Properties

 Label 169.2.b.b.168.5 Level $169$ Weight $2$ Character 169.168 Analytic conductor $1.349$ Analytic rank $0$ Dimension $6$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(168,169)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 168.5 Root $$1.24698i$$ of defining polynomial Character $$\chi$$ $$=$$ 169.168 Dual form 169.2.b.b.168.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} +0.246980i q^{5} -1.80194i q^{6} +2.35690i q^{7} +2.69202i q^{8} +2.04892 q^{9} +O(q^{10})$$ $$q+0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} +0.246980i q^{5} -1.80194i q^{6} +2.35690i q^{7} +2.69202i q^{8} +2.04892 q^{9} -0.198062 q^{10} +4.24698i q^{11} -3.04892 q^{12} -1.89008 q^{14} -0.554958i q^{15} +0.554958 q^{16} -2.15883 q^{17} +1.64310i q^{18} -0.0881460i q^{19} +0.335126i q^{20} -5.29590i q^{21} -3.40581 q^{22} -1.49396 q^{23} -6.04892i q^{24} +4.93900 q^{25} +2.13706 q^{27} +3.19806i q^{28} +4.63102 q^{29} +0.445042 q^{30} -6.63102i q^{31} +5.82908i q^{32} -9.54288i q^{33} -1.73125i q^{34} -0.582105 q^{35} +2.78017 q^{36} -5.69202i q^{37} +0.0706876 q^{38} -0.664874 q^{40} -11.5918i q^{41} +4.24698 q^{42} +0.295897 q^{43} +5.76271i q^{44} +0.506041i q^{45} -1.19806i q^{46} +7.35690i q^{47} -1.24698 q^{48} +1.44504 q^{49} +3.96077i q^{50} +4.85086 q^{51} -10.3937 q^{53} +1.71379i q^{54} -1.04892 q^{55} -6.34481 q^{56} +0.198062i q^{57} +3.71379i q^{58} +6.78017i q^{59} -0.753020i q^{60} +3.47219 q^{61} +5.31767 q^{62} +4.82908i q^{63} -3.56465 q^{64} +7.65279 q^{66} +7.67994i q^{67} -2.92931 q^{68} +3.35690 q^{69} -0.466812i q^{70} -8.66487i q^{71} +5.51573i q^{72} -6.73556i q^{73} +4.56465 q^{74} -11.0978 q^{75} -0.119605i q^{76} -10.0097 q^{77} +9.97046 q^{79} +0.137063i q^{80} -10.9487 q^{81} +9.29590 q^{82} +1.60925i q^{83} -7.18598i q^{84} -0.533188i q^{85} +0.237291i q^{86} -10.4058 q^{87} -11.4330 q^{88} +2.88471i q^{89} -0.405813 q^{90} -2.02715 q^{92} +14.8998i q^{93} -5.89977 q^{94} +0.0217703 q^{95} -13.0978i q^{96} -8.05861i q^{97} +1.15883i q^{98} +8.70171i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} - 6 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 - 6 * q^9 $$6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100})$$ 6 * q - 4 * q^3 - 6 * q^9 - 10 * q^10 - 10 * q^14 + 4 * q^16 + 4 * q^17 + 6 * q^22 + 10 * q^23 + 10 * q^25 + 2 * q^27 - 2 * q^29 + 2 * q^30 + 8 * q^35 + 14 * q^36 - 24 * q^38 - 6 * q^40 + 16 * q^42 - 26 * q^43 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 2 * q^53 + 12 * q^55 + 8 * q^56 + 8 * q^61 - 2 * q^62 + 22 * q^64 + 10 * q^66 - 42 * q^68 + 12 * q^69 - 16 * q^74 - 30 * q^75 - 16 * q^77 - 10 * q^79 - 2 * q^81 + 28 * q^82 - 36 * q^87 - 30 * q^88 + 24 * q^90 + 10 * q^94 - 6 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.801938i 0.567056i 0.958964 + 0.283528i $$0.0915048\pi$$
−0.958964 + 0.283528i $$0.908495\pi$$
$$3$$ −2.24698 −1.29729 −0.648647 0.761089i $$-0.724665\pi$$
−0.648647 + 0.761089i $$0.724665\pi$$
$$4$$ 1.35690 0.678448
$$5$$ 0.246980i 0.110453i 0.998474 + 0.0552263i $$0.0175880\pi$$
−0.998474 + 0.0552263i $$0.982412\pi$$
$$6$$ − 1.80194i − 0.735638i
$$7$$ 2.35690i 0.890823i 0.895326 + 0.445411i $$0.146943\pi$$
−0.895326 + 0.445411i $$0.853057\pi$$
$$8$$ 2.69202i 0.951773i
$$9$$ 2.04892 0.682972
$$10$$ −0.198062 −0.0626328
$$11$$ 4.24698i 1.28051i 0.768161 + 0.640256i $$0.221172\pi$$
−0.768161 + 0.640256i $$0.778828\pi$$
$$12$$ −3.04892 −0.880147
$$13$$ 0 0
$$14$$ −1.89008 −0.505146
$$15$$ − 0.554958i − 0.143290i
$$16$$ 0.554958 0.138740
$$17$$ −2.15883 −0.523594 −0.261797 0.965123i $$-0.584315\pi$$
−0.261797 + 0.965123i $$0.584315\pi$$
$$18$$ 1.64310i 0.387283i
$$19$$ − 0.0881460i − 0.0202221i −0.999949 0.0101110i $$-0.996782\pi$$
0.999949 0.0101110i $$-0.00321850\pi$$
$$20$$ 0.335126i 0.0749364i
$$21$$ − 5.29590i − 1.15566i
$$22$$ −3.40581 −0.726122
$$23$$ −1.49396 −0.311512 −0.155756 0.987796i $$-0.549781\pi$$
−0.155756 + 0.987796i $$0.549781\pi$$
$$24$$ − 6.04892i − 1.23473i
$$25$$ 4.93900 0.987800
$$26$$ 0 0
$$27$$ 2.13706 0.411278
$$28$$ 3.19806i 0.604377i
$$29$$ 4.63102 0.859959 0.429980 0.902839i $$-0.358521\pi$$
0.429980 + 0.902839i $$0.358521\pi$$
$$30$$ 0.445042 0.0812532
$$31$$ − 6.63102i − 1.19097i −0.803368 0.595483i $$-0.796961\pi$$
0.803368 0.595483i $$-0.203039\pi$$
$$32$$ 5.82908i 1.03045i
$$33$$ − 9.54288i − 1.66120i
$$34$$ − 1.73125i − 0.296907i
$$35$$ −0.582105 −0.0983937
$$36$$ 2.78017 0.463361
$$37$$ − 5.69202i − 0.935763i −0.883791 0.467881i $$-0.845017\pi$$
0.883791 0.467881i $$-0.154983\pi$$
$$38$$ 0.0706876 0.0114670
$$39$$ 0 0
$$40$$ −0.664874 −0.105126
$$41$$ − 11.5918i − 1.81033i −0.425056 0.905167i $$-0.639746\pi$$
0.425056 0.905167i $$-0.360254\pi$$
$$42$$ 4.24698 0.655323
$$43$$ 0.295897 0.0451239 0.0225619 0.999745i $$-0.492818\pi$$
0.0225619 + 0.999745i $$0.492818\pi$$
$$44$$ 5.76271i 0.868761i
$$45$$ 0.506041i 0.0754361i
$$46$$ − 1.19806i − 0.176645i
$$47$$ 7.35690i 1.07311i 0.843864 + 0.536557i $$0.180275\pi$$
−0.843864 + 0.536557i $$0.819725\pi$$
$$48$$ −1.24698 −0.179986
$$49$$ 1.44504 0.206435
$$50$$ 3.96077i 0.560138i
$$51$$ 4.85086 0.679256
$$52$$ 0 0
$$53$$ −10.3937 −1.42769 −0.713844 0.700304i $$-0.753048\pi$$
−0.713844 + 0.700304i $$0.753048\pi$$
$$54$$ 1.71379i 0.233218i
$$55$$ −1.04892 −0.141436
$$56$$ −6.34481 −0.847861
$$57$$ 0.198062i 0.0262340i
$$58$$ 3.71379i 0.487645i
$$59$$ 6.78017i 0.882703i 0.897334 + 0.441351i $$0.145501\pi$$
−0.897334 + 0.441351i $$0.854499\pi$$
$$60$$ − 0.753020i − 0.0972145i
$$61$$ 3.47219 0.444568 0.222284 0.974982i $$-0.428649\pi$$
0.222284 + 0.974982i $$0.428649\pi$$
$$62$$ 5.31767 0.675344
$$63$$ 4.82908i 0.608407i
$$64$$ −3.56465 −0.445581
$$65$$ 0 0
$$66$$ 7.65279 0.941994
$$67$$ 7.67994i 0.938254i 0.883131 + 0.469127i $$0.155431\pi$$
−0.883131 + 0.469127i $$0.844569\pi$$
$$68$$ −2.92931 −0.355231
$$69$$ 3.35690 0.404123
$$70$$ − 0.466812i − 0.0557947i
$$71$$ − 8.66487i − 1.02833i −0.857691 0.514166i $$-0.828102\pi$$
0.857691 0.514166i $$-0.171898\pi$$
$$72$$ 5.51573i 0.650035i
$$73$$ − 6.73556i − 0.788338i −0.919038 0.394169i $$-0.871032\pi$$
0.919038 0.394169i $$-0.128968\pi$$
$$74$$ 4.56465 0.530629
$$75$$ −11.0978 −1.28147
$$76$$ − 0.119605i − 0.0137196i
$$77$$ −10.0097 −1.14071
$$78$$ 0 0
$$79$$ 9.97046 1.12176 0.560882 0.827896i $$-0.310462\pi$$
0.560882 + 0.827896i $$0.310462\pi$$
$$80$$ 0.137063i 0.0153241i
$$81$$ −10.9487 −1.21652
$$82$$ 9.29590 1.02656
$$83$$ 1.60925i 0.176638i 0.996092 + 0.0883192i $$0.0281495\pi$$
−0.996092 + 0.0883192i $$0.971850\pi$$
$$84$$ − 7.18598i − 0.784055i
$$85$$ − 0.533188i − 0.0578323i
$$86$$ 0.237291i 0.0255877i
$$87$$ −10.4058 −1.11562
$$88$$ −11.4330 −1.21876
$$89$$ 2.88471i 0.305778i 0.988243 + 0.152889i $$0.0488577\pi$$
−0.988243 + 0.152889i $$0.951142\pi$$
$$90$$ −0.405813 −0.0427765
$$91$$ 0 0
$$92$$ −2.02715 −0.211345
$$93$$ 14.8998i 1.54503i
$$94$$ −5.89977 −0.608515
$$95$$ 0.0217703 0.00223358
$$96$$ − 13.0978i − 1.33679i
$$97$$ − 8.05861i − 0.818227i −0.912483 0.409114i $$-0.865838\pi$$
0.912483 0.409114i $$-0.134162\pi$$
$$98$$ 1.15883i 0.117060i
$$99$$ 8.70171i 0.874555i
$$100$$ 6.70171 0.670171
$$101$$ 13.3545 1.32882 0.664411 0.747367i $$-0.268682\pi$$
0.664411 + 0.747367i $$0.268682\pi$$
$$102$$ 3.89008i 0.385176i
$$103$$ −1.36227 −0.134229 −0.0671144 0.997745i $$-0.521379\pi$$
−0.0671144 + 0.997745i $$0.521379\pi$$
$$104$$ 0 0
$$105$$ 1.30798 0.127646
$$106$$ − 8.33513i − 0.809579i
$$107$$ 3.26875 0.316002 0.158001 0.987439i $$-0.449495\pi$$
0.158001 + 0.987439i $$0.449495\pi$$
$$108$$ 2.89977 0.279031
$$109$$ 15.7017i 1.50395i 0.659191 + 0.751976i $$0.270899\pi$$
−0.659191 + 0.751976i $$0.729101\pi$$
$$110$$ − 0.841166i − 0.0802021i
$$111$$ 12.7899i 1.21396i
$$112$$ 1.30798i 0.123592i
$$113$$ 12.0489 1.13347 0.566733 0.823901i $$-0.308207\pi$$
0.566733 + 0.823901i $$0.308207\pi$$
$$114$$ −0.158834 −0.0148761
$$115$$ − 0.368977i − 0.0344073i
$$116$$ 6.28382 0.583438
$$117$$ 0 0
$$118$$ −5.43727 −0.500541
$$119$$ − 5.08815i − 0.466430i
$$120$$ 1.49396 0.136379
$$121$$ −7.03684 −0.639712
$$122$$ 2.78448i 0.252095i
$$123$$ 26.0465i 2.34854i
$$124$$ − 8.99761i − 0.808009i
$$125$$ 2.45473i 0.219558i
$$126$$ −3.87263 −0.345001
$$127$$ 9.80731 0.870258 0.435129 0.900368i $$-0.356703\pi$$
0.435129 + 0.900368i $$0.356703\pi$$
$$128$$ 8.79954i 0.777777i
$$129$$ −0.664874 −0.0585389
$$130$$ 0 0
$$131$$ −6.57673 −0.574611 −0.287306 0.957839i $$-0.592760\pi$$
−0.287306 + 0.957839i $$0.592760\pi$$
$$132$$ − 12.9487i − 1.12704i
$$133$$ 0.207751 0.0180143
$$134$$ −6.15883 −0.532042
$$135$$ 0.527811i 0.0454267i
$$136$$ − 5.81163i − 0.498343i
$$137$$ − 6.21983i − 0.531396i −0.964056 0.265698i $$-0.914398\pi$$
0.964056 0.265698i $$-0.0856024\pi$$
$$138$$ 2.69202i 0.229160i
$$139$$ −14.7071 −1.24744 −0.623719 0.781648i $$-0.714379\pi$$
−0.623719 + 0.781648i $$0.714379\pi$$
$$140$$ −0.789856 −0.0667550
$$141$$ − 16.5308i − 1.39214i
$$142$$ 6.94869 0.583121
$$143$$ 0 0
$$144$$ 1.13706 0.0947553
$$145$$ 1.14377i 0.0949848i
$$146$$ 5.40150 0.447031
$$147$$ −3.24698 −0.267806
$$148$$ − 7.72348i − 0.634866i
$$149$$ 4.33513i 0.355147i 0.984108 + 0.177574i $$0.0568248\pi$$
−0.984108 + 0.177574i $$0.943175\pi$$
$$150$$ − 8.89977i − 0.726663i
$$151$$ − 3.94438i − 0.320989i −0.987037 0.160494i $$-0.948691\pi$$
0.987037 0.160494i $$-0.0513089\pi$$
$$152$$ 0.237291 0.0192468
$$153$$ −4.42327 −0.357600
$$154$$ − 8.02715i − 0.646846i
$$155$$ 1.63773 0.131545
$$156$$ 0 0
$$157$$ 4.45473 0.355526 0.177763 0.984073i $$-0.443114\pi$$
0.177763 + 0.984073i $$0.443114\pi$$
$$158$$ 7.99569i 0.636103i
$$159$$ 23.3545 1.85213
$$160$$ −1.43967 −0.113816
$$161$$ − 3.52111i − 0.277502i
$$162$$ − 8.78017i − 0.689835i
$$163$$ − 16.1588i − 1.26566i −0.774292 0.632829i $$-0.781894\pi$$
0.774292 0.632829i $$-0.218106\pi$$
$$164$$ − 15.7289i − 1.22822i
$$165$$ 2.35690 0.183484
$$166$$ −1.29052 −0.100164
$$167$$ − 16.1172i − 1.24719i −0.781749 0.623594i $$-0.785672\pi$$
0.781749 0.623594i $$-0.214328\pi$$
$$168$$ 14.2567 1.09993
$$169$$ 0 0
$$170$$ 0.427583 0.0327942
$$171$$ − 0.180604i − 0.0138111i
$$172$$ 0.401501 0.0306142
$$173$$ 21.5362 1.63736 0.818682 0.574247i $$-0.194705\pi$$
0.818682 + 0.574247i $$0.194705\pi$$
$$174$$ − 8.34481i − 0.632619i
$$175$$ 11.6407i 0.879955i
$$176$$ 2.35690i 0.177658i
$$177$$ − 15.2349i − 1.14513i
$$178$$ −2.31336 −0.173393
$$179$$ −11.4330 −0.854540 −0.427270 0.904124i $$-0.640525\pi$$
−0.427270 + 0.904124i $$0.640525\pi$$
$$180$$ 0.686645i 0.0511795i
$$181$$ −20.9705 −1.55872 −0.779361 0.626575i $$-0.784456\pi$$
−0.779361 + 0.626575i $$0.784456\pi$$
$$182$$ 0 0
$$183$$ −7.80194 −0.576736
$$184$$ − 4.02177i − 0.296489i
$$185$$ 1.40581 0.103357
$$186$$ −11.9487 −0.876120
$$187$$ − 9.16852i − 0.670469i
$$188$$ 9.98254i 0.728052i
$$189$$ 5.03684i 0.366376i
$$190$$ 0.0174584i 0.00126657i
$$191$$ −14.4373 −1.04464 −0.522322 0.852748i $$-0.674934\pi$$
−0.522322 + 0.852748i $$0.674934\pi$$
$$192$$ 8.00969 0.578049
$$193$$ 13.5797i 0.977489i 0.872427 + 0.488745i $$0.162545\pi$$
−0.872427 + 0.488745i $$0.837455\pi$$
$$194$$ 6.46250 0.463980
$$195$$ 0 0
$$196$$ 1.96077 0.140055
$$197$$ − 0.560335i − 0.0399222i −0.999801 0.0199611i $$-0.993646\pi$$
0.999801 0.0199611i $$-0.00635424\pi$$
$$198$$ −6.97823 −0.495921
$$199$$ −11.4916 −0.814616 −0.407308 0.913291i $$-0.633532\pi$$
−0.407308 + 0.913291i $$0.633532\pi$$
$$200$$ 13.2959i 0.940162i
$$201$$ − 17.2567i − 1.21719i
$$202$$ 10.7095i 0.753516i
$$203$$ 10.9148i 0.766071i
$$204$$ 6.58211 0.460840
$$205$$ 2.86294 0.199956
$$206$$ − 1.09246i − 0.0761151i
$$207$$ −3.06100 −0.212754
$$208$$ 0 0
$$209$$ 0.374354 0.0258946
$$210$$ 1.04892i 0.0723822i
$$211$$ 8.78448 0.604748 0.302374 0.953189i $$-0.402221\pi$$
0.302374 + 0.953189i $$0.402221\pi$$
$$212$$ −14.1032 −0.968613
$$213$$ 19.4698i 1.33405i
$$214$$ 2.62133i 0.179191i
$$215$$ 0.0730805i 0.00498405i
$$216$$ 5.75302i 0.391443i
$$217$$ 15.6286 1.06094
$$218$$ −12.5918 −0.852824
$$219$$ 15.1347i 1.02271i
$$220$$ −1.42327 −0.0959570
$$221$$ 0 0
$$222$$ −10.2567 −0.688383
$$223$$ − 2.25906i − 0.151278i −0.997135 0.0756390i $$-0.975900\pi$$
0.997135 0.0756390i $$-0.0240996\pi$$
$$224$$ −13.7385 −0.917945
$$225$$ 10.1196 0.674640
$$226$$ 9.66248i 0.642739i
$$227$$ − 6.96615i − 0.462359i −0.972911 0.231180i $$-0.925741\pi$$
0.972911 0.231180i $$-0.0742585\pi$$
$$228$$ 0.268750i 0.0177984i
$$229$$ 24.1739i 1.59746i 0.601692 + 0.798728i $$0.294493\pi$$
−0.601692 + 0.798728i $$0.705507\pi$$
$$230$$ 0.295897 0.0195109
$$231$$ 22.4916 1.47984
$$232$$ 12.4668i 0.818486i
$$233$$ 3.06100 0.200533 0.100266 0.994961i $$-0.468031\pi$$
0.100266 + 0.994961i $$0.468031\pi$$
$$234$$ 0 0
$$235$$ −1.81700 −0.118528
$$236$$ 9.19998i 0.598868i
$$237$$ −22.4034 −1.45526
$$238$$ 4.08038 0.264492
$$239$$ − 25.1468i − 1.62661i −0.581839 0.813304i $$-0.697667\pi$$
0.581839 0.813304i $$-0.302333\pi$$
$$240$$ − 0.307979i − 0.0198799i
$$241$$ 20.2664i 1.30547i 0.757586 + 0.652735i $$0.226379\pi$$
−0.757586 + 0.652735i $$0.773621\pi$$
$$242$$ − 5.64310i − 0.362752i
$$243$$ 18.1903 1.16691
$$244$$ 4.71140 0.301616
$$245$$ 0.356896i 0.0228012i
$$246$$ −20.8877 −1.33175
$$247$$ 0 0
$$248$$ 17.8509 1.13353
$$249$$ − 3.61596i − 0.229152i
$$250$$ −1.96854 −0.124501
$$251$$ 23.7211 1.49726 0.748631 0.662987i $$-0.230712\pi$$
0.748631 + 0.662987i $$0.230712\pi$$
$$252$$ 6.55257i 0.412773i
$$253$$ − 6.34481i − 0.398895i
$$254$$ 7.86486i 0.493485i
$$255$$ 1.19806i 0.0750256i
$$256$$ −14.1860 −0.886624
$$257$$ −14.2241 −0.887278 −0.443639 0.896206i $$-0.646313\pi$$
−0.443639 + 0.896206i $$0.646313\pi$$
$$258$$ − 0.533188i − 0.0331948i
$$259$$ 13.4155 0.833599
$$260$$ 0 0
$$261$$ 9.48858 0.587329
$$262$$ − 5.27413i − 0.325837i
$$263$$ −17.0954 −1.05415 −0.527075 0.849819i $$-0.676711\pi$$
−0.527075 + 0.849819i $$0.676711\pi$$
$$264$$ 25.6896 1.58109
$$265$$ − 2.56704i − 0.157692i
$$266$$ 0.166603i 0.0102151i
$$267$$ − 6.48188i − 0.396684i
$$268$$ 10.4209i 0.636556i
$$269$$ −6.46681 −0.394288 −0.197144 0.980374i $$-0.563167\pi$$
−0.197144 + 0.980374i $$0.563167\pi$$
$$270$$ −0.423272 −0.0257595
$$271$$ − 6.44803i − 0.391690i −0.980635 0.195845i $$-0.937255\pi$$
0.980635 0.195845i $$-0.0627449\pi$$
$$272$$ −1.19806 −0.0726432
$$273$$ 0 0
$$274$$ 4.98792 0.301331
$$275$$ 20.9758i 1.26489i
$$276$$ 4.55496 0.274176
$$277$$ −13.4601 −0.808739 −0.404370 0.914596i $$-0.632509\pi$$
−0.404370 + 0.914596i $$0.632509\pi$$
$$278$$ − 11.7942i − 0.707367i
$$279$$ − 13.5864i − 0.813398i
$$280$$ − 1.56704i − 0.0936485i
$$281$$ − 5.03684i − 0.300472i −0.988650 0.150236i $$-0.951997\pi$$
0.988650 0.150236i $$-0.0480034\pi$$
$$282$$ 13.2567 0.789423
$$283$$ −22.1280 −1.31537 −0.657686 0.753293i $$-0.728464\pi$$
−0.657686 + 0.753293i $$0.728464\pi$$
$$284$$ − 11.7573i − 0.697669i
$$285$$ −0.0489173 −0.00289761
$$286$$ 0 0
$$287$$ 27.3207 1.61269
$$288$$ 11.9433i 0.703766i
$$289$$ −12.3394 −0.725849
$$290$$ −0.917231 −0.0538616
$$291$$ 18.1075i 1.06148i
$$292$$ − 9.13946i − 0.534846i
$$293$$ − 14.9463i − 0.873172i −0.899663 0.436586i $$-0.856187\pi$$
0.899663 0.436586i $$-0.143813\pi$$
$$294$$ − 2.60388i − 0.151861i
$$295$$ −1.67456 −0.0974968
$$296$$ 15.3230 0.890634
$$297$$ 9.07606i 0.526647i
$$298$$ −3.47650 −0.201388
$$299$$ 0 0
$$300$$ −15.0586 −0.869409
$$301$$ 0.697398i 0.0401974i
$$302$$ 3.16315 0.182019
$$303$$ −30.0073 −1.72387
$$304$$ − 0.0489173i − 0.00280560i
$$305$$ 0.857560i 0.0491037i
$$306$$ − 3.54719i − 0.202779i
$$307$$ − 19.1293i − 1.09177i −0.837861 0.545883i $$-0.816194\pi$$
0.837861 0.545883i $$-0.183806\pi$$
$$308$$ −13.5821 −0.773912
$$309$$ 3.06100 0.174134
$$310$$ 1.31336i 0.0745936i
$$311$$ 0.269815 0.0152998 0.00764990 0.999971i $$-0.497565\pi$$
0.00764990 + 0.999971i $$0.497565\pi$$
$$312$$ 0 0
$$313$$ −23.3937 −1.32229 −0.661146 0.750257i $$-0.729930\pi$$
−0.661146 + 0.750257i $$0.729930\pi$$
$$314$$ 3.57242i 0.201603i
$$315$$ −1.19269 −0.0672002
$$316$$ 13.5289 0.761059
$$317$$ − 13.9952i − 0.786050i −0.919528 0.393025i $$-0.871429\pi$$
0.919528 0.393025i $$-0.128571\pi$$
$$318$$ 18.7289i 1.05026i
$$319$$ 19.6679i 1.10119i
$$320$$ − 0.880395i − 0.0492156i
$$321$$ −7.34481 −0.409948
$$322$$ 2.82371 0.157359
$$323$$ 0.190293i 0.0105882i
$$324$$ −14.8562 −0.825346
$$325$$ 0 0
$$326$$ 12.9584 0.717698
$$327$$ − 35.2814i − 1.95107i
$$328$$ 31.2054 1.72303
$$329$$ −17.3394 −0.955954
$$330$$ 1.89008i 0.104046i
$$331$$ 17.8213i 0.979548i 0.871849 + 0.489774i $$0.162921\pi$$
−0.871849 + 0.489774i $$0.837079\pi$$
$$332$$ 2.18359i 0.119840i
$$333$$ − 11.6625i − 0.639100i
$$334$$ 12.9250 0.707225
$$335$$ −1.89679 −0.103633
$$336$$ − 2.93900i − 0.160336i
$$337$$ 27.8485 1.51700 0.758501 0.651672i $$-0.225932\pi$$
0.758501 + 0.651672i $$0.225932\pi$$
$$338$$ 0 0
$$339$$ −27.0737 −1.47044
$$340$$ − 0.723480i − 0.0392362i
$$341$$ 28.1618 1.52505
$$342$$ 0.144833 0.00783167
$$343$$ 19.9041i 1.07472i
$$344$$ 0.796561i 0.0429477i
$$345$$ 0.829085i 0.0446364i
$$346$$ 17.2707i 0.928477i
$$347$$ 1.50365 0.0807200 0.0403600 0.999185i $$-0.487150\pi$$
0.0403600 + 0.999185i $$0.487150\pi$$
$$348$$ −14.1196 −0.756890
$$349$$ 14.1860i 0.759358i 0.925118 + 0.379679i $$0.123966\pi$$
−0.925118 + 0.379679i $$0.876034\pi$$
$$350$$ −9.33513 −0.498983
$$351$$ 0 0
$$352$$ −24.7560 −1.31950
$$353$$ − 7.16852i − 0.381542i −0.981635 0.190771i $$-0.938901\pi$$
0.981635 0.190771i $$-0.0610988\pi$$
$$354$$ 12.2174 0.649350
$$355$$ 2.14005 0.113582
$$356$$ 3.91425i 0.207455i
$$357$$ 11.4330i 0.605096i
$$358$$ − 9.16852i − 0.484571i
$$359$$ − 19.8853i − 1.04951i −0.851255 0.524753i $$-0.824158\pi$$
0.851255 0.524753i $$-0.175842\pi$$
$$360$$ −1.36227 −0.0717981
$$361$$ 18.9922 0.999591
$$362$$ − 16.8170i − 0.883882i
$$363$$ 15.8116 0.829895
$$364$$ 0 0
$$365$$ 1.66355 0.0870740
$$366$$ − 6.25667i − 0.327041i
$$367$$ 1.08383 0.0565757 0.0282878 0.999600i $$-0.490994\pi$$
0.0282878 + 0.999600i $$0.490994\pi$$
$$368$$ −0.829085 −0.0432190
$$369$$ − 23.7506i − 1.23641i
$$370$$ 1.12737i 0.0586094i
$$371$$ − 24.4969i − 1.27182i
$$372$$ 20.2174i 1.04823i
$$373$$ −6.13036 −0.317418 −0.158709 0.987325i $$-0.550733\pi$$
−0.158709 + 0.987325i $$0.550733\pi$$
$$374$$ 7.35258 0.380193
$$375$$ − 5.51573i − 0.284831i
$$376$$ −19.8049 −1.02136
$$377$$ 0 0
$$378$$ −4.03923 −0.207756
$$379$$ − 2.40880i − 0.123732i −0.998084 0.0618658i $$-0.980295\pi$$
0.998084 0.0618658i $$-0.0197051\pi$$
$$380$$ 0.0295400 0.00151537
$$381$$ −22.0368 −1.12898
$$382$$ − 11.5778i − 0.592371i
$$383$$ − 30.3913i − 1.55292i −0.630164 0.776462i $$-0.717012\pi$$
0.630164 0.776462i $$-0.282988\pi$$
$$384$$ − 19.7724i − 1.00901i
$$385$$ − 2.47219i − 0.125994i
$$386$$ −10.8901 −0.554291
$$387$$ 0.606268 0.0308184
$$388$$ − 10.9347i − 0.555125i
$$389$$ 15.9409 0.808237 0.404118 0.914707i $$-0.367578\pi$$
0.404118 + 0.914707i $$0.367578\pi$$
$$390$$ 0 0
$$391$$ 3.22521 0.163106
$$392$$ 3.89008i 0.196479i
$$393$$ 14.7778 0.745440
$$394$$ 0.449354 0.0226381
$$395$$ 2.46250i 0.123902i
$$396$$ 11.8073i 0.593340i
$$397$$ − 16.9148i − 0.848931i −0.905444 0.424466i $$-0.860462\pi$$
0.905444 0.424466i $$-0.139538\pi$$
$$398$$ − 9.21552i − 0.461932i
$$399$$ −0.466812 −0.0233698
$$400$$ 2.74094 0.137047
$$401$$ − 26.6625i − 1.33146i −0.746192 0.665730i $$-0.768120\pi$$
0.746192 0.665730i $$-0.231880\pi$$
$$402$$ 13.8388 0.690215
$$403$$ 0 0
$$404$$ 18.1207 0.901537
$$405$$ − 2.70410i − 0.134368i
$$406$$ −8.75302 −0.434405
$$407$$ 24.1739 1.19826
$$408$$ 13.0586i 0.646497i
$$409$$ 28.5163i 1.41004i 0.709187 + 0.705021i $$0.249062\pi$$
−0.709187 + 0.705021i $$0.750938\pi$$
$$410$$ 2.29590i 0.113386i
$$411$$ 13.9758i 0.689377i
$$412$$ −1.84846 −0.0910672
$$413$$ −15.9801 −0.786332
$$414$$ − 2.45473i − 0.120643i
$$415$$ −0.397452 −0.0195102
$$416$$ 0 0
$$417$$ 33.0465 1.61830
$$418$$ 0.300209i 0.0146837i
$$419$$ −29.6093 −1.44651 −0.723253 0.690583i $$-0.757354\pi$$
−0.723253 + 0.690583i $$0.757354\pi$$
$$420$$ 1.77479 0.0866009
$$421$$ − 11.6606i − 0.568301i −0.958780 0.284151i $$-0.908288\pi$$
0.958780 0.284151i $$-0.0917115\pi$$
$$422$$ 7.04461i 0.342926i
$$423$$ 15.0737i 0.732907i
$$424$$ − 27.9801i − 1.35884i
$$425$$ −10.6625 −0.517206
$$426$$ −15.6136 −0.756480
$$427$$ 8.18359i 0.396032i
$$428$$ 4.43535 0.214391
$$429$$ 0 0
$$430$$ −0.0586060 −0.00282623
$$431$$ 4.34913i 0.209490i 0.994499 + 0.104745i $$0.0334026\pi$$
−0.994499 + 0.104745i $$0.966597\pi$$
$$432$$ 1.18598 0.0570605
$$433$$ 14.3884 0.691460 0.345730 0.938334i $$-0.387631\pi$$
0.345730 + 0.938334i $$0.387631\pi$$
$$434$$ 12.5332i 0.601612i
$$435$$ − 2.57002i − 0.123223i
$$436$$ 21.3056i 1.02035i
$$437$$ 0.131687i 0.00629942i
$$438$$ −12.1371 −0.579931
$$439$$ 20.2325 0.965645 0.482822 0.875718i $$-0.339612\pi$$
0.482822 + 0.875718i $$0.339612\pi$$
$$440$$ − 2.82371i − 0.134615i
$$441$$ 2.96077 0.140989
$$442$$ 0 0
$$443$$ 8.12200 0.385888 0.192944 0.981210i $$-0.438196\pi$$
0.192944 + 0.981210i $$0.438196\pi$$
$$444$$ 17.3545i 0.823608i
$$445$$ −0.712464 −0.0337740
$$446$$ 1.81163 0.0857830
$$447$$ − 9.74094i − 0.460731i
$$448$$ − 8.40150i − 0.396934i
$$449$$ − 12.4916i − 0.589513i −0.955572 0.294757i $$-0.904761\pi$$
0.955572 0.294757i $$-0.0952386\pi$$
$$450$$ 8.11529i 0.382559i
$$451$$ 49.2301 2.31816
$$452$$ 16.3491 0.768998
$$453$$ 8.86294i 0.416417i
$$454$$ 5.58642 0.262184
$$455$$ 0 0
$$456$$ −0.533188 −0.0249688
$$457$$ 5.98121i 0.279789i 0.990166 + 0.139895i $$0.0446764\pi$$
−0.990166 + 0.139895i $$0.955324\pi$$
$$458$$ −19.3860 −0.905847
$$459$$ −4.61356 −0.215343
$$460$$ − 0.500664i − 0.0233436i
$$461$$ − 2.05669i − 0.0957895i −0.998852 0.0478947i $$-0.984749\pi$$
0.998852 0.0478947i $$-0.0152512\pi$$
$$462$$ 18.0368i 0.839150i
$$463$$ 8.44935i 0.392675i 0.980536 + 0.196337i $$0.0629048\pi$$
−0.980536 + 0.196337i $$0.937095\pi$$
$$464$$ 2.57002 0.119310
$$465$$ −3.67994 −0.170653
$$466$$ 2.45473i 0.113713i
$$467$$ −33.5139 −1.55084 −0.775420 0.631446i $$-0.782462\pi$$
−0.775420 + 0.631446i $$0.782462\pi$$
$$468$$ 0 0
$$469$$ −18.1008 −0.835818
$$470$$ − 1.45712i − 0.0672121i
$$471$$ −10.0097 −0.461222
$$472$$ −18.2524 −0.840133
$$473$$ 1.25667i 0.0577817i
$$474$$ − 17.9661i − 0.825213i
$$475$$ − 0.435353i − 0.0199754i
$$476$$ − 6.90408i − 0.316448i
$$477$$ −21.2959 −0.975072
$$478$$ 20.1661 0.922377
$$479$$ 24.7313i 1.13000i 0.825091 + 0.565000i $$0.191124\pi$$
−0.825091 + 0.565000i $$0.808876\pi$$
$$480$$ 3.23490 0.147652
$$481$$ 0 0
$$482$$ −16.2524 −0.740275
$$483$$ 7.91185i 0.360002i
$$484$$ −9.54825 −0.434012
$$485$$ 1.99031 0.0903754
$$486$$ 14.5875i 0.661702i
$$487$$ − 37.7555i − 1.71087i −0.517913 0.855433i $$-0.673291\pi$$
0.517913 0.855433i $$-0.326709\pi$$
$$488$$ 9.34721i 0.423128i
$$489$$ 36.3086i 1.64193i
$$490$$ −0.286208 −0.0129296
$$491$$ −31.3110 −1.41304 −0.706522 0.707691i $$-0.749737\pi$$
−0.706522 + 0.707691i $$0.749737\pi$$
$$492$$ 35.3424i 1.59336i
$$493$$ −9.99761 −0.450270
$$494$$ 0 0
$$495$$ −2.14914 −0.0965969
$$496$$ − 3.67994i − 0.165234i
$$497$$ 20.4222 0.916061
$$498$$ 2.89977 0.129942
$$499$$ 21.4873i 0.961902i 0.876748 + 0.480951i $$0.159708\pi$$
−0.876748 + 0.480951i $$0.840292\pi$$
$$500$$ 3.33081i 0.148959i
$$501$$ 36.2150i 1.61797i
$$502$$ 19.0228i 0.849031i
$$503$$ 37.5924 1.67616 0.838081 0.545546i $$-0.183678\pi$$
0.838081 + 0.545546i $$0.183678\pi$$
$$504$$ −13.0000 −0.579066
$$505$$ 3.29829i 0.146772i
$$506$$ 5.08815 0.226196
$$507$$ 0 0
$$508$$ 13.3075 0.590425
$$509$$ − 17.1075i − 0.758278i −0.925340 0.379139i $$-0.876220\pi$$
0.925340 0.379139i $$-0.123780\pi$$
$$510$$ −0.960771 −0.0425437
$$511$$ 15.8750 0.702269
$$512$$ 6.22282i 0.275012i
$$513$$ − 0.188374i − 0.00831690i
$$514$$ − 11.4069i − 0.503136i
$$515$$ − 0.336454i − 0.0148259i
$$516$$ −0.902165 −0.0397156
$$517$$ −31.2446 −1.37414
$$518$$ 10.7584i 0.472697i
$$519$$ −48.3913 −2.12414
$$520$$ 0 0
$$521$$ −19.8465 −0.869493 −0.434746 0.900553i $$-0.643162\pi$$
−0.434746 + 0.900553i $$0.643162\pi$$
$$522$$ 7.60925i 0.333048i
$$523$$ −11.4300 −0.499798 −0.249899 0.968272i $$-0.580397\pi$$
−0.249899 + 0.968272i $$0.580397\pi$$
$$524$$ −8.92394 −0.389844
$$525$$ − 26.1564i − 1.14156i
$$526$$ − 13.7095i − 0.597762i
$$527$$ 14.3153i 0.623583i
$$528$$ − 5.29590i − 0.230474i
$$529$$ −20.7681 −0.902960
$$530$$ 2.05861 0.0894201
$$531$$ 13.8920i 0.602862i
$$532$$ 0.281896 0.0122218
$$533$$ 0 0
$$534$$ 5.19806 0.224942
$$535$$ 0.807315i 0.0349033i
$$536$$ −20.6746 −0.893005
$$537$$ 25.6896 1.10859
$$538$$ − 5.18598i − 0.223584i
$$539$$ 6.13706i 0.264342i
$$540$$ 0.716185i 0.0308197i
$$541$$ − 16.1884i − 0.695993i −0.937496 0.347996i $$-0.886862\pi$$
0.937496 0.347996i $$-0.113138\pi$$
$$542$$ 5.17092 0.222110
$$543$$ 47.1202 2.02212
$$544$$ − 12.5840i − 0.539536i
$$545$$ −3.87800 −0.166115
$$546$$ 0 0
$$547$$ 5.33081 0.227929 0.113965 0.993485i $$-0.463645\pi$$
0.113965 + 0.993485i $$0.463645\pi$$
$$548$$ − 8.43967i − 0.360525i
$$549$$ 7.11423 0.303628
$$550$$ −16.8213 −0.717263
$$551$$ − 0.408206i − 0.0173902i
$$552$$ 9.03684i 0.384633i
$$553$$ 23.4993i 0.999293i
$$554$$ − 10.7942i − 0.458600i
$$555$$ −3.15883 −0.134085
$$556$$ −19.9560 −0.846322
$$557$$ − 7.39075i − 0.313156i −0.987666 0.156578i $$-0.949954\pi$$
0.987666 0.156578i $$-0.0500463\pi$$
$$558$$ 10.8955 0.461242
$$559$$ 0 0
$$560$$ −0.323044 −0.0136511
$$561$$ 20.6015i 0.869795i
$$562$$ 4.03923 0.170385
$$563$$ 9.47889 0.399488 0.199744 0.979848i $$-0.435989\pi$$
0.199744 + 0.979848i $$0.435989\pi$$
$$564$$ − 22.4306i − 0.944497i
$$565$$ 2.97584i 0.125194i
$$566$$ − 17.7453i − 0.745889i
$$567$$ − 25.8049i − 1.08370i
$$568$$ 23.3260 0.978738
$$569$$ 10.1438 0.425249 0.212624 0.977134i $$-0.431799\pi$$
0.212624 + 0.977134i $$0.431799\pi$$
$$570$$ − 0.0392287i − 0.00164311i
$$571$$ 14.0925 0.589751 0.294876 0.955536i $$-0.404722\pi$$
0.294876 + 0.955536i $$0.404722\pi$$
$$572$$ 0 0
$$573$$ 32.4403 1.35521
$$574$$ 21.9095i 0.914483i
$$575$$ −7.37867 −0.307712
$$576$$ −7.30367 −0.304319
$$577$$ 25.1545i 1.04720i 0.851965 + 0.523598i $$0.175411\pi$$
−0.851965 + 0.523598i $$0.824589\pi$$
$$578$$ − 9.89546i − 0.411597i
$$579$$ − 30.5133i − 1.26809i
$$580$$ 1.55197i 0.0644422i
$$581$$ −3.79284 −0.157354
$$582$$ −14.5211 −0.601919
$$583$$ − 44.1420i − 1.82817i
$$584$$ 18.1323 0.750319
$$585$$ 0 0
$$586$$ 11.9860 0.495137
$$587$$ 43.8353i 1.80928i 0.426180 + 0.904639i $$0.359859\pi$$
−0.426180 + 0.904639i $$0.640141\pi$$
$$588$$ −4.40581 −0.181693
$$589$$ −0.584498 −0.0240838
$$590$$ − 1.34290i − 0.0552861i
$$591$$ 1.25906i 0.0517909i
$$592$$ − 3.15883i − 0.129827i
$$593$$ 24.9965i 1.02648i 0.858244 + 0.513242i $$0.171556\pi$$
−0.858244 + 0.513242i $$0.828444\pi$$
$$594$$ −7.27844 −0.298638
$$595$$ 1.25667 0.0515184
$$596$$ 5.88231i 0.240949i
$$597$$ 25.8213 1.05680
$$598$$ 0 0
$$599$$ −6.24027 −0.254971 −0.127485 0.991840i $$-0.540691\pi$$
−0.127485 + 0.991840i $$0.540691\pi$$
$$600$$ − 29.8756i − 1.21967i
$$601$$ 6.32975 0.258196 0.129098 0.991632i $$-0.458792\pi$$
0.129098 + 0.991632i $$0.458792\pi$$
$$602$$ −0.559270 −0.0227941
$$603$$ 15.7356i 0.640802i
$$604$$ − 5.35211i − 0.217774i
$$605$$ − 1.73795i − 0.0706579i
$$606$$ − 24.0640i − 0.977532i
$$607$$ −43.6480 −1.77162 −0.885809 0.464050i $$-0.846396\pi$$
−0.885809 + 0.464050i $$0.846396\pi$$
$$608$$ 0.513811 0.0208378
$$609$$ − 24.5254i − 0.993820i
$$610$$ −0.687710 −0.0278445
$$611$$ 0 0
$$612$$ −6.00192 −0.242613
$$613$$ − 25.9541i − 1.04827i −0.851634 0.524137i $$-0.824388\pi$$
0.851634 0.524137i $$-0.175612\pi$$
$$614$$ 15.3405 0.619092
$$615$$ −6.43296 −0.259402
$$616$$ − 26.9463i − 1.08570i
$$617$$ 45.9396i 1.84946i 0.380626 + 0.924729i $$0.375709\pi$$
−0.380626 + 0.924729i $$0.624291\pi$$
$$618$$ 2.45473i 0.0987437i
$$619$$ − 6.73556i − 0.270725i −0.990796 0.135363i $$-0.956780\pi$$
0.990796 0.135363i $$-0.0432199\pi$$
$$620$$ 2.22223 0.0892467
$$621$$ −3.19269 −0.128118
$$622$$ 0.216375i 0.00867583i
$$623$$ −6.79895 −0.272394
$$624$$ 0 0
$$625$$ 24.0887 0.963549
$$626$$ − 18.7603i − 0.749813i
$$627$$ −0.841166 −0.0335930
$$628$$ 6.04461 0.241206
$$629$$ 12.2881i 0.489960i
$$630$$ − 0.956459i − 0.0381063i
$$631$$ − 45.0998i − 1.79539i −0.440614 0.897696i $$-0.645239\pi$$
0.440614 0.897696i $$-0.354761\pi$$
$$632$$ 26.8407i 1.06767i
$$633$$ −19.7385 −0.784537
$$634$$ 11.2233 0.445734
$$635$$ 2.42221i 0.0961223i
$$636$$ 31.6896 1.25658
$$637$$ 0 0
$$638$$ −15.7724 −0.624435
$$639$$ − 17.7536i − 0.702322i
$$640$$ −2.17331 −0.0859075
$$641$$ −32.5821 −1.28692 −0.643458 0.765482i $$-0.722501\pi$$
−0.643458 + 0.765482i $$0.722501\pi$$
$$642$$ − 5.89008i − 0.232463i
$$643$$ 25.5754i 1.00860i 0.863530 + 0.504298i $$0.168249\pi$$
−0.863530 + 0.504298i $$0.831751\pi$$
$$644$$ − 4.77777i − 0.188271i
$$645$$ − 0.164210i − 0.00646578i
$$646$$ −0.152603 −0.00600408
$$647$$ 30.1715 1.18616 0.593082 0.805142i $$-0.297911\pi$$
0.593082 + 0.805142i $$0.297911\pi$$
$$648$$ − 29.4741i − 1.15785i
$$649$$ −28.7952 −1.13031
$$650$$ 0 0
$$651$$ −35.1172 −1.37635
$$652$$ − 21.9259i − 0.858683i
$$653$$ 36.9028 1.44412 0.722058 0.691832i $$-0.243196\pi$$
0.722058 + 0.691832i $$0.243196\pi$$
$$654$$ 28.2935 1.10636
$$655$$ − 1.62432i − 0.0634673i
$$656$$ − 6.43296i − 0.251165i
$$657$$ − 13.8006i − 0.538413i
$$658$$ − 13.9051i − 0.542079i
$$659$$ 23.6866 0.922701 0.461350 0.887218i $$-0.347365\pi$$
0.461350 + 0.887218i $$0.347365\pi$$
$$660$$ 3.19806 0.124484
$$661$$ 31.7590i 1.23528i 0.786460 + 0.617641i $$0.211911\pi$$
−0.786460 + 0.617641i $$0.788089\pi$$
$$662$$ −14.2916 −0.555458
$$663$$ 0 0
$$664$$ −4.33214 −0.168120
$$665$$ 0.0513102i 0.00198973i
$$666$$ 9.35258 0.362405
$$667$$ −6.91856 −0.267888
$$668$$ − 21.8694i − 0.846152i
$$669$$ 5.07606i 0.196252i
$$670$$ − 1.52111i − 0.0587655i
$$671$$ 14.7463i 0.569275i
$$672$$ 30.8702 1.19085
$$673$$ 7.50232 0.289193 0.144597 0.989491i $$-0.453812\pi$$
0.144597 + 0.989491i $$0.453812\pi$$
$$674$$ 22.3327i 0.860225i
$$675$$ 10.5550 0.406261
$$676$$ 0 0
$$677$$ −35.0315 −1.34637 −0.673184 0.739475i $$-0.735074\pi$$
−0.673184 + 0.739475i $$0.735074\pi$$
$$678$$ − 21.7114i − 0.833821i
$$679$$ 18.9933 0.728896
$$680$$ 1.43535 0.0550433
$$681$$ 15.6528i 0.599816i
$$682$$ 22.5840i 0.864787i
$$683$$ 24.0834i 0.921524i 0.887524 + 0.460762i $$0.152424\pi$$
−0.887524 + 0.460762i $$0.847576\pi$$
$$684$$ − 0.245061i − 0.00937013i
$$685$$ 1.53617 0.0586941
$$686$$ −15.9618 −0.609426
$$687$$ − 54.3183i − 2.07237i
$$688$$ 0.164210 0.00626046
$$689$$ 0 0
$$690$$ −0.664874 −0.0253113
$$691$$ 2.01447i 0.0766342i 0.999266 + 0.0383171i $$0.0121997\pi$$
−0.999266 + 0.0383171i $$0.987800\pi$$
$$692$$ 29.2223 1.11087
$$693$$ −20.5090 −0.779073
$$694$$ 1.20583i 0.0457728i
$$695$$ − 3.63235i − 0.137783i
$$696$$ − 28.0127i − 1.06182i
$$697$$ 25.0248i 0.947880i
$$698$$ −11.3763 −0.430598
$$699$$ −6.87800 −0.260150
$$700$$ 15.7952i 0.597004i
$$701$$ 48.8189 1.84387 0.921933 0.387350i $$-0.126610\pi$$
0.921933 + 0.387350i $$0.126610\pi$$
$$702$$ 0 0
$$703$$ −0.501729 −0.0189231
$$704$$ − 15.1390i − 0.570572i
$$705$$ 4.08277 0.153766
$$706$$ 5.74871 0.216355
$$707$$ 31.4752i 1.18375i
$$708$$ − 20.6722i − 0.776908i
$$709$$ 20.8060i 0.781385i 0.920521 + 0.390693i $$0.127764\pi$$
−0.920521 + 0.390693i $$0.872236\pi$$
$$710$$ 1.71618i 0.0644073i
$$711$$ 20.4286 0.766134
$$712$$ −7.76569 −0.291032
$$713$$ 9.90648i 0.371000i
$$714$$ −9.16852 −0.343123
$$715$$ 0 0
$$716$$ −15.5133 −0.579761
$$717$$ 56.5042i 2.11019i
$$718$$ 15.9468 0.595128
$$719$$ −21.4306 −0.799225 −0.399613 0.916684i $$-0.630855\pi$$
−0.399613 + 0.916684i $$0.630855\pi$$
$$720$$ 0.280831i 0.0104660i
$$721$$ − 3.21073i − 0.119574i
$$722$$ 15.2306i 0.566824i
$$723$$ − 45.5381i − 1.69358i
$$724$$ −28.4547 −1.05751
$$725$$ 22.8726 0.849468
$$726$$ 12.6799i 0.470597i
$$727$$ −13.4862 −0.500175 −0.250088 0.968223i $$-0.580459\pi$$
−0.250088 + 0.968223i $$0.580459\pi$$
$$728$$ 0 0
$$729$$ −8.02715 −0.297302
$$730$$ 1.33406i 0.0493758i
$$731$$ −0.638792 −0.0236266
$$732$$ −10.5864 −0.391285
$$733$$ − 43.5424i − 1.60828i −0.594443 0.804138i $$-0.702627\pi$$
0.594443 0.804138i $$-0.297373\pi$$
$$734$$ 0.869167i 0.0320816i
$$735$$ − 0.801938i − 0.0295799i
$$736$$ − 8.70841i − 0.320996i
$$737$$ −32.6165 −1.20145
$$738$$ 19.0465 0.701112
$$739$$ − 20.0543i − 0.737709i −0.929487 0.368855i $$-0.879750\pi$$
0.929487 0.368855i $$-0.120250\pi$$
$$740$$ 1.90754 0.0701226
$$741$$ 0 0
$$742$$ 19.6450 0.721191
$$743$$ − 33.1685i − 1.21684i −0.793617 0.608418i $$-0.791805\pi$$
0.793617 0.608418i $$-0.208195\pi$$
$$744$$ −40.1105 −1.47052
$$745$$ −1.07069 −0.0392270
$$746$$ − 4.91617i − 0.179994i
$$747$$ 3.29722i 0.120639i
$$748$$ − 12.4407i − 0.454878i
$$749$$ 7.70410i 0.281502i
$$750$$ 4.42327 0.161515
$$751$$ −39.2814 −1.43340 −0.716700 0.697382i $$-0.754348\pi$$
−0.716700 + 0.697382i $$0.754348\pi$$
$$752$$ 4.08277i 0.148883i
$$753$$ −53.3008 −1.94239
$$754$$ 0 0
$$755$$ 0.974181 0.0354541
$$756$$ 6.83446i 0.248567i
$$757$$ −46.6426 −1.69526 −0.847628 0.530592i $$-0.821970\pi$$
−0.847628 + 0.530592i $$0.821970\pi$$
$$758$$ 1.93171 0.0701627
$$759$$ 14.2567i 0.517484i
$$760$$ 0.0586060i 0.00212586i
$$761$$ 21.8984i 0.793818i 0.917858 + 0.396909i $$0.129917\pi$$
−0.917858 + 0.396909i $$0.870083\pi$$
$$762$$ − 17.6722i − 0.640195i
$$763$$ −37.0073 −1.33975
$$764$$ −19.5899 −0.708737
$$765$$ − 1.09246i − 0.0394979i
$$766$$ 24.3720 0.880595
$$767$$ 0 0
$$768$$ 31.8756 1.15021
$$769$$ 46.7096i 1.68439i 0.539172 + 0.842196i $$0.318737\pi$$
−0.539172 + 0.842196i $$0.681263\pi$$
$$770$$ 1.98254 0.0714458
$$771$$ 31.9614 1.15106
$$772$$ 18.4263i 0.663175i
$$773$$ − 30.2416i − 1.08771i −0.839178 0.543857i $$-0.816963\pi$$
0.839178 0.543857i $$-0.183037\pi$$
$$774$$ 0.486189i 0.0174757i
$$775$$ − 32.7506i − 1.17644i
$$776$$ 21.6939 0.778767
$$777$$ −30.1444 −1.08142
$$778$$ 12.7836i 0.458315i
$$779$$ −1.02177 −0.0366087
$$780$$ 0 0
$$781$$ 36.7995 1.31679
$$782$$ 2.58642i 0.0924901i
$$783$$ 9.89679 0.353682
$$784$$ 0.801938 0.0286406
$$785$$ 1.10023i 0.0392688i
$$786$$ 11.8509i 0.422706i
$$787$$ − 28.7023i − 1.02313i −0.859246 0.511563i $$-0.829067\pi$$
0.859246 0.511563i $$-0.170933\pi$$
$$788$$ − 0.760316i − 0.0270851i
$$789$$ 38.4131 1.36754
$$790$$ −1.97477 −0.0702592
$$791$$ 28.3980i 1.00972i
$$792$$ −23.4252 −0.832378
$$793$$ 0 0
$$794$$ 13.5646 0.481391
$$795$$ 5.76809i 0.204573i
$$796$$ −15.5929 −0.552674
$$797$$ 18.5418 0.656785 0.328392 0.944541i $$-0.393493\pi$$
0.328392 + 0.944541i $$0.393493\pi$$
$$798$$ − 0.374354i − 0.0132520i
$$799$$ − 15.8823i − 0.561876i
$$800$$ 28.7899i 1.01788i
$$801$$ 5.91053i 0.208838i
$$802$$ 21.3817 0.755012
$$803$$ 28.6058 1.00948
$$804$$ − 23.4155i − 0.825801i
$$805$$ 0.869641 0.0306508
$$806$$ 0 0
$$807$$ 14.5308 0.511508
$$808$$ 35.9506i 1.26474i
$$809$$ −10.0677 −0.353962 −0.176981 0.984214i $$-0.556633\pi$$
−0.176981 + 0.984214i $$0.556633\pi$$
$$810$$ 2.16852 0.0761941
$$811$$ − 10.0285i − 0.352147i −0.984377 0.176074i $$-0.943660\pi$$
0.984377 0.176074i $$-0.0563397\pi$$
$$812$$ 14.8103i 0.519740i
$$813$$ 14.4886i 0.508137i
$$814$$ 19.3860i 0.679478i
$$815$$ 3.99090 0.139795
$$816$$ 2.69202 0.0942396
$$817$$ − 0.0260821i 0 0.000912498i
$$818$$ −22.8683 −0.799572
$$819$$ 0 0
$$820$$ 3.88471 0.135660
$$821$$ − 26.1704i − 0.913355i −0.889632 0.456677i $$-0.849039\pi$$
0.889632 0.456677i $$-0.150961\pi$$
$$822$$ −11.2078 −0.390915
$$823$$ −1.82238 −0.0635242 −0.0317621 0.999495i $$-0.510112\pi$$
−0.0317621 + 0.999495i $$0.510112\pi$$
$$824$$ − 3.66727i − 0.127755i
$$825$$ − 47.1323i − 1.64094i
$$826$$ − 12.8151i − 0.445894i
$$827$$ 32.2941i 1.12298i 0.827485 + 0.561488i $$0.189771\pi$$
−0.827485 + 0.561488i $$0.810229\pi$$
$$828$$ −4.15346 −0.144343
$$829$$ −15.1002 −0.524453 −0.262226 0.965006i $$-0.584457\pi$$
−0.262226 + 0.965006i $$0.584457\pi$$
$$830$$ − 0.318732i − 0.0110634i
$$831$$ 30.2446 1.04917
$$832$$ 0 0
$$833$$ −3.11960 −0.108088
$$834$$ 26.5013i 0.917663i
$$835$$ 3.98062 0.137755
$$836$$ 0.507960 0.0175682
$$837$$ − 14.1709i − 0.489818i
$$838$$ − 23.7448i − 0.820250i
$$839$$ 32.9965i 1.13917i 0.821933 + 0.569584i $$0.192896\pi$$
−0.821933 + 0.569584i $$0.807104\pi$$
$$840$$ 3.52111i 0.121490i
$$841$$ −7.55363 −0.260470
$$842$$ 9.35105 0.322258
$$843$$ 11.3177i 0.389801i
$$844$$ 11.9196 0.410290
$$845$$ 0 0
$$846$$ −12.0881 −0.415599
$$847$$ − 16.5851i − 0.569870i
$$848$$ −5.76809 −0.198077
$$849$$ 49.7211 1.70642
$$850$$ − 8.55065i − 0.293285i
$$851$$ 8.50365i 0.291501i
$$852$$ 26.4185i 0.905082i
$$853$$ 37.7802i 1.29357i 0.762673 + 0.646784i $$0.223887\pi$$
−0.762673 + 0.646784i $$0.776113\pi$$
$$854$$ −6.56273 −0.224572
$$855$$ 0.0446055 0.00152547
$$856$$ 8.79954i 0.300762i
$$857$$ −27.3623 −0.934677 −0.467339 0.884078i $$-0.654787\pi$$
−0.467339 + 0.884078i $$0.654787\pi$$
$$858$$ 0 0
$$859$$ −20.0629 −0.684538 −0.342269 0.939602i $$-0.611195\pi$$
−0.342269 + 0.939602i $$0.611195\pi$$
$$860$$ 0.0991626i 0.00338142i
$$861$$ −61.3889 −2.09213
$$862$$ −3.48773 −0.118792
$$863$$ − 6.14483i − 0.209173i −0.994516 0.104586i $$-0.966648\pi$$
0.994516 0.104586i $$-0.0333518\pi$$
$$864$$ 12.4571i 0.423800i
$$865$$ 5.31900i 0.180851i
$$866$$ 11.5386i 0.392096i
$$867$$ 27.7265 0.941640
$$868$$ 21.2064 0.719793
$$869$$ 42.3443i 1.43643i
$$870$$ 2.06100 0.0698744
$$871$$ 0 0
$$872$$ −42.2693 −1.43142
$$873$$ − 16.5114i − 0.558827i
$$874$$ −0.105604 −0.00357212
$$875$$ −5.78554 −0.195587
$$876$$ 20.5362i 0.693853i
$$877$$ − 13.5077i − 0.456123i −0.973647 0.228061i $$-0.926761\pi$$
0.973647 0.228061i $$-0.0732386\pi$$
$$878$$ 16.2252i 0.547574i
$$879$$ 33.5840i 1.13276i
$$880$$ −0.582105 −0.0196228
$$881$$ 5.23431 0.176348 0.0881741 0.996105i $$-0.471897\pi$$
0.0881741 + 0.996105i $$0.471897\pi$$
$$882$$ 2.37435i 0.0799487i
$$883$$ 4.57301 0.153894 0.0769470 0.997035i $$-0.475483\pi$$
0.0769470 + 0.997035i $$0.475483\pi$$
$$884$$ 0 0
$$885$$ 3.76271 0.126482
$$886$$ 6.51334i 0.218820i
$$887$$ −1.64071 −0.0550897 −0.0275448 0.999621i $$-0.508769\pi$$
−0.0275448 + 0.999621i $$0.508769\pi$$
$$888$$ −34.4306 −1.15541
$$889$$ 23.1148i 0.775246i
$$890$$ − 0.571352i − 0.0191517i
$$891$$ − 46.4989i − 1.55777i
$$892$$ − 3.06531i − 0.102634i
$$893$$ 0.648481 0.0217006
$$894$$ 7.81163 0.261260
$$895$$ − 2.82371i − 0.0943861i
$$896$$ −20.7396 −0.692862
$$897$$ 0 0
$$898$$ 10.0175 0.334287
$$899$$ − 30.7084i − 1.02418i
$$900$$ 13.7313 0.457708
$$901$$ 22.4383 0.747529
$$902$$ 39.4795i 1.31452i
$$903$$ − 1.56704i − 0.0521478i
$$904$$ 32.4359i 1.07880i
$$905$$ − 5.17928i − 0.172165i
$$906$$ −7.10752 −0.236132
$$907$$ −8.10215 −0.269027 −0.134514 0.990912i $$-0.542947\pi$$
−0.134514 + 0.990912i $$0.542947\pi$$
$$908$$ − 9.45234i − 0.313687i
$$909$$ 27.3623 0.907549
$$910$$ 0 0
$$911$$ −9.18119 −0.304187 −0.152093 0.988366i $$-0.548601\pi$$
−0.152093 + 0.988366i $$0.548601\pi$$
$$912$$ 0.109916i 0.00363969i
$$913$$ −6.83446 −0.226188
$$914$$ −4.79656 −0.158656
$$915$$ − 1.92692i − 0.0637020i
$$916$$ 32.8015i 1.08379i
$$917$$ − 15.5007i − 0.511877i
$$918$$ − 3.69979i − 0.122111i
$$919$$ 27.5036 0.907262 0.453631 0.891190i $$-0.350128\pi$$
0.453631 + 0.891190i $$0.350128\pi$$
$$920$$ 0.993295 0.0327480
$$921$$ 42.9831i 1.41634i
$$922$$ 1.64933 0.0543180
$$923$$ 0 0
$$924$$ 30.5187 1.00399
$$925$$ − 28.1129i − 0.924346i
$$926$$ −6.77586 −0.222668
$$927$$ −2.79118 −0.0916745
$$928$$ 26.9946i 0.886142i
$$929$$ 24.2131i 0.794407i 0.917731 + 0.397203i $$0.130019\pi$$
−0.917731 + 0.397203i $$0.869981\pi$$
$$930$$ − 2.95108i − 0.0967698i
$$931$$ − 0.127375i − 0.00417454i
$$932$$ 4.15346 0.136051
$$933$$ −0.606268 −0.0198483
$$934$$ − 26.8761i − 0.879412i
$$935$$ 2.26444 0.0740550
$$936$$ 0 0
$$937$$ 11.1830 0.365333 0.182666 0.983175i $$-0.441527\pi$$
0.182666 + 0.983175i $$0.441527\pi$$
$$938$$ − 14.5157i − 0.473955i
$$939$$ 52.5652 1.71540
$$940$$ −2.46548 −0.0804152
$$941$$ − 15.9638i − 0.520404i −0.965554 0.260202i $$-0.916211\pi$$
0.965554 0.260202i $$-0.0837891\pi$$
$$942$$ − 8.02715i − 0.261539i
$$943$$ 17.3177i 0.563941i
$$944$$ 3.76271i 0.122466i
$$945$$ −1.24400 −0.0404672
$$946$$ −1.00777 −0.0327654
$$947$$ − 6.51466i − 0.211698i −0.994382 0.105849i $$-0.966244\pi$$
0.994382 0.105849i $$-0.0337560\pi$$
$$948$$ −30.3991 −0.987317
$$949$$ 0 0
$$950$$ 0.349126 0.0113271
$$951$$ 31.4470i 1.01974i
$$952$$ 13.6974 0.443935
$$953$$ −47.6469 −1.54344 −0.771718 0.635965i $$-0.780602\pi$$
−0.771718 + 0.635965i $$0.780602\pi$$
$$954$$ − 17.0780i − 0.552920i
$$955$$ − 3.56571i − 0.115384i
$$956$$ − 34.1215i − 1.10357i
$$957$$ − 44.1933i − 1.42857i
$$958$$ −19.8329 −0.640773
$$959$$ 14.6595 0.473380
$$960$$ 1.97823i 0.0638471i
$$961$$ −12.9705 −0.418402
$$962$$ 0 0
$$963$$ 6.69740 0.215821
$$964$$ 27.4993i 0.885694i
$$965$$ −3.35391 −0.107966
$$966$$ −6.34481 −0.204141
$$967$$ − 43.8122i − 1.40891i −0.709751 0.704453i $$-0.751192\pi$$
0.709751 0.704453i $$-0.248808\pi$$
$$968$$ − 18.9433i − 0.608861i
$$969$$ − 0.427583i − 0.0137360i
$$970$$ 1.59611i 0.0512479i
$$971$$ 4.29483 0.137828 0.0689139 0.997623i $$-0.478047\pi$$
0.0689139 + 0.997623i $$0.478047\pi$$
$$972$$ 24.6823 0.791686
$$973$$ − 34.6631i − 1.11125i
$$974$$ 30.2776 0.970156
$$975$$ 0 0
$$976$$ 1.92692 0.0616792
$$977$$ 26.8019i 0.857470i 0.903430 + 0.428735i $$0.141041\pi$$
−0.903430 + 0.428735i $$0.858959\pi$$
$$978$$ −29.1172 −0.931066
$$979$$ −12.2513 −0.391553
$$980$$ 0.484271i 0.0154695i
$$981$$ 32.1715i 1.02716i
$$982$$ − 25.1094i − 0.801275i
$$983$$ − 27.2495i − 0.869124i −0.900642 0.434562i $$-0.856903\pi$$
0.900642 0.434562i $$-0.143097\pi$$
$$984$$ −70.1178 −2.23527
$$985$$ 0.138391 0.00440951
$$986$$ − 8.01746i − 0.255328i
$$987$$ 38.9614 1.24015
$$988$$ 0 0
$$989$$ −0.442058 −0.0140566
$$990$$ − 1.72348i − 0.0547758i
$$991$$ 24.3889 0.774740 0.387370 0.921924i $$-0.373384\pi$$
0.387370 + 0.921924i $$0.373384\pi$$
$$992$$ 38.6528 1.22723
$$993$$ − 40.0441i − 1.27076i
$$994$$ 16.3773i 0.519458i
$$995$$ − 2.83818i − 0.0899764i
$$996$$ − 4.90648i − 0.155468i
$$997$$ 31.3207 0.991935 0.495967 0.868341i $$-0.334814\pi$$
0.495967 + 0.868341i $$0.334814\pi$$
$$998$$ −17.2314 −0.545452
$$999$$ − 12.1642i − 0.384859i
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 169.2.b.b.168.5 6
3.2 odd 2 1521.2.b.l.1351.2 6
4.3 odd 2 2704.2.f.o.337.6 6
13.2 odd 12 169.2.c.c.22.1 6
13.3 even 3 169.2.e.b.147.2 12
13.4 even 6 169.2.e.b.23.2 12
13.5 odd 4 169.2.a.b.1.3 3
13.6 odd 12 169.2.c.c.146.1 6
13.7 odd 12 169.2.c.b.146.3 6
13.8 odd 4 169.2.a.c.1.1 yes 3
13.9 even 3 169.2.e.b.23.5 12
13.10 even 6 169.2.e.b.147.5 12
13.11 odd 12 169.2.c.b.22.3 6
13.12 even 2 inner 169.2.b.b.168.2 6
39.5 even 4 1521.2.a.r.1.1 3
39.8 even 4 1521.2.a.o.1.3 3
39.38 odd 2 1521.2.b.l.1351.5 6
52.31 even 4 2704.2.a.z.1.3 3
52.47 even 4 2704.2.a.ba.1.3 3
52.51 odd 2 2704.2.f.o.337.5 6
65.34 odd 4 4225.2.a.bb.1.3 3
65.44 odd 4 4225.2.a.bg.1.1 3
91.34 even 4 8281.2.a.bj.1.1 3
91.83 even 4 8281.2.a.bf.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 13.5 odd 4
169.2.a.c.1.1 yes 3 13.8 odd 4
169.2.b.b.168.2 6 13.12 even 2 inner
169.2.b.b.168.5 6 1.1 even 1 trivial
169.2.c.b.22.3 6 13.11 odd 12
169.2.c.b.146.3 6 13.7 odd 12
169.2.c.c.22.1 6 13.2 odd 12
169.2.c.c.146.1 6 13.6 odd 12
169.2.e.b.23.2 12 13.4 even 6
169.2.e.b.23.5 12 13.9 even 3
169.2.e.b.147.2 12 13.3 even 3
169.2.e.b.147.5 12 13.10 even 6
1521.2.a.o.1.3 3 39.8 even 4
1521.2.a.r.1.1 3 39.5 even 4
1521.2.b.l.1351.2 6 3.2 odd 2
1521.2.b.l.1351.5 6 39.38 odd 2
2704.2.a.z.1.3 3 52.31 even 4
2704.2.a.ba.1.3 3 52.47 even 4
2704.2.f.o.337.5 6 52.51 odd 2
2704.2.f.o.337.6 6 4.3 odd 2
4225.2.a.bb.1.3 3 65.34 odd 4
4225.2.a.bg.1.1 3 65.44 odd 4
8281.2.a.bf.1.3 3 91.83 even 4
8281.2.a.bj.1.1 3 91.34 even 4