659i $$-0.941530\pi$$
0.983176 0.182659i $$-0.0584705\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 10.1962 0.395687
$$665$$ − 0.248711i − 0.00964461i
$$666$$ 10.4641 0.405476
$$667$$ 15.2679 0.591177
$$668$$ 2.53590i 0.0981169i
$$669$$ 13.0718i 0.505385i
$$670$$ − 2.98076i − 0.115157i
$$671$$ 5.66025i 0.218512i
$$672$$ 0.732051 0.0282395
$$673$$ −14.0718 −0.542428 −0.271214 0.962519i $$-0.587425\pi$$
−0.271214 + 0.962519i $$0.587425\pi$$
$$674$$ 6.85641i 0.264099i
$$675$$ 4.92820 0.189687
$$676$$ 0 0
$$677$$ −38.5359 −1.48105 −0.740527 0.672026i $$-0.765424\pi$$
−0.740527 + 0.672026i $$0.765424\pi$$
$$678$$ 18.6603i 0.716643i
$$679$$ −4.39230 −0.168561
$$680$$ 0.607695 0.0233040
$$681$$ 1.80385i 0.0691236i
$$682$$ 25.8564i 0.990093i
$$683$$ − 37.8564i − 1.44854i −0.689519 0.724268i $$-0.742178\pi$$
0.689519 0.724268i $$-0.257822\pi$$
$$684$$ 1.26795i 0.0484812i
$$685$$ −0.516660 −0.0197406
$$686$$ 9.85641 0.376319
$$687$$ 15.8564i 0.604960i
$$688$$ −7.66025 −0.292044
$$689$$ 0 0
$$690$$ −1.66025 −0.0632048
$$691$$ − 26.3397i − 1.00201i −0.865444 0.501006i $$-0.832964\pi$$
0.865444 0.501006i $$-0.167036\pi$$
$$692$$ −16.3923 −0.623142
$$693$$ −3.46410 −0.131590
$$694$$ − 8.87564i − 0.336915i
$$695$$ 2.64102i 0.100179i
$$696$$ 2.46410i 0.0934015i
$$697$$ − 25.8372i − 0.978653i
$$698$$ −19.3205 −0.731292
$$699$$ −19.8564 −0.751038
$$700$$ − 3.60770i − 0.136358i
$$701$$ −31.3205 −1.18296 −0.591480 0.806320i $$-0.701456\pi$$
−0.591480 + 0.806320i $$0.701456\pi$$
$$702$$ 0 0
$$703$$ 13.2679 0.500410
$$704$$ − 4.73205i − 0.178346i
$$705$$ −2.19615 −0.0827119
$$706$$ −19.7846 −0.744604
$$707$$ 8.73205i 0.328403i
$$708$$ − 8.00000i − 0.300658i
$$709$$ 40.8564i 1.53439i 0.641411 + 0.767197i $$0.278349\pi$$
−0.641411 + 0.767197i $$0.721651\pi$$
$$710$$ 0.339746i 0.0127504i
$$711$$ −9.46410 −0.354932
$$712$$ 2.53590 0.0950368
$$713$$ 33.8564i 1.26793i
$$714$$ 1.66025 0.0621334
$$715$$ 0 0
$$716$$ 22.0526 0.824143
$$717$$ 9.66025i 0.360769i
$$718$$ 23.1244 0.862993
$$719$$ 22.5359 0.840447 0.420224 0.907421i $$-0.361952\pi$$
0.420224 + 0.907421i $$0.361952\pi$$
$$720$$ − 0.267949i − 0.00998588i
$$721$$ 13.7128i 0.510692i
$$722$$ − 17.3923i − 0.647275i
$$723$$ − 17.5885i − 0.654122i
$$724$$ 8.80385 0.327192
$$725$$ 12.1436 0.451002
$$726$$ 11.3923i 0.422808i
$$727$$ −20.9808 −0.778133 −0.389067 0.921210i $$-0.627202\pi$$
−0.389067 + 0.921210i $$0.627202\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.60770i 0.0965151i
$$731$$ −17.3731 −0.642566
$$732$$ −1.19615 −0.0442111
$$733$$ − 19.0000i − 0.701781i −0.936416 0.350891i $$-0.885879\pi$$
0.936416 0.350891i $$-0.114121\pi$$
$$734$$ 14.7321i 0.543770i
$$735$$ − 1.73205i − 0.0638877i
$$736$$ − 6.19615i − 0.228393i
$$737$$ −52.6410 −1.93906
$$738$$ −11.3923 −0.419357
$$739$$ − 10.9282i − 0.402000i −0.979591 0.201000i $$-0.935581\pi$$
0.979591 0.201000i $$-0.0644192\pi$$
$$740$$ −2.80385 −0.103071
$$741$$ 0 0
$$742$$ 0.339746 0.0124725
$$743$$ 27.6077i 1.01283i 0.862290 + 0.506414i $$0.169029\pi$$
−0.862290 + 0.506414i $$0.830971\pi$$
$$744$$ −5.46410 −0.200324
$$745$$ −0.751289 −0.0275251
$$746$$ 10.2679i 0.375936i
$$747$$ − 10.1962i − 0.373058i
$$748$$ − 10.7321i − 0.392403i
$$749$$ − 0.143594i − 0.00524679i
$$750$$ −2.66025 −0.0971387
$$751$$ −15.9090 −0.580526 −0.290263 0.956947i $$-0.593743\pi$$
−0.290263 + 0.956947i $$0.593743\pi$$
$$752$$ − 8.19615i − 0.298883i
$$753$$ −6.53590 −0.238181
$$754$$ 0 0
$$755$$ 0.875644 0.0318680
$$756$$ − 0.732051i − 0.0266244i
$$757$$ 7.07180 0.257029 0.128514 0.991708i $$-0.458979\pi$$
0.128514 + 0.991708i $$0.458979\pi$$
$$758$$ −1.46410 −0.0531786
$$759$$ 29.3205i 1.06427i
$$760$$ − 0.339746i − 0.0123239i
$$761$$ − 23.3205i − 0.845368i −0.906277 0.422684i $$-0.861088\pi$$
0.906277 0.422684i $$-0.138912\pi$$
$$762$$ 17.8564i 0.646869i
$$763$$ −4.00000 −0.144810
$$764$$ −6.92820 −0.250654
$$765$$ − 0.607695i − 0.0219713i
$$766$$ −5.46410 −0.197426
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 16.1436i 0.582153i 0.956700 + 0.291076i $$0.0940134\pi$$
−0.956700 + 0.291076i $$0.905987\pi$$
$$770$$ 0.928203 0.0334501
$$771$$ −26.6603 −0.960146
$$772$$ 8.26795i 0.297570i
$$773$$ 35.0718i 1.26144i 0.776009 + 0.630722i $$0.217241\pi$$
−0.776009 + 0.630722i $$0.782759\pi$$
$$774$$ 7.66025i 0.275342i
$$775$$ 26.9282i 0.967290i
$$776$$ −6.00000 −0.215387
$$777$$ −7.66025 −0.274810
$$778$$ − 29.7846i − 1.06783i
$$779$$ −14.4449 −0.517541
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ − 14.0526i − 0.502518i
$$783$$ 2.46410 0.0880598
$$784$$ 6.46410 0.230861
$$785$$ 6.32051i 0.225589i
$$786$$ − 13.4641i − 0.480249i
$$787$$ 39.3205i 1.40162i 0.713346 + 0.700812i $$0.247179\pi$$
−0.713346 + 0.700812i $$0.752821\pi$$
$$788$$ 9.85641i 0.351120i
$$789$$ 28.0526 0.998698
$$790$$ 2.53590 0.0902232
$$791$$ − 13.6603i − 0.485703i
$$792$$ −4.73205 −0.168146
$$793$$ 0 0
$$794$$ −0.392305 −0.0139224
$$795$$ − 0.124356i − 0.00441044i
$$796$$ −3.80385 −0.134824
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ − 0.928203i − 0.0328580i
$$799$$ − 18.5885i − 0.657612i
$$800$$ − 4.92820i − 0.174238i
$$801$$ − 2.53590i − 0.0896016i
$$802$$ 21.9282 0.774312
$$803$$ 46.0526 1.62516
$$804$$ − 11.1244i − 0.392326i
$$805$$ 1.21539 0.0428369
$$806$$ 0 0
$$807$$ −1.46410 −0.0515388
$$808$$ 11.9282i 0.419633i
$$809$$ −22.4115 −0.787948 −0.393974 0.919122i $$-0.628900\pi$$
−0.393974 + 0.919122i $$0.628900\pi$$
$$810$$ −0.267949 −0.00941477
$$811$$ 45.1769i 1.58638i 0.608977 + 0.793188i $$0.291580\pi$$
−0.608977 + 0.793188i $$0.708420\pi$$
$$812$$ − 1.80385i − 0.0633026i
$$813$$ 5.85641i 0.205393i
$$814$$ 49.5167i 1.73556i
$$815$$ −1.75129 −0.0613450
$$816$$ 2.26795 0.0793941
$$817$$ 9.71281i 0.339808i
$$818$$ −14.2679 −0.498867
$$819$$ 0 0
$$820$$ 3.05256 0.106600
$$821$$ − 12.9282i − 0.451197i −0.974220 0.225599i $$-0.927566\pi$$
0.974220 0.225599i $$-0.0724338\pi$$
$$822$$ −1.92820 −0.0672538
$$823$$ 41.5692 1.44901 0.724506 0.689269i $$-0.242068\pi$$
0.724506 + 0.689269i $$0.242068\pi$$
$$824$$ 18.7321i 0.652562i
$$825$$ 23.3205i 0.811916i
$$826$$ 5.85641i 0.203770i
$$827$$ 33.4641i 1.16366i 0.813310 + 0.581830i $$0.197663\pi$$
−0.813310 + 0.581830i $$0.802337\pi$$
$$828$$ −6.19615 −0.215331
$$829$$ −12.1244 −0.421096 −0.210548 0.977583i $$-0.567525\pi$$
−0.210548 + 0.977583i $$0.567525\pi$$
$$830$$ 2.73205i 0.0948309i
$$831$$ 2.26795 0.0786743
$$832$$ 0 0
$$833$$ 14.6603 0.507948
$$834$$ 9.85641i 0.341299i
$$835$$ −0.679492 −0.0235148
$$836$$ −6.00000 −0.207514
$$837$$ 5.46410i 0.188867i
$$838$$ 10.5359i 0.363957i
$$839$$ 14.1436i 0.488291i 0.969739 + 0.244146i $$0.0785075\pi$$
−0.969739 + 0.244146i $$0.921493\pi$$
$$840$$ 0.196152i 0.00676790i
$$841$$ −22.9282 −0.790628
$$842$$ −32.7128 −1.12736
$$843$$ − 22.3205i − 0.768759i
$$844$$ 4.39230 0.151189
$$845$$ 0 0
$$846$$ −8.19615 −0.281790
$$847$$ − 8.33975i − 0.286557i
$$848$$ 0.464102 0.0159373
$$849$$ −8.33975 −0.286219
$$850$$ − 11.1769i − 0.383365i
$$851$$ 64.8372i 2.22259i
$$852$$ 1.26795i 0.0434392i
$$853$$ 8.17691i 0.279972i 0.990153 + 0.139986i $$0.0447058\pi$$
−0.990153 + 0.139986i $$0.955294\pi$$
$$854$$ 0.875644 0.0299639
$$855$$ −0.339746 −0.0116191
$$856$$ − 0.196152i − 0.00670435i
$$857$$ 19.4449 0.664224 0.332112 0.943240i $$-0.392239\pi$$
0.332112 + 0.943240i $$0.392239\pi$$
$$858$$ 0 0
$$859$$ −22.8756 −0.780507 −0.390253 0.920707i $$-0.627613\pi$$
−0.390253 + 0.920707i $$0.627613\pi$$
$$860$$ − 2.05256i − 0.0699917i
$$861$$ 8.33975 0.284218
$$862$$ 11.1244 0.378897
$$863$$ − 7.12436i − 0.242516i −0.992621 0.121258i $$-0.961307\pi$$
0.992621 0.121258i $$-0.0386928\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ − 4.39230i − 0.149343i
$$866$$ − 14.8564i − 0.504841i
$$867$$ −11.8564 −0.402665
$$868$$ 4.00000 0.135769
$$869$$ − 44.7846i − 1.51921i
$$870$$ −0.660254 −0.0223847
$$871$$ 0 0
$$872$$ −5.46410 −0.185038
$$873$$ 6.00000i 0.203069i
$$874$$ −7.85641 −0.265747
$$875$$ 1.94744 0.0658355
$$876$$ 9.73205i 0.328816i
$$877$$ − 10.0718i − 0.340100i −0.985435 0.170050i $$-0.945607\pi$$
0.985435 0.170050i $$-0.0543930\pi$$
$$878$$ − 17.6603i − 0.596005i
$$879$$ − 14.5167i − 0.489635i
$$880$$ 1.26795 0.0427426
$$881$$ −51.8372 −1.74644 −0.873219 0.487327i $$-0.837972\pi$$
−0.873219 + 0.487327i $$0.837972\pi$$
$$882$$ − 6.46410i − 0.217658i
$$883$$ −29.0718 −0.978344 −0.489172 0.872187i $$-0.662701\pi$$
−0.489172 + 0.872187i $$0.662701\pi$$
$$884$$ 0 0
$$885$$ 2.14359 0.0720561
$$886$$ − 36.3923i − 1.22262i
$$887$$ 10.1436 0.340589 0.170294 0.985393i $$-0.445528\pi$$
0.170294 + 0.985393i $$0.445528\pi$$
$$888$$ −10.4641 −0.351152
$$889$$ − 13.0718i − 0.438414i
$$890$$ 0.679492i 0.0227766i
$$891$$ 4.73205i 0.158530i
$$892$$ − 13.0718i − 0.437676i
$$893$$ −10.3923 −0.347765
$$894$$ −2.80385 −0.0937747
$$895$$ 5.90897i 0.197515i
$$896$$ −0.732051 −0.0244561
$$897$$ 0 0
$$898$$ −23.3205 −0.778215
$$899$$ 13.4641i 0.449053i
$$900$$ −4.92820 −0.164273
$$901$$ 1.05256 0.0350658
$$902$$ − 53.9090i − 1.79497i
$$903$$ − 5.60770i − 0.186612i
$$904$$ − 18.6603i − 0.620631i
$$905$$ 2.35898i 0.0784153i
$$906$$ 3.26795 0.108570
$$907$$ −15.6077 −0.518245 −0.259123 0.965844i $$-0.583433\pi$$
−0.259123 + 0.965844i $$0.583433\pi$$
$$908$$ − 1.80385i − 0.0598628i
$$909$$ 11.9282 0.395634
$$910$$ 0 0
$$911$$ −9.46410 −0.313560 −0.156780 0.987634i $$-0.550111\pi$$
−0.156780 + 0.987634i $$0.550111\pi$$
$$912$$ − 1.26795i − 0.0419860i
$$913$$ 48.2487 1.59680
$$914$$ −18.6603 −0.617226
$$915$$ − 0.320508i − 0.0105957i
$$916$$ − 15.8564i − 0.523910i
$$917$$ 9.85641i 0.325487i
$$918$$ − 2.26795i − 0.0748535i
$$919$$ −57.9615 −1.91197 −0.955987 0.293409i $$-0.905210\pi$$
−0.955987 + 0.293409i $$0.905210\pi$$
$$920$$ 1.66025 0.0547370
$$921$$ 8.58846i 0.282999i
$$922$$ −25.7321 −0.847440
$$923$$ 0 0
$$924$$ 3.46410 0.113961
$$925$$ 51.5692i 1.69559i
$$926$$ −28.0526 −0.921864
$$927$$ 18.7321 0.615241
$$928$$ − 2.46410i − 0.0808881i
$$929$$ − 9.24871i − 0.303440i −0.988423 0.151720i $$-0.951519\pi$$
0.988423 0.151720i $$-0.0484813\pi$$
$$930$$ − 1.46410i − 0.0480098i
$$931$$ − 8.19615i − 0.268618i
$$932$$ 19.8564 0.650418
$$933$$ −15.6603 −0.512694
$$934$$ − 12.5885i − 0.411907i
$$935$$ 2.87564 0.0940436
$$936$$ 0 0
$$937$$ 43.2487 1.41287 0.706437 0.707776i $$-0.250301\pi$$
0.706437 + 0.707776i $$0.250301\pi$$
$$938$$ 8.14359i 0.265898i
$$939$$ 13.4641 0.439384
$$940$$ 2.19615 0.0716306
$$941$$ 56.6410i 1.84644i 0.384267 + 0.923222i $$0.374454\pi$$
−0.384267 + 0.923222i $$0.625546\pi$$
$$942$$ 23.5885i 0.768553i
$$943$$ − 70.5885i − 2.29868i
$$944$$ 8.00000i 0.260378i
$$945$$ 0.196152 0.00638084
$$946$$ −36.2487 −1.17855
$$947$$ 34.9282i 1.13501i 0.823369 + 0.567507i $$0.192092\pi$$
−0.823369 + 0.567507i $$0.807908\pi$$
$$948$$ 9.46410 0.307380
$$949$$ 0 0
$$950$$ −6.24871 −0.202735
$$951$$ − 3.33975i − 0.108299i
$$952$$ −1.66025 −0.0538091
$$953$$ 41.5692 1.34656 0.673280 0.739388i $$-0.264885\pi$$
0.673280 + 0.739388i $$0.264885\pi$$
$$954$$ − 0.464102i − 0.0150258i
$$955$$ − 1.85641i − 0.0600719i
$$956$$ − 9.66025i − 0.312435i
$$957$$ 11.6603i 0.376922i
$$958$$ 26.5359 0.857336
$$959$$ 1.41154 0.0455811
$$960$$ 0.267949i 0.00864802i
$$961$$ 1.14359 0.0368901
$$962$$ 0 0
$$963$$ −0.196152 −0.00632092
$$964$$ 17.5885i 0.566486i
$$965$$ −2.21539 −0.0713159
$$966$$ 4.53590 0.145940
$$967$$ 18.8756i 0.607000i 0.952831 + 0.303500i $$0.0981552\pi$$
−0.952831 + 0.303500i $$0.901845\pi$$
$$968$$ − 11.3923i − 0.366163i
$$969$$ − 2.87564i − 0.0923790i
$$970$$ − 1.60770i − 0.0516200i
$$971$$ −18.2487 −0.585629 −0.292815 0.956169i $$-0.594592\pi$$
−0.292815 + 0.956169i $$0.594592\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ − 7.21539i − 0.231315i
$$974$$ −21.1244 −0.676868
$$975$$ 0 0
$$976$$ 1.19615 0.0382879
$$977$$ − 32.0718i − 1.02607i −0.858368 0.513034i $$-0.828522\pi$$
0.858368 0.513034i $$-0.171478\pi$$
$$978$$ −6.53590 −0.208995
$$979$$ 12.0000 0.383522
$$980$$ 1.73205i 0.0553283i
$$981$$ 5.46410i 0.174455i
$$982$$ 5.26795i 0.168107i
$$983$$ − 20.7846i − 0.662926i −0.943468 0.331463i $$-0.892458\pi$$
0.943468 0.331463i $$-0.107542\pi$$
$$984$$ 11.3923 0.363173
$$985$$ −2.64102 −0.0841498
$$986$$ − 5.58846i − 0.177973i
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ −47.4641 −1.50927
$$990$$ − 1.26795i − 0.0402981i
$$991$$ −8.58846 −0.272821 −0.136411 0.990652i $$-0.543557\pi$$
−0.136411 + 0.990652i $$0.543557\pi$$
$$992$$ 5.46410 0.173485
$$993$$ − 20.0000i − 0.634681i
$$994$$ − 0.928203i − 0.0294408i
$$995$$ − 1.01924i − 0.0323120i
$$996$$ 10.1962i 0.323077i
$$997$$ 38.6603 1.22438 0.612191 0.790710i $$-0.290288\pi$$
0.612191 + 0.790710i $$0.290288\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 10.4641i 0.331070i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1014.2.b.e.337.2 4
3.2 odd 2 3042.2.b.i.1351.3 4
13.2 odd 12 1014.2.e.i.529.2 4
13.3 even 3 78.2.i.a.43.2 4
13.4 even 6 78.2.i.a.49.2 yes 4
13.5 odd 4 1014.2.a.i.1.2 2
13.6 odd 12 1014.2.e.i.991.2 4
13.7 odd 12 1014.2.e.g.991.1 4
13.8 odd 4 1014.2.a.k.1.1 2
13.9 even 3 1014.2.i.a.361.1 4
13.10 even 6 1014.2.i.a.823.1 4
13.11 odd 12 1014.2.e.g.529.1 4
13.12 even 2 inner 1014.2.b.e.337.3 4
39.5 even 4 3042.2.a.y.1.1 2
39.8 even 4 3042.2.a.p.1.2 2
39.17 odd 6 234.2.l.c.127.1 4
39.29 odd 6 234.2.l.c.199.1 4
39.38 odd 2 3042.2.b.i.1351.2 4
52.3 odd 6 624.2.bv.e.433.1 4
52.31 even 4 8112.2.a.bj.1.2 2
52.43 odd 6 624.2.bv.e.49.2 4
52.47 even 4 8112.2.a.bp.1.1 2
65.3 odd 12 1950.2.y.g.199.1 4
65.4 even 6 1950.2.bc.d.751.1 4
65.17 odd 12 1950.2.y.g.49.1 4
65.29 even 6 1950.2.bc.d.901.1 4
65.42 odd 12 1950.2.y.b.199.2 4
65.43 odd 12 1950.2.y.b.49.2 4
156.95 even 6 1872.2.by.h.1297.1 4
156.107 even 6 1872.2.by.h.433.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.2 4 13.3 even 3
78.2.i.a.49.2 yes 4 13.4 even 6
234.2.l.c.127.1 4 39.17 odd 6
234.2.l.c.199.1 4 39.29 odd 6
624.2.bv.e.49.2 4 52.43 odd 6
624.2.bv.e.433.1 4 52.3 odd 6
1014.2.a.i.1.2 2 13.5 odd 4
1014.2.a.k.1.1 2 13.8 odd 4
1014.2.b.e.337.2 4 1.1 even 1 trivial
1014.2.b.e.337.3 4 13.12 even 2 inner
1014.2.e.g.529.1 4 13.11 odd 12
1014.2.e.g.991.1 4 13.7 odd 12
1014.2.e.i.529.2 4 13.2 odd 12
1014.2.e.i.991.2 4 13.6 odd 12
1014.2.i.a.361.1 4 13.9 even 3
1014.2.i.a.823.1 4 13.10 even 6
1872.2.by.h.433.2 4 156.107 even 6
1872.2.by.h.1297.1 4 156.95 even 6
1950.2.y.b.49.2 4 65.43 odd 12
1950.2.y.b.199.2 4 65.42 odd 12
1950.2.y.g.49.1 4 65.17 odd 12
1950.2.y.g.199.1 4 65.3 odd 12
1950.2.bc.d.751.1 4 65.4 even 6
1950.2.bc.d.901.1 4 65.29 even 6
3042.2.a.p.1.2 2 39.8 even 4
3042.2.a.y.1.1 2 39.5 even 4
3042.2.b.i.1351.2 4 39.38 odd 2
3042.2.b.i.1351.3 4 3.2 odd 2
8112.2.a.bj.1.2 2 52.31 even 4
8112.2.a.bp.1.1 2 52.47 even 4