# Properties

 Label 9405.2.a.z.1.3 Level $9405$ Weight $2$ Character 9405.1 Self dual yes Analytic conductor $75.099$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9405.a (trivial)

## 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: yes Analytic conductor: $$75.0993031010$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7281497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ x^6 - 2*x^5 - 5*x^4 + 7*x^3 + 6*x^2 - 2*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.77015$$ of defining polynomial Character $$\chi$$ $$=$$ 9405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.205229 q^{2} -1.95788 q^{4} +1.00000 q^{5} +2.33231 q^{7} +0.812271 q^{8} +O(q^{10})$$ $$q-0.205229 q^{2} -1.95788 q^{4} +1.00000 q^{5} +2.33231 q^{7} +0.812271 q^{8} -0.205229 q^{10} +1.00000 q^{11} -4.30769 q^{13} -0.478657 q^{14} +3.74906 q^{16} -6.71096 q^{17} -1.00000 q^{19} -1.95788 q^{20} -0.205229 q^{22} -8.24907 q^{23} +1.00000 q^{25} +0.884062 q^{26} -4.56638 q^{28} -3.02355 q^{29} -10.2077 q^{31} -2.39396 q^{32} +1.37728 q^{34} +2.33231 q^{35} -5.06393 q^{37} +0.205229 q^{38} +0.812271 q^{40} +6.47966 q^{41} +1.64130 q^{43} -1.95788 q^{44} +1.69295 q^{46} +8.15593 q^{47} -1.56033 q^{49} -0.205229 q^{50} +8.43395 q^{52} +2.35693 q^{53} +1.00000 q^{55} +1.89447 q^{56} +0.620519 q^{58} +4.65880 q^{59} +8.93722 q^{61} +2.09491 q^{62} -7.00681 q^{64} -4.30769 q^{65} +13.2994 q^{67} +13.1393 q^{68} -0.478657 q^{70} +8.76859 q^{71} -2.66639 q^{73} +1.03926 q^{74} +1.95788 q^{76} +2.33231 q^{77} -12.1246 q^{79} +3.74906 q^{80} -1.32981 q^{82} +17.8672 q^{83} -6.71096 q^{85} -0.336843 q^{86} +0.812271 q^{88} +1.27875 q^{89} -10.0469 q^{91} +16.1507 q^{92} -1.67383 q^{94} -1.00000 q^{95} +14.9748 q^{97} +0.320225 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10})$$ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 $$6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100})$$ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.205229 −0.145119 −0.0725593 0.997364i $$-0.523117\pi$$
−0.0725593 + 0.997364i $$0.523117\pi$$
$$3$$ 0 0
$$4$$ −1.95788 −0.978941
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.33231 0.881530 0.440765 0.897623i $$-0.354707\pi$$
0.440765 + 0.897623i $$0.354707\pi$$
$$8$$ 0.812271 0.287181
$$9$$ 0 0
$$10$$ −0.205229 −0.0648990
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −4.30769 −1.19474 −0.597369 0.801966i $$-0.703787\pi$$
−0.597369 + 0.801966i $$0.703787\pi$$
$$14$$ −0.478657 −0.127926
$$15$$ 0 0
$$16$$ 3.74906 0.937265
$$17$$ −6.71096 −1.62765 −0.813824 0.581111i $$-0.802618\pi$$
−0.813824 + 0.581111i $$0.802618\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ −1.95788 −0.437796
$$21$$ 0 0
$$22$$ −0.205229 −0.0437549
$$23$$ −8.24907 −1.72005 −0.860025 0.510252i $$-0.829552\pi$$
−0.860025 + 0.510252i $$0.829552\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.884062 0.173379
$$27$$ 0 0
$$28$$ −4.56638 −0.862966
$$29$$ −3.02355 −0.561459 −0.280730 0.959787i $$-0.590576\pi$$
−0.280730 + 0.959787i $$0.590576\pi$$
$$30$$ 0 0
$$31$$ −10.2077 −1.83335 −0.916677 0.399628i $$-0.869139\pi$$
−0.916677 + 0.399628i $$0.869139\pi$$
$$32$$ −2.39396 −0.423196
$$33$$ 0 0
$$34$$ 1.37728 0.236202
$$35$$ 2.33231 0.394232
$$36$$ 0 0
$$37$$ −5.06393 −0.832506 −0.416253 0.909249i $$-0.636657\pi$$
−0.416253 + 0.909249i $$0.636657\pi$$
$$38$$ 0.205229 0.0332925
$$39$$ 0 0
$$40$$ 0.812271 0.128431
$$41$$ 6.47966 1.01195 0.505976 0.862548i $$-0.331132\pi$$
0.505976 + 0.862548i $$0.331132\pi$$
$$42$$ 0 0
$$43$$ 1.64130 0.250296 0.125148 0.992138i $$-0.460059\pi$$
0.125148 + 0.992138i $$0.460059\pi$$
$$44$$ −1.95788 −0.295162
$$45$$ 0 0
$$46$$ 1.69295 0.249611
$$47$$ 8.15593 1.18966 0.594832 0.803850i $$-0.297219\pi$$
0.594832 + 0.803850i $$0.297219\pi$$
$$48$$ 0 0
$$49$$ −1.56033 −0.222905
$$50$$ −0.205229 −0.0290237
$$51$$ 0 0
$$52$$ 8.43395 1.16958
$$53$$ 2.35693 0.323749 0.161875 0.986811i $$-0.448246\pi$$
0.161875 + 0.986811i $$0.448246\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 1.89447 0.253159
$$57$$ 0 0
$$58$$ 0.620519 0.0814782
$$59$$ 4.65880 0.606525 0.303262 0.952907i $$-0.401924\pi$$
0.303262 + 0.952907i $$0.401924\pi$$
$$60$$ 0 0
$$61$$ 8.93722 1.14429 0.572147 0.820151i $$-0.306111\pi$$
0.572147 + 0.820151i $$0.306111\pi$$
$$62$$ 2.09491 0.266054
$$63$$ 0 0
$$64$$ −7.00681 −0.875852
$$65$$ −4.30769 −0.534303
$$66$$ 0 0
$$67$$ 13.2994 1.62478 0.812392 0.583111i $$-0.198165\pi$$
0.812392 + 0.583111i $$0.198165\pi$$
$$68$$ 13.1393 1.59337
$$69$$ 0 0
$$70$$ −0.478657 −0.0572104
$$71$$ 8.76859 1.04064 0.520320 0.853971i $$-0.325813\pi$$
0.520320 + 0.853971i $$0.325813\pi$$
$$72$$ 0 0
$$73$$ −2.66639 −0.312077 −0.156038 0.987751i $$-0.549872\pi$$
−0.156038 + 0.987751i $$0.549872\pi$$
$$74$$ 1.03926 0.120812
$$75$$ 0 0
$$76$$ 1.95788 0.224584
$$77$$ 2.33231 0.265791
$$78$$ 0 0
$$79$$ −12.1246 −1.36413 −0.682065 0.731292i $$-0.738918\pi$$
−0.682065 + 0.731292i $$0.738918\pi$$
$$80$$ 3.74906 0.419158
$$81$$ 0 0
$$82$$ −1.32981 −0.146853
$$83$$ 17.8672 1.96117 0.980587 0.196082i $$-0.0628220\pi$$
0.980587 + 0.196082i $$0.0628220\pi$$
$$84$$ 0 0
$$85$$ −6.71096 −0.727906
$$86$$ −0.336843 −0.0363227
$$87$$ 0 0
$$88$$ 0.812271 0.0865884
$$89$$ 1.27875 0.135548 0.0677739 0.997701i $$-0.478410\pi$$
0.0677739 + 0.997701i $$0.478410\pi$$
$$90$$ 0 0
$$91$$ −10.0469 −1.05320
$$92$$ 16.1507 1.68383
$$93$$ 0 0
$$94$$ −1.67383 −0.172642
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 14.9748 1.52046 0.760230 0.649654i $$-0.225086\pi$$
0.760230 + 0.649654i $$0.225086\pi$$
$$98$$ 0.320225 0.0323476
$$99$$ 0 0
$$100$$ −1.95788 −0.195788
$$101$$ −0.465796 −0.0463484 −0.0231742 0.999731i $$-0.507377\pi$$
−0.0231742 + 0.999731i $$0.507377\pi$$
$$102$$ 0 0
$$103$$ 12.7767 1.25892 0.629462 0.777031i $$-0.283275\pi$$
0.629462 + 0.777031i $$0.283275\pi$$
$$104$$ −3.49901 −0.343106
$$105$$ 0 0
$$106$$ −0.483709 −0.0469820
$$107$$ −3.81929 −0.369225 −0.184612 0.982811i $$-0.559103\pi$$
−0.184612 + 0.982811i $$0.559103\pi$$
$$108$$ 0 0
$$109$$ 9.14480 0.875913 0.437956 0.898996i $$-0.355702\pi$$
0.437956 + 0.898996i $$0.355702\pi$$
$$110$$ −0.205229 −0.0195678
$$111$$ 0 0
$$112$$ 8.74397 0.826228
$$113$$ −1.55378 −0.146167 −0.0730837 0.997326i $$-0.523284\pi$$
−0.0730837 + 0.997326i $$0.523284\pi$$
$$114$$ 0 0
$$115$$ −8.24907 −0.769230
$$116$$ 5.91975 0.549635
$$117$$ 0 0
$$118$$ −0.956120 −0.0880180
$$119$$ −15.6520 −1.43482
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −1.83417 −0.166058
$$123$$ 0 0
$$124$$ 19.9854 1.79475
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −6.95455 −0.617117 −0.308558 0.951205i $$-0.599846\pi$$
−0.308558 + 0.951205i $$0.599846\pi$$
$$128$$ 6.22591 0.550298
$$129$$ 0 0
$$130$$ 0.884062 0.0775373
$$131$$ 12.3544 1.07941 0.539703 0.841855i $$-0.318537\pi$$
0.539703 + 0.841855i $$0.318537\pi$$
$$132$$ 0 0
$$133$$ −2.33231 −0.202237
$$134$$ −2.72943 −0.235787
$$135$$ 0 0
$$136$$ −5.45112 −0.467430
$$137$$ 5.14851 0.439867 0.219934 0.975515i $$-0.429416\pi$$
0.219934 + 0.975515i $$0.429416\pi$$
$$138$$ 0 0
$$139$$ −6.46045 −0.547968 −0.273984 0.961734i $$-0.588342\pi$$
−0.273984 + 0.961734i $$0.588342\pi$$
$$140$$ −4.56638 −0.385930
$$141$$ 0 0
$$142$$ −1.79957 −0.151016
$$143$$ −4.30769 −0.360227
$$144$$ 0 0
$$145$$ −3.02355 −0.251092
$$146$$ 0.547219 0.0452882
$$147$$ 0 0
$$148$$ 9.91458 0.814974
$$149$$ −1.25732 −0.103004 −0.0515019 0.998673i $$-0.516401\pi$$
−0.0515019 + 0.998673i $$0.516401\pi$$
$$150$$ 0 0
$$151$$ −3.48878 −0.283913 −0.141956 0.989873i $$-0.545339\pi$$
−0.141956 + 0.989873i $$0.545339\pi$$
$$152$$ −0.812271 −0.0658839
$$153$$ 0 0
$$154$$ −0.478657 −0.0385713
$$155$$ −10.2077 −0.819901
$$156$$ 0 0
$$157$$ 6.34241 0.506180 0.253090 0.967443i $$-0.418553\pi$$
0.253090 + 0.967443i $$0.418553\pi$$
$$158$$ 2.48832 0.197960
$$159$$ 0 0
$$160$$ −2.39396 −0.189259
$$161$$ −19.2394 −1.51628
$$162$$ 0 0
$$163$$ −23.4566 −1.83726 −0.918630 0.395119i $$-0.870704\pi$$
−0.918630 + 0.395119i $$0.870704\pi$$
$$164$$ −12.6864 −0.990641
$$165$$ 0 0
$$166$$ −3.66685 −0.284603
$$167$$ −6.70854 −0.519122 −0.259561 0.965727i $$-0.583578\pi$$
−0.259561 + 0.965727i $$0.583578\pi$$
$$168$$ 0 0
$$169$$ 5.55619 0.427399
$$170$$ 1.37728 0.105633
$$171$$ 0 0
$$172$$ −3.21348 −0.245025
$$173$$ 8.18749 0.622483 0.311242 0.950331i $$-0.399255\pi$$
0.311242 + 0.950331i $$0.399255\pi$$
$$174$$ 0 0
$$175$$ 2.33231 0.176306
$$176$$ 3.74906 0.282596
$$177$$ 0 0
$$178$$ −0.262437 −0.0196705
$$179$$ 13.0297 0.973885 0.486943 0.873434i $$-0.338112\pi$$
0.486943 + 0.873434i $$0.338112\pi$$
$$180$$ 0 0
$$181$$ −13.6260 −1.01281 −0.506407 0.862294i $$-0.669027\pi$$
−0.506407 + 0.862294i $$0.669027\pi$$
$$182$$ 2.06190 0.152839
$$183$$ 0 0
$$184$$ −6.70048 −0.493966
$$185$$ −5.06393 −0.372308
$$186$$ 0 0
$$187$$ −6.71096 −0.490754
$$188$$ −15.9683 −1.16461
$$189$$ 0 0
$$190$$ 0.205229 0.0148889
$$191$$ −2.87940 −0.208346 −0.104173 0.994559i $$-0.533220\pi$$
−0.104173 + 0.994559i $$0.533220\pi$$
$$192$$ 0 0
$$193$$ 23.3069 1.67767 0.838834 0.544387i $$-0.183238\pi$$
0.838834 + 0.544387i $$0.183238\pi$$
$$194$$ −3.07326 −0.220647
$$195$$ 0 0
$$196$$ 3.05495 0.218211
$$197$$ 21.4910 1.53117 0.765585 0.643335i $$-0.222450\pi$$
0.765585 + 0.643335i $$0.222450\pi$$
$$198$$ 0 0
$$199$$ −11.8592 −0.840678 −0.420339 0.907367i $$-0.638089\pi$$
−0.420339 + 0.907367i $$0.638089\pi$$
$$200$$ 0.812271 0.0574362
$$201$$ 0 0
$$202$$ 0.0955946 0.00672601
$$203$$ −7.05185 −0.494943
$$204$$ 0 0
$$205$$ 6.47966 0.452559
$$206$$ −2.62214 −0.182693
$$207$$ 0 0
$$208$$ −16.1498 −1.11979
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −12.2769 −0.845175 −0.422587 0.906322i $$-0.638878\pi$$
−0.422587 + 0.906322i $$0.638878\pi$$
$$212$$ −4.61459 −0.316931
$$213$$ 0 0
$$214$$ 0.783828 0.0535814
$$215$$ 1.64130 0.111936
$$216$$ 0 0
$$217$$ −23.8075 −1.61616
$$218$$ −1.87678 −0.127111
$$219$$ 0 0
$$220$$ −1.95788 −0.132000
$$221$$ 28.9088 1.94461
$$222$$ 0 0
$$223$$ 18.1532 1.21563 0.607813 0.794080i $$-0.292047\pi$$
0.607813 + 0.794080i $$0.292047\pi$$
$$224$$ −5.58345 −0.373060
$$225$$ 0 0
$$226$$ 0.318880 0.0212116
$$227$$ 18.9555 1.25812 0.629061 0.777356i $$-0.283440\pi$$
0.629061 + 0.777356i $$0.283440\pi$$
$$228$$ 0 0
$$229$$ −16.3134 −1.07802 −0.539009 0.842300i $$-0.681201\pi$$
−0.539009 + 0.842300i $$0.681201\pi$$
$$230$$ 1.69295 0.111630
$$231$$ 0 0
$$232$$ −2.45594 −0.161240
$$233$$ 22.9853 1.50582 0.752910 0.658124i $$-0.228650\pi$$
0.752910 + 0.658124i $$0.228650\pi$$
$$234$$ 0 0
$$235$$ 8.15593 0.532034
$$236$$ −9.12138 −0.593751
$$237$$ 0 0
$$238$$ 3.21225 0.208219
$$239$$ 17.2793 1.11771 0.558853 0.829267i $$-0.311241\pi$$
0.558853 + 0.829267i $$0.311241\pi$$
$$240$$ 0 0
$$241$$ 13.4260 0.864843 0.432422 0.901672i $$-0.357659\pi$$
0.432422 + 0.901672i $$0.357659\pi$$
$$242$$ −0.205229 −0.0131926
$$243$$ 0 0
$$244$$ −17.4980 −1.12020
$$245$$ −1.56033 −0.0996860
$$246$$ 0 0
$$247$$ 4.30769 0.274092
$$248$$ −8.29141 −0.526505
$$249$$ 0 0
$$250$$ −0.205229 −0.0129798
$$251$$ −1.20422 −0.0760099 −0.0380050 0.999278i $$-0.512100\pi$$
−0.0380050 + 0.999278i $$0.512100\pi$$
$$252$$ 0 0
$$253$$ −8.24907 −0.518615
$$254$$ 1.42727 0.0895551
$$255$$ 0 0
$$256$$ 12.7359 0.795993
$$257$$ 17.1513 1.06987 0.534935 0.844893i $$-0.320336\pi$$
0.534935 + 0.844893i $$0.320336\pi$$
$$258$$ 0 0
$$259$$ −11.8107 −0.733879
$$260$$ 8.43395 0.523051
$$261$$ 0 0
$$262$$ −2.53547 −0.156642
$$263$$ −6.60962 −0.407567 −0.203783 0.979016i $$-0.565324\pi$$
−0.203783 + 0.979016i $$0.565324\pi$$
$$264$$ 0 0
$$265$$ 2.35693 0.144785
$$266$$ 0.478657 0.0293483
$$267$$ 0 0
$$268$$ −26.0387 −1.59057
$$269$$ 2.85724 0.174209 0.0871046 0.996199i $$-0.472239\pi$$
0.0871046 + 0.996199i $$0.472239\pi$$
$$270$$ 0 0
$$271$$ −1.28250 −0.0779062 −0.0389531 0.999241i $$-0.512402\pi$$
−0.0389531 + 0.999241i $$0.512402\pi$$
$$272$$ −25.1598 −1.52554
$$273$$ 0 0
$$274$$ −1.05662 −0.0638329
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 16.2492 0.976322 0.488161 0.872753i $$-0.337668\pi$$
0.488161 + 0.872753i $$0.337668\pi$$
$$278$$ 1.32587 0.0795204
$$279$$ 0 0
$$280$$ 1.89447 0.113216
$$281$$ 29.4723 1.75817 0.879085 0.476664i $$-0.158154\pi$$
0.879085 + 0.476664i $$0.158154\pi$$
$$282$$ 0 0
$$283$$ −32.9008 −1.95575 −0.977874 0.209196i $$-0.932915\pi$$
−0.977874 + 0.209196i $$0.932915\pi$$
$$284$$ −17.1679 −1.01872
$$285$$ 0 0
$$286$$ 0.884062 0.0522757
$$287$$ 15.1126 0.892066
$$288$$ 0 0
$$289$$ 28.0370 1.64924
$$290$$ 0.620519 0.0364381
$$291$$ 0 0
$$292$$ 5.22047 0.305505
$$293$$ −27.9704 −1.63405 −0.817023 0.576605i $$-0.804377\pi$$
−0.817023 + 0.576605i $$0.804377\pi$$
$$294$$ 0 0
$$295$$ 4.65880 0.271246
$$296$$ −4.11329 −0.239080
$$297$$ 0 0
$$298$$ 0.258038 0.0149478
$$299$$ 35.5344 2.05501
$$300$$ 0 0
$$301$$ 3.82803 0.220644
$$302$$ 0.715997 0.0412010
$$303$$ 0 0
$$304$$ −3.74906 −0.215023
$$305$$ 8.93722 0.511744
$$306$$ 0 0
$$307$$ −0.783038 −0.0446903 −0.0223452 0.999750i $$-0.507113\pi$$
−0.0223452 + 0.999750i $$0.507113\pi$$
$$308$$ −4.56638 −0.260194
$$309$$ 0 0
$$310$$ 2.09491 0.118983
$$311$$ 10.0442 0.569555 0.284777 0.958594i $$-0.408080\pi$$
0.284777 + 0.958594i $$0.408080\pi$$
$$312$$ 0 0
$$313$$ −1.80765 −0.102174 −0.0510872 0.998694i $$-0.516269\pi$$
−0.0510872 + 0.998694i $$0.516269\pi$$
$$314$$ −1.30165 −0.0734561
$$315$$ 0 0
$$316$$ 23.7386 1.33540
$$317$$ −14.2824 −0.802182 −0.401091 0.916038i $$-0.631369\pi$$
−0.401091 + 0.916038i $$0.631369\pi$$
$$318$$ 0 0
$$319$$ −3.02355 −0.169286
$$320$$ −7.00681 −0.391693
$$321$$ 0 0
$$322$$ 3.94847 0.220040
$$323$$ 6.71096 0.373408
$$324$$ 0 0
$$325$$ −4.30769 −0.238948
$$326$$ 4.81396 0.266621
$$327$$ 0 0
$$328$$ 5.26323 0.290614
$$329$$ 19.0221 1.04872
$$330$$ 0 0
$$331$$ −14.4996 −0.796969 −0.398484 0.917175i $$-0.630464\pi$$
−0.398484 + 0.917175i $$0.630464\pi$$
$$332$$ −34.9818 −1.91987
$$333$$ 0 0
$$334$$ 1.37678 0.0753343
$$335$$ 13.2994 0.726626
$$336$$ 0 0
$$337$$ 17.0292 0.927638 0.463819 0.885930i $$-0.346479\pi$$
0.463819 + 0.885930i $$0.346479\pi$$
$$338$$ −1.14029 −0.0620236
$$339$$ 0 0
$$340$$ 13.1393 0.712577
$$341$$ −10.2077 −0.552777
$$342$$ 0 0
$$343$$ −19.9653 −1.07803
$$344$$ 1.33318 0.0718804
$$345$$ 0 0
$$346$$ −1.68031 −0.0903339
$$347$$ 12.4889 0.670439 0.335220 0.942140i $$-0.391189\pi$$
0.335220 + 0.942140i $$0.391189\pi$$
$$348$$ 0 0
$$349$$ −20.9482 −1.12133 −0.560667 0.828042i $$-0.689455\pi$$
−0.560667 + 0.828042i $$0.689455\pi$$
$$350$$ −0.478657 −0.0255853
$$351$$ 0 0
$$352$$ −2.39396 −0.127598
$$353$$ 11.3626 0.604769 0.302384 0.953186i $$-0.402217\pi$$
0.302384 + 0.953186i $$0.402217\pi$$
$$354$$ 0 0
$$355$$ 8.76859 0.465388
$$356$$ −2.50365 −0.132693
$$357$$ 0 0
$$358$$ −2.67407 −0.141329
$$359$$ 3.59897 0.189946 0.0949732 0.995480i $$-0.469723\pi$$
0.0949732 + 0.995480i $$0.469723\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 2.79645 0.146978
$$363$$ 0 0
$$364$$ 19.6706 1.03102
$$365$$ −2.66639 −0.139565
$$366$$ 0 0
$$367$$ 22.8321 1.19183 0.595914 0.803048i $$-0.296790\pi$$
0.595914 + 0.803048i $$0.296790\pi$$
$$368$$ −30.9263 −1.61214
$$369$$ 0 0
$$370$$ 1.03926 0.0540288
$$371$$ 5.49709 0.285395
$$372$$ 0 0
$$373$$ −14.5973 −0.755822 −0.377911 0.925842i $$-0.623357\pi$$
−0.377911 + 0.925842i $$0.623357\pi$$
$$374$$ 1.37728 0.0712176
$$375$$ 0 0
$$376$$ 6.62482 0.341649
$$377$$ 13.0245 0.670797
$$378$$ 0 0
$$379$$ 11.8706 0.609752 0.304876 0.952392i $$-0.401385\pi$$
0.304876 + 0.952392i $$0.401385\pi$$
$$380$$ 1.95788 0.100437
$$381$$ 0 0
$$382$$ 0.590936 0.0302349
$$383$$ −12.6086 −0.644268 −0.322134 0.946694i $$-0.604400\pi$$
−0.322134 + 0.946694i $$0.604400\pi$$
$$384$$ 0 0
$$385$$ 2.33231 0.118865
$$386$$ −4.78325 −0.243461
$$387$$ 0 0
$$388$$ −29.3189 −1.48844
$$389$$ −35.3815 −1.79391 −0.896956 0.442120i $$-0.854227\pi$$
−0.896956 + 0.442120i $$0.854227\pi$$
$$390$$ 0 0
$$391$$ 55.3592 2.79964
$$392$$ −1.26741 −0.0640140
$$393$$ 0 0
$$394$$ −4.41057 −0.222201
$$395$$ −12.1246 −0.610057
$$396$$ 0 0
$$397$$ −3.65851 −0.183615 −0.0918076 0.995777i $$-0.529264\pi$$
−0.0918076 + 0.995777i $$0.529264\pi$$
$$398$$ 2.43385 0.121998
$$399$$ 0 0
$$400$$ 3.74906 0.187453
$$401$$ −7.80707 −0.389867 −0.194933 0.980816i $$-0.562449\pi$$
−0.194933 + 0.980816i $$0.562449\pi$$
$$402$$ 0 0
$$403$$ 43.9716 2.19038
$$404$$ 0.911972 0.0453723
$$405$$ 0 0
$$406$$ 1.44724 0.0718255
$$407$$ −5.06393 −0.251010
$$408$$ 0 0
$$409$$ 18.6176 0.920582 0.460291 0.887768i $$-0.347745\pi$$
0.460291 + 0.887768i $$0.347745\pi$$
$$410$$ −1.32981 −0.0656747
$$411$$ 0 0
$$412$$ −25.0152 −1.23241
$$413$$ 10.8658 0.534670
$$414$$ 0 0
$$415$$ 17.8672 0.877064
$$416$$ 10.3124 0.505608
$$417$$ 0 0
$$418$$ 0.205229 0.0100381
$$419$$ −4.36059 −0.213029 −0.106514 0.994311i $$-0.533969\pi$$
−0.106514 + 0.994311i $$0.533969\pi$$
$$420$$ 0 0
$$421$$ 13.8050 0.672813 0.336407 0.941717i $$-0.390788\pi$$
0.336407 + 0.941717i $$0.390788\pi$$
$$422$$ 2.51957 0.122651
$$423$$ 0 0
$$424$$ 1.91446 0.0929746
$$425$$ −6.71096 −0.325530
$$426$$ 0 0
$$427$$ 20.8444 1.00873
$$428$$ 7.47771 0.361449
$$429$$ 0 0
$$430$$ −0.336843 −0.0162440
$$431$$ 22.3219 1.07521 0.537603 0.843198i $$-0.319330\pi$$
0.537603 + 0.843198i $$0.319330\pi$$
$$432$$ 0 0
$$433$$ −32.5287 −1.56323 −0.781614 0.623762i $$-0.785603\pi$$
−0.781614 + 0.623762i $$0.785603\pi$$
$$434$$ 4.88598 0.234535
$$435$$ 0 0
$$436$$ −17.9044 −0.857467
$$437$$ 8.24907 0.394607
$$438$$ 0 0
$$439$$ −11.6856 −0.557723 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$440$$ 0.812271 0.0387235
$$441$$ 0 0
$$442$$ −5.93291 −0.282200
$$443$$ 24.6321 1.17031 0.585154 0.810923i $$-0.301034\pi$$
0.585154 + 0.810923i $$0.301034\pi$$
$$444$$ 0 0
$$445$$ 1.27875 0.0606188
$$446$$ −3.72555 −0.176410
$$447$$ 0 0
$$448$$ −16.3421 −0.772090
$$449$$ 0.271967 0.0128349 0.00641746 0.999979i $$-0.497957\pi$$
0.00641746 + 0.999979i $$0.497957\pi$$
$$450$$ 0 0
$$451$$ 6.47966 0.305115
$$452$$ 3.04212 0.143089
$$453$$ 0 0
$$454$$ −3.89021 −0.182577
$$455$$ −10.0469 −0.471004
$$456$$ 0 0
$$457$$ −26.6203 −1.24524 −0.622622 0.782523i $$-0.713933\pi$$
−0.622622 + 0.782523i $$0.713933\pi$$
$$458$$ 3.34797 0.156440
$$459$$ 0 0
$$460$$ 16.1507 0.753030
$$461$$ −18.5239 −0.862742 −0.431371 0.902175i $$-0.641970\pi$$
−0.431371 + 0.902175i $$0.641970\pi$$
$$462$$ 0 0
$$463$$ 14.9154 0.693178 0.346589 0.938017i $$-0.387340\pi$$
0.346589 + 0.938017i $$0.387340\pi$$
$$464$$ −11.3355 −0.526236
$$465$$ 0 0
$$466$$ −4.71725 −0.218522
$$467$$ −24.4100 −1.12956 −0.564780 0.825242i $$-0.691039\pi$$
−0.564780 + 0.825242i $$0.691039\pi$$
$$468$$ 0 0
$$469$$ 31.0184 1.43230
$$470$$ −1.67383 −0.0772080
$$471$$ 0 0
$$472$$ 3.78421 0.174182
$$473$$ 1.64130 0.0754672
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 30.6448 1.40460
$$477$$ 0 0
$$478$$ −3.54621 −0.162200
$$479$$ −12.0531 −0.550722 −0.275361 0.961341i $$-0.588797\pi$$
−0.275361 + 0.961341i $$0.588797\pi$$
$$480$$ 0 0
$$481$$ 21.8139 0.994626
$$482$$ −2.75540 −0.125505
$$483$$ 0 0
$$484$$ −1.95788 −0.0889946
$$485$$ 14.9748 0.679971
$$486$$ 0 0
$$487$$ 27.5058 1.24641 0.623204 0.782059i $$-0.285831\pi$$
0.623204 + 0.782059i $$0.285831\pi$$
$$488$$ 7.25944 0.328620
$$489$$ 0 0
$$490$$ 0.320225 0.0144663
$$491$$ −33.0788 −1.49283 −0.746413 0.665483i $$-0.768226\pi$$
−0.746413 + 0.665483i $$0.768226\pi$$
$$492$$ 0 0
$$493$$ 20.2909 0.913858
$$494$$ −0.884062 −0.0397758
$$495$$ 0 0
$$496$$ −38.2692 −1.71834
$$497$$ 20.4511 0.917356
$$498$$ 0 0
$$499$$ 9.58033 0.428874 0.214437 0.976738i $$-0.431208\pi$$
0.214437 + 0.976738i $$0.431208\pi$$
$$500$$ −1.95788 −0.0875591
$$501$$ 0 0
$$502$$ 0.247141 0.0110305
$$503$$ −16.0390 −0.715142 −0.357571 0.933886i $$-0.616395\pi$$
−0.357571 + 0.933886i $$0.616395\pi$$
$$504$$ 0 0
$$505$$ −0.465796 −0.0207276
$$506$$ 1.69295 0.0752606
$$507$$ 0 0
$$508$$ 13.6162 0.604121
$$509$$ −11.7844 −0.522334 −0.261167 0.965294i $$-0.584107\pi$$
−0.261167 + 0.965294i $$0.584107\pi$$
$$510$$ 0 0
$$511$$ −6.21884 −0.275105
$$512$$ −15.0656 −0.665811
$$513$$ 0 0
$$514$$ −3.51994 −0.155258
$$515$$ 12.7767 0.563008
$$516$$ 0 0
$$517$$ 8.15593 0.358697
$$518$$ 2.42389 0.106499
$$519$$ 0 0
$$520$$ −3.49901 −0.153442
$$521$$ 7.41167 0.324711 0.162356 0.986732i $$-0.448091\pi$$
0.162356 + 0.986732i $$0.448091\pi$$
$$522$$ 0 0
$$523$$ 5.89974 0.257977 0.128989 0.991646i $$-0.458827\pi$$
0.128989 + 0.991646i $$0.458827\pi$$
$$524$$ −24.1884 −1.05667
$$525$$ 0 0
$$526$$ 1.35648 0.0591455
$$527$$ 68.5034 2.98406
$$528$$ 0 0
$$529$$ 45.0472 1.95857
$$530$$ −0.483709 −0.0210110
$$531$$ 0 0
$$532$$ 4.56638 0.197978
$$533$$ −27.9123 −1.20902
$$534$$ 0 0
$$535$$ −3.81929 −0.165122
$$536$$ 10.8027 0.466608
$$537$$ 0 0
$$538$$ −0.586388 −0.0252810
$$539$$ −1.56033 −0.0672083
$$540$$ 0 0
$$541$$ 23.5417 1.01214 0.506068 0.862493i $$-0.331098\pi$$
0.506068 + 0.862493i $$0.331098\pi$$
$$542$$ 0.263205 0.0113056
$$543$$ 0 0
$$544$$ 16.0658 0.688814
$$545$$ 9.14480 0.391720
$$546$$ 0 0
$$547$$ 1.63464 0.0698923 0.0349461 0.999389i $$-0.488874\pi$$
0.0349461 + 0.999389i $$0.488874\pi$$
$$548$$ −10.0802 −0.430604
$$549$$ 0 0
$$550$$ −0.205229 −0.00875098
$$551$$ 3.02355 0.128808
$$552$$ 0 0
$$553$$ −28.2784 −1.20252
$$554$$ −3.33481 −0.141683
$$555$$ 0 0
$$556$$ 12.6488 0.536428
$$557$$ 9.30739 0.394367 0.197183 0.980367i $$-0.436821\pi$$
0.197183 + 0.980367i $$0.436821\pi$$
$$558$$ 0 0
$$559$$ −7.07023 −0.299039
$$560$$ 8.74397 0.369500
$$561$$ 0 0
$$562$$ −6.04856 −0.255143
$$563$$ 1.61027 0.0678646 0.0339323 0.999424i $$-0.489197\pi$$
0.0339323 + 0.999424i $$0.489197\pi$$
$$564$$ 0 0
$$565$$ −1.55378 −0.0653681
$$566$$ 6.75218 0.283815
$$567$$ 0 0
$$568$$ 7.12247 0.298852
$$569$$ 17.0331 0.714064 0.357032 0.934092i $$-0.383789\pi$$
0.357032 + 0.934092i $$0.383789\pi$$
$$570$$ 0 0
$$571$$ −5.32472 −0.222833 −0.111416 0.993774i $$-0.535539\pi$$
−0.111416 + 0.993774i $$0.535539\pi$$
$$572$$ 8.43395 0.352641
$$573$$ 0 0
$$574$$ −3.10153 −0.129455
$$575$$ −8.24907 −0.344010
$$576$$ 0 0
$$577$$ −12.6976 −0.528608 −0.264304 0.964439i $$-0.585142\pi$$
−0.264304 + 0.964439i $$0.585142\pi$$
$$578$$ −5.75401 −0.239335
$$579$$ 0 0
$$580$$ 5.91975 0.245804
$$581$$ 41.6717 1.72883
$$582$$ 0 0
$$583$$ 2.35693 0.0976140
$$584$$ −2.16583 −0.0896226
$$585$$ 0 0
$$586$$ 5.74032 0.237131
$$587$$ 28.4780 1.17541 0.587707 0.809074i $$-0.300031\pi$$
0.587707 + 0.809074i $$0.300031\pi$$
$$588$$ 0 0
$$589$$ 10.2077 0.420600
$$590$$ −0.956120 −0.0393628
$$591$$ 0 0
$$592$$ −18.9850 −0.780279
$$593$$ 34.7604 1.42744 0.713719 0.700433i $$-0.247010\pi$$
0.713719 + 0.700433i $$0.247010\pi$$
$$594$$ 0 0
$$595$$ −15.6520 −0.641671
$$596$$ 2.46169 0.100835
$$597$$ 0 0
$$598$$ −7.29269 −0.298220
$$599$$ −28.1663 −1.15084 −0.575422 0.817857i $$-0.695162\pi$$
−0.575422 + 0.817857i $$0.695162\pi$$
$$600$$ 0 0
$$601$$ 35.3257 1.44097 0.720484 0.693472i $$-0.243920\pi$$
0.720484 + 0.693472i $$0.243920\pi$$
$$602$$ −0.785621 −0.0320195
$$603$$ 0 0
$$604$$ 6.83061 0.277934
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −20.4938 −0.831816 −0.415908 0.909407i $$-0.636536\pi$$
−0.415908 + 0.909407i $$0.636536\pi$$
$$608$$ 2.39396 0.0970878
$$609$$ 0 0
$$610$$ −1.83417 −0.0742635
$$611$$ −35.1332 −1.42134
$$612$$ 0 0
$$613$$ −2.87465 −0.116106 −0.0580530 0.998314i $$-0.518489\pi$$
−0.0580530 + 0.998314i $$0.518489\pi$$
$$614$$ 0.160702 0.00648540
$$615$$ 0 0
$$616$$ 1.89447 0.0763302
$$617$$ −13.0363 −0.524820 −0.262410 0.964956i $$-0.584517\pi$$
−0.262410 + 0.964956i $$0.584517\pi$$
$$618$$ 0 0
$$619$$ −36.8449 −1.48092 −0.740461 0.672099i $$-0.765393\pi$$
−0.740461 + 0.672099i $$0.765393\pi$$
$$620$$ 19.9854 0.802635
$$621$$ 0 0
$$622$$ −2.06136 −0.0826530
$$623$$ 2.98245 0.119489
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0.370981 0.0148274
$$627$$ 0 0
$$628$$ −12.4177 −0.495520
$$629$$ 33.9839 1.35503
$$630$$ 0 0
$$631$$ −31.5517 −1.25605 −0.628026 0.778192i $$-0.716137\pi$$
−0.628026 + 0.778192i $$0.716137\pi$$
$$632$$ −9.84849 −0.391752
$$633$$ 0 0
$$634$$ 2.93117 0.116412
$$635$$ −6.95455 −0.275983
$$636$$ 0 0
$$637$$ 6.72143 0.266313
$$638$$ 0.620519 0.0245666
$$639$$ 0 0
$$640$$ 6.22591 0.246101
$$641$$ 13.4739 0.532188 0.266094 0.963947i $$-0.414267\pi$$
0.266094 + 0.963947i $$0.414267\pi$$
$$642$$ 0 0
$$643$$ −25.9686 −1.02410 −0.512051 0.858955i $$-0.671114\pi$$
−0.512051 + 0.858955i $$0.671114\pi$$
$$644$$ 37.6684 1.48434
$$645$$ 0 0
$$646$$ −1.37728 −0.0541885
$$647$$ 48.7918 1.91820 0.959101 0.283063i $$-0.0913505\pi$$
0.959101 + 0.283063i $$0.0913505\pi$$
$$648$$ 0 0
$$649$$ 4.65880 0.182874
$$650$$ 0.884062 0.0346757
$$651$$ 0 0
$$652$$ 45.9252 1.79857
$$653$$ −5.61760 −0.219834 −0.109917 0.993941i $$-0.535058\pi$$
−0.109917 + 0.993941i $$0.535058\pi$$
$$654$$ 0 0
$$655$$ 12.3544 0.482725
$$656$$ 24.2926 0.948468
$$657$$ 0 0
$$658$$ −3.90389 −0.152189
$$659$$ 8.87841 0.345854 0.172927 0.984935i $$-0.444678\pi$$
0.172927 + 0.984935i $$0.444678\pi$$
$$660$$ 0 0
$$661$$ −19.6584 −0.764625 −0.382312 0.924033i $$-0.624872\pi$$
−0.382312 + 0.924033i $$0.624872\pi$$
$$662$$ 2.97573 0.115655
$$663$$ 0 0
$$664$$ 14.5130 0.563212
$$665$$ −2.33231 −0.0904431
$$666$$ 0 0
$$667$$ 24.9415 0.965738
$$668$$ 13.1345 0.508190
$$669$$ 0 0
$$670$$ −2.72943 −0.105447
$$671$$ 8.93722 0.345018
$$672$$ 0 0
$$673$$ −14.2191 −0.548106 −0.274053 0.961715i $$-0.588364\pi$$
−0.274053 + 0.961715i $$0.588364\pi$$
$$674$$ −3.49487 −0.134618
$$675$$ 0 0
$$676$$ −10.8784 −0.418399
$$677$$ −51.0566 −1.96226 −0.981132 0.193339i $$-0.938068\pi$$
−0.981132 + 0.193339i $$0.938068\pi$$
$$678$$ 0 0
$$679$$ 34.9259 1.34033
$$680$$ −5.45112 −0.209041
$$681$$ 0 0
$$682$$ 2.09491 0.0802183
$$683$$ 36.1772 1.38428 0.692142 0.721762i $$-0.256667\pi$$
0.692142 + 0.721762i $$0.256667\pi$$
$$684$$ 0 0
$$685$$ 5.14851 0.196715
$$686$$ 4.09746 0.156442
$$687$$ 0 0
$$688$$ 6.15335 0.234594
$$689$$ −10.1529 −0.386795
$$690$$ 0 0
$$691$$ 44.3403 1.68678 0.843392 0.537299i $$-0.180555\pi$$
0.843392 + 0.537299i $$0.180555\pi$$
$$692$$ −16.0301 −0.609374
$$693$$ 0 0
$$694$$ −2.56308 −0.0972932
$$695$$ −6.46045 −0.245059
$$696$$ 0 0
$$697$$ −43.4847 −1.64710
$$698$$ 4.29918 0.162726
$$699$$ 0 0
$$700$$ −4.56638 −0.172593
$$701$$ 19.4759 0.735594 0.367797 0.929906i $$-0.380112\pi$$
0.367797 + 0.929906i $$0.380112\pi$$
$$702$$ 0 0
$$703$$ 5.06393 0.190990
$$704$$ −7.00681 −0.264079
$$705$$ 0 0
$$706$$ −2.33193 −0.0877632
$$707$$ −1.08638 −0.0408575
$$708$$ 0 0
$$709$$ 34.6294 1.30053 0.650266 0.759706i $$-0.274657\pi$$
0.650266 + 0.759706i $$0.274657\pi$$
$$710$$ −1.79957 −0.0675365
$$711$$ 0 0
$$712$$ 1.03870 0.0389268
$$713$$ 84.2039 3.15346
$$714$$ 0 0
$$715$$ −4.30769 −0.161098
$$716$$ −25.5106 −0.953376
$$717$$ 0 0
$$718$$ −0.738612 −0.0275648
$$719$$ 38.7823 1.44633 0.723167 0.690673i $$-0.242686\pi$$
0.723167 + 0.690673i $$0.242686\pi$$
$$720$$ 0 0
$$721$$ 29.7992 1.10978
$$722$$ −0.205229 −0.00763782
$$723$$ 0 0
$$724$$ 26.6781 0.991485
$$725$$ −3.02355 −0.112292
$$726$$ 0 0
$$727$$ −11.1719 −0.414343 −0.207171 0.978305i $$-0.566426\pi$$
−0.207171 + 0.978305i $$0.566426\pi$$
$$728$$ −8.16077 −0.302458
$$729$$ 0 0
$$730$$ 0.547219 0.0202535
$$731$$ −11.0147 −0.407395
$$732$$ 0 0
$$733$$ −34.8431 −1.28696 −0.643479 0.765464i $$-0.722510\pi$$
−0.643479 + 0.765464i $$0.722510\pi$$
$$734$$ −4.68581 −0.172956
$$735$$ 0 0
$$736$$ 19.7479 0.727918
$$737$$ 13.2994 0.489891
$$738$$ 0 0
$$739$$ 32.1054 1.18102 0.590508 0.807032i $$-0.298928\pi$$
0.590508 + 0.807032i $$0.298928\pi$$
$$740$$ 9.91458 0.364467
$$741$$ 0 0
$$742$$ −1.12816 −0.0414161
$$743$$ 27.4635 1.00754 0.503769 0.863838i $$-0.331946\pi$$
0.503769 + 0.863838i $$0.331946\pi$$
$$744$$ 0 0
$$745$$ −1.25732 −0.0460647
$$746$$ 2.99579 0.109684
$$747$$ 0 0
$$748$$ 13.1393 0.480419
$$749$$ −8.90776 −0.325483
$$750$$ 0 0
$$751$$ −13.0173 −0.475007 −0.237504 0.971387i $$-0.576329\pi$$
−0.237504 + 0.971387i $$0.576329\pi$$
$$752$$ 30.5771 1.11503
$$753$$ 0 0
$$754$$ −2.67300 −0.0973451
$$755$$ −3.48878 −0.126970
$$756$$ 0 0
$$757$$ 7.87306 0.286151 0.143076 0.989712i $$-0.454301\pi$$
0.143076 + 0.989712i $$0.454301\pi$$
$$758$$ −2.43619 −0.0884864
$$759$$ 0 0
$$760$$ −0.812271 −0.0294642
$$761$$ −49.9368 −1.81021 −0.905104 0.425189i $$-0.860208\pi$$
−0.905104 + 0.425189i $$0.860208\pi$$
$$762$$ 0 0
$$763$$ 21.3285 0.772144
$$764$$ 5.63753 0.203959
$$765$$ 0 0
$$766$$ 2.58764 0.0934953
$$767$$ −20.0687 −0.724638
$$768$$ 0 0
$$769$$ 20.1374 0.726173 0.363086 0.931755i $$-0.381723\pi$$
0.363086 + 0.931755i $$0.381723\pi$$
$$770$$ −0.478657 −0.0172496
$$771$$ 0 0
$$772$$ −45.6322 −1.64234
$$773$$ −6.99436 −0.251570 −0.125785 0.992058i $$-0.540145\pi$$
−0.125785 + 0.992058i $$0.540145\pi$$
$$774$$ 0 0
$$775$$ −10.2077 −0.366671
$$776$$ 12.1636 0.436648
$$777$$ 0 0
$$778$$ 7.26129 0.260330
$$779$$ −6.47966 −0.232158
$$780$$ 0 0
$$781$$ 8.76859 0.313765
$$782$$ −11.3613 −0.406279
$$783$$ 0 0
$$784$$ −5.84979 −0.208921
$$785$$ 6.34241 0.226370
$$786$$ 0 0
$$787$$ 49.1498 1.75200 0.876000 0.482311i $$-0.160203\pi$$
0.876000 + 0.482311i $$0.160203\pi$$
$$788$$ −42.0768 −1.49892
$$789$$ 0 0
$$790$$ 2.48832 0.0885306
$$791$$ −3.62390 −0.128851
$$792$$ 0 0
$$793$$ −38.4988 −1.36713
$$794$$ 0.750831 0.0266460
$$795$$ 0 0
$$796$$ 23.2190 0.822974
$$797$$ −29.6783 −1.05126 −0.525630 0.850713i $$-0.676170\pi$$
−0.525630 + 0.850713i $$0.676170\pi$$
$$798$$ 0 0
$$799$$ −54.7341 −1.93635
$$800$$ −2.39396 −0.0846391
$$801$$ 0 0
$$802$$ 1.60224 0.0565769
$$803$$ −2.66639 −0.0940947
$$804$$ 0 0
$$805$$ −19.2394 −0.678099
$$806$$ −9.02422 −0.317865
$$807$$ 0 0
$$808$$ −0.378352 −0.0133104
$$809$$ −11.9905 −0.421562 −0.210781 0.977533i $$-0.567601\pi$$
−0.210781 + 0.977533i $$0.567601\pi$$
$$810$$ 0 0
$$811$$ −17.0389 −0.598318 −0.299159 0.954203i $$-0.596706\pi$$
−0.299159 + 0.954203i $$0.596706\pi$$
$$812$$ 13.8067 0.484520
$$813$$ 0 0
$$814$$ 1.03926 0.0364262
$$815$$ −23.4566 −0.821648
$$816$$ 0 0
$$817$$ −1.64130 −0.0574220
$$818$$ −3.82087 −0.133594
$$819$$ 0 0
$$820$$ −12.6864 −0.443028
$$821$$ 4.91990 0.171706 0.0858529 0.996308i $$-0.472638\pi$$
0.0858529 + 0.996308i $$0.472638\pi$$
$$822$$ 0 0
$$823$$ 14.9864 0.522392 0.261196 0.965286i $$-0.415883\pi$$
0.261196 + 0.965286i $$0.415883\pi$$
$$824$$ 10.3781 0.361539
$$825$$ 0 0
$$826$$ −2.22997 −0.0775905
$$827$$ −18.0947 −0.629213 −0.314607 0.949222i $$-0.601873\pi$$
−0.314607 + 0.949222i $$0.601873\pi$$
$$828$$ 0 0
$$829$$ −20.2361 −0.702827 −0.351414 0.936220i $$-0.614299\pi$$
−0.351414 + 0.936220i $$0.614299\pi$$
$$830$$ −3.66685 −0.127278
$$831$$ 0 0
$$832$$ 30.1832 1.04641
$$833$$ 10.4713 0.362810
$$834$$ 0 0
$$835$$ −6.70854 −0.232159
$$836$$ 1.95788 0.0677147
$$837$$ 0 0
$$838$$ 0.894919 0.0309145
$$839$$ 11.4706 0.396008 0.198004 0.980201i $$-0.436554\pi$$
0.198004 + 0.980201i $$0.436554\pi$$
$$840$$ 0 0
$$841$$ −19.8581 −0.684764
$$842$$ −2.83318 −0.0976377
$$843$$ 0 0
$$844$$ 24.0367 0.827376
$$845$$ 5.55619 0.191139
$$846$$ 0 0
$$847$$ 2.33231 0.0801391
$$848$$ 8.83627 0.303439
$$849$$ 0 0
$$850$$ 1.37728 0.0472404
$$851$$ 41.7727 1.43195
$$852$$ 0 0
$$853$$ −48.1136 −1.64738 −0.823690 0.567040i $$-0.808088\pi$$
−0.823690 + 0.567040i $$0.808088\pi$$
$$854$$ −4.27786 −0.146385
$$855$$ 0 0
$$856$$ −3.10230 −0.106034
$$857$$ −16.5752 −0.566197 −0.283099 0.959091i $$-0.591362\pi$$
−0.283099 + 0.959091i $$0.591362\pi$$
$$858$$ 0 0
$$859$$ −8.14570 −0.277928 −0.138964 0.990297i $$-0.544377\pi$$
−0.138964 + 0.990297i $$0.544377\pi$$
$$860$$ −3.21348 −0.109579
$$861$$ 0 0
$$862$$ −4.58109 −0.156032
$$863$$ 2.28140 0.0776597 0.0388298 0.999246i $$-0.487637\pi$$
0.0388298 + 0.999246i $$0.487637\pi$$
$$864$$ 0 0
$$865$$ 8.18749 0.278383
$$866$$ 6.67582 0.226853
$$867$$ 0 0
$$868$$ 46.6122 1.58212
$$869$$ −12.1246 −0.411300
$$870$$ 0 0
$$871$$ −57.2899 −1.94119
$$872$$ 7.42805 0.251546
$$873$$ 0 0
$$874$$ −1.69295 −0.0572647
$$875$$ 2.33231 0.0788464
$$876$$ 0 0
$$877$$ 31.4497 1.06198 0.530991 0.847378i $$-0.321820\pi$$
0.530991 + 0.847378i $$0.321820\pi$$
$$878$$ 2.39822 0.0809360
$$879$$ 0 0
$$880$$ 3.74906 0.126381
$$881$$ 29.6609 0.999301 0.499651 0.866227i $$-0.333462\pi$$
0.499651 + 0.866227i $$0.333462\pi$$
$$882$$ 0 0
$$883$$ 44.7897 1.50729 0.753647 0.657279i $$-0.228293\pi$$
0.753647 + 0.657279i $$0.228293\pi$$
$$884$$ −56.5999 −1.90366
$$885$$ 0 0
$$886$$ −5.05522 −0.169833
$$887$$ 51.6475 1.73415 0.867077 0.498175i $$-0.165996\pi$$
0.867077 + 0.498175i $$0.165996\pi$$
$$888$$ 0 0
$$889$$ −16.2202 −0.544007
$$890$$ −0.262437 −0.00879692
$$891$$ 0 0
$$892$$ −35.5417 −1.19003
$$893$$ −8.15593 −0.272928
$$894$$ 0 0
$$895$$ 13.0297 0.435535
$$896$$ 14.5208 0.485104
$$897$$ 0 0
$$898$$ −0.0558154 −0.00186258
$$899$$ 30.8635 1.02935
$$900$$ 0 0
$$901$$ −15.8173 −0.526949
$$902$$ −1.32981 −0.0442779
$$903$$ 0 0
$$904$$ −1.26209 −0.0419765
$$905$$ −13.6260 −0.452945
$$906$$ 0 0
$$907$$ 24.5110 0.813874 0.406937 0.913456i $$-0.366597\pi$$
0.406937 + 0.913456i $$0.366597\pi$$
$$908$$ −37.1126 −1.23163
$$909$$ 0 0
$$910$$ 2.06190 0.0683515
$$911$$ −50.0214 −1.65728 −0.828641 0.559781i $$-0.810885\pi$$
−0.828641 + 0.559781i $$0.810885\pi$$
$$912$$ 0 0
$$913$$ 17.8672 0.591316
$$914$$ 5.46324 0.180708
$$915$$ 0 0
$$916$$ 31.9396 1.05532
$$917$$ 28.8142 0.951529
$$918$$ 0 0
$$919$$ −30.0880 −0.992511 −0.496255 0.868177i $$-0.665292\pi$$
−0.496255 + 0.868177i $$0.665292\pi$$
$$920$$ −6.70048 −0.220908
$$921$$ 0 0
$$922$$ 3.80163 0.125200
$$923$$ −37.7724 −1.24329
$$924$$ 0 0
$$925$$ −5.06393 −0.166501
$$926$$ −3.06107 −0.100593
$$927$$ 0 0
$$928$$ 7.23825 0.237607
$$929$$ −13.6447 −0.447669 −0.223834 0.974627i $$-0.571857\pi$$
−0.223834 + 0.974627i $$0.571857\pi$$
$$930$$ 0 0
$$931$$ 1.56033 0.0511379
$$932$$ −45.0026 −1.47411
$$933$$ 0 0
$$934$$ 5.00963 0.163920
$$935$$ −6.71096 −0.219472
$$936$$ 0 0
$$937$$ 18.1311 0.592318 0.296159 0.955139i $$-0.404294\pi$$
0.296159 + 0.955139i $$0.404294\pi$$
$$938$$ −6.36587 −0.207853
$$939$$ 0 0
$$940$$ −15.9683 −0.520830
$$941$$ 48.2974 1.57445 0.787224 0.616667i $$-0.211517\pi$$
0.787224 + 0.616667i $$0.211517\pi$$
$$942$$ 0 0
$$943$$ −53.4511 −1.74061
$$944$$ 17.4661 0.568474
$$945$$ 0 0
$$946$$ −0.336843 −0.0109517
$$947$$ −14.7816 −0.480337 −0.240169 0.970731i $$-0.577203\pi$$
−0.240169 + 0.970731i $$0.577203\pi$$
$$948$$ 0 0
$$949$$ 11.4860 0.372850
$$950$$ 0.205229 0.00665850
$$951$$ 0 0
$$952$$ −12.7137 −0.412053
$$953$$ 16.9908 0.550387 0.275193 0.961389i $$-0.411258\pi$$
0.275193 + 0.961389i $$0.411258\pi$$
$$954$$ 0 0
$$955$$ −2.87940 −0.0931752
$$956$$ −33.8309 −1.09417
$$957$$ 0 0
$$958$$ 2.47365 0.0799200
$$959$$ 12.0079 0.387756
$$960$$ 0 0
$$961$$ 73.1969 2.36119
$$962$$ −4.47683 −0.144339
$$963$$ 0 0
$$964$$ −26.2865 −0.846630
$$965$$ 23.3069 0.750276
$$966$$ 0 0
$$967$$ −27.7887 −0.893626 −0.446813 0.894627i $$-0.647441\pi$$
−0.446813 + 0.894627i $$0.647441\pi$$
$$968$$ 0.812271 0.0261074
$$969$$ 0 0
$$970$$ −3.07326 −0.0986764
$$971$$ −24.0365 −0.771367 −0.385683 0.922631i $$-0.626034\pi$$
−0.385683 + 0.922631i $$0.626034\pi$$
$$972$$ 0 0
$$973$$ −15.0678 −0.483051
$$974$$ −5.64499 −0.180877
$$975$$ 0 0
$$976$$ 33.5062 1.07251
$$977$$ 13.6409 0.436412 0.218206 0.975903i $$-0.429980\pi$$
0.218206 + 0.975903i $$0.429980\pi$$
$$978$$ 0 0
$$979$$ 1.27875 0.0408692
$$980$$ 3.05495 0.0975867
$$981$$ 0 0
$$982$$ 6.78872 0.216637
$$983$$ −7.81613 −0.249296 −0.124648 0.992201i $$-0.539780\pi$$
−0.124648 + 0.992201i $$0.539780\pi$$
$$984$$ 0 0
$$985$$ 21.4910 0.684760
$$986$$ −4.16428 −0.132618
$$987$$ 0 0
$$988$$ −8.43395 −0.268320
$$989$$ −13.5392 −0.430522
$$990$$ 0 0
$$991$$ −44.7494 −1.42151 −0.710756 0.703439i $$-0.751647\pi$$
−0.710756 + 0.703439i $$0.751647\pi$$
$$992$$ 24.4368 0.775868
$$993$$ 0 0
$$994$$ −4.19714 −0.133125
$$995$$ −11.8592 −0.375963
$$996$$ 0 0
$$997$$ 31.3614 0.993227 0.496613 0.867972i $$-0.334577\pi$$
0.496613 + 0.867972i $$0.334577\pi$$
$$998$$ −1.96616 −0.0622376
$$999$$ 0 0
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 9405.2.a.z.1.3 6
3.2 odd 2 1045.2.a.f.1.4 6
15.14 odd 2 5225.2.a.l.1.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.f.1.4 6 3.2 odd 2
5225.2.a.l.1.3 6 15.14 odd 2
9405.2.a.z.1.3 6 1.1 even 1 trivial