Properties

 Label 475.2.b.b.324.2 Level $475$ Weight $2$ Character 475.324 Analytic conductor $3.793$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.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: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.1827904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 14x^{2} + 1$$ x^6 + 9*x^4 + 14*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 324.2 Root $$-1.37720i$$ of defining polynomial Character $$\chi$$ $$=$$ 475.324 Dual form 475.2.b.b.324.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.65109i q^{2} -2.37720i q^{3} -0.726109 q^{4} -3.92498 q^{6} -0.377203i q^{7} -2.10331i q^{8} -2.65109 q^{9} +O(q^{10})$$ $$q-1.65109i q^{2} -2.37720i q^{3} -0.726109 q^{4} -3.92498 q^{6} -0.377203i q^{7} -2.10331i q^{8} -2.65109 q^{9} -1.37720 q^{11} +1.72611i q^{12} -2.82167i q^{13} -0.622797 q^{14} -4.92498 q^{16} +6.37720i q^{17} +4.37720i q^{18} -1.00000 q^{19} -0.896688 q^{21} +2.27389i q^{22} -6.19887i q^{23} -5.00000 q^{24} -4.65884 q^{26} -0.829422i q^{27} +0.273891i q^{28} +3.37720 q^{29} +2.48052 q^{31} +3.92498i q^{32} +3.27389i q^{33} +10.5294 q^{34} +1.92498 q^{36} +5.58383i q^{37} +1.65109i q^{38} -6.70769 q^{39} +8.50106 q^{41} +1.48052i q^{42} +12.1522i q^{43} +1.00000 q^{44} -10.2349 q^{46} -6.87826i q^{47} +11.7077i q^{48} +6.85772 q^{49} +15.1599 q^{51} +2.04884i q^{52} -11.5478i q^{53} -1.36945 q^{54} -0.793375 q^{56} +2.37720i q^{57} -5.57608i q^{58} -6.05659 q^{59} +5.02830 q^{61} -4.09556i q^{62} +1.00000i q^{63} -3.36945 q^{64} +5.40550 q^{66} +3.22717i q^{67} -4.63055i q^{68} -14.7360 q^{69} -2.30219 q^{71} +5.57608i q^{72} +3.19887i q^{73} +9.21942 q^{74} +0.726109 q^{76} +0.519485i q^{77} +11.0750i q^{78} +6.71836 q^{79} -9.92498 q^{81} -14.0360i q^{82} -18.2165i q^{83} +0.651093 q^{84} +20.0643 q^{86} -8.02830i q^{87} +2.89669i q^{88} -1.50106 q^{89} -1.06434 q^{91} +4.50106i q^{92} -5.89669i q^{93} -11.3567 q^{94} +9.33048 q^{96} -11.7827i q^{97} -11.3227i q^{98} +3.65109 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10})$$ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 $$6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100})$$ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.65109i − 1.16750i −0.811934 0.583750i $$-0.801585\pi$$
0.811934 0.583750i $$-0.198415\pi$$
$$3$$ − 2.37720i − 1.37248i −0.727376 0.686239i $$-0.759260\pi$$
0.727376 0.686239i $$-0.240740\pi$$
$$4$$ −0.726109 −0.363055
$$5$$ 0 0
$$6$$ −3.92498 −1.60237
$$7$$ − 0.377203i − 0.142569i −0.997456 0.0712846i $$-0.977290\pi$$
0.997456 0.0712846i $$-0.0227099\pi$$
$$8$$ − 2.10331i − 0.743633i
$$9$$ −2.65109 −0.883698
$$10$$ 0 0
$$11$$ −1.37720 −0.415242 −0.207621 0.978209i $$-0.566572\pi$$
−0.207621 + 0.978209i $$0.566572\pi$$
$$12$$ 1.72611i 0.498285i
$$13$$ − 2.82167i − 0.782591i −0.920265 0.391295i $$-0.872027\pi$$
0.920265 0.391295i $$-0.127973\pi$$
$$14$$ −0.622797 −0.166450
$$15$$ 0 0
$$16$$ −4.92498 −1.23125
$$17$$ 6.37720i 1.54670i 0.633980 + 0.773349i $$0.281420\pi$$
−0.633980 + 0.773349i $$0.718580\pi$$
$$18$$ 4.37720i 1.03172i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −0.896688 −0.195673
$$22$$ 2.27389i 0.484795i
$$23$$ − 6.19887i − 1.29255i −0.763103 0.646277i $$-0.776325\pi$$
0.763103 0.646277i $$-0.223675\pi$$
$$24$$ −5.00000 −1.02062
$$25$$ 0 0
$$26$$ −4.65884 −0.913674
$$27$$ − 0.829422i − 0.159622i
$$28$$ 0.273891i 0.0517604i
$$29$$ 3.37720 0.627131 0.313565 0.949567i $$-0.398476\pi$$
0.313565 + 0.949567i $$0.398476\pi$$
$$30$$ 0 0
$$31$$ 2.48052 0.445514 0.222757 0.974874i $$-0.428494\pi$$
0.222757 + 0.974874i $$0.428494\pi$$
$$32$$ 3.92498i 0.693846i
$$33$$ 3.27389i 0.569911i
$$34$$ 10.5294 1.80577
$$35$$ 0 0
$$36$$ 1.92498 0.320831
$$37$$ 5.58383i 0.917976i 0.888443 + 0.458988i $$0.151788\pi$$
−0.888443 + 0.458988i $$0.848212\pi$$
$$38$$ 1.65109i 0.267843i
$$39$$ −6.70769 −1.07409
$$40$$ 0 0
$$41$$ 8.50106 1.32764 0.663821 0.747891i $$-0.268934\pi$$
0.663821 + 0.747891i $$0.268934\pi$$
$$42$$ 1.48052i 0.228448i
$$43$$ 12.1522i 1.85319i 0.376065 + 0.926593i $$0.377277\pi$$
−0.376065 + 0.926593i $$0.622723\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −10.2349 −1.50906
$$47$$ − 6.87826i − 1.00330i −0.865071 0.501649i $$-0.832727\pi$$
0.865071 0.501649i $$-0.167273\pi$$
$$48$$ 11.7077i 1.68986i
$$49$$ 6.85772 0.979674
$$50$$ 0 0
$$51$$ 15.1599 2.12281
$$52$$ 2.04884i 0.284123i
$$53$$ − 11.5478i − 1.58621i −0.609085 0.793105i $$-0.708463\pi$$
0.609085 0.793105i $$-0.291537\pi$$
$$54$$ −1.36945 −0.186359
$$55$$ 0 0
$$56$$ −0.793375 −0.106019
$$57$$ 2.37720i 0.314868i
$$58$$ − 5.57608i − 0.732175i
$$59$$ −6.05659 −0.788501 −0.394251 0.919003i $$-0.628996\pi$$
−0.394251 + 0.919003i $$0.628996\pi$$
$$60$$ 0 0
$$61$$ 5.02830 0.643807 0.321904 0.946772i $$-0.395677\pi$$
0.321904 + 0.946772i $$0.395677\pi$$
$$62$$ − 4.09556i − 0.520137i
$$63$$ 1.00000i 0.125988i
$$64$$ −3.36945 −0.421182
$$65$$ 0 0
$$66$$ 5.40550 0.665371
$$67$$ 3.22717i 0.394262i 0.980377 + 0.197131i $$0.0631624\pi$$
−0.980377 + 0.197131i $$0.936838\pi$$
$$68$$ − 4.63055i − 0.561536i
$$69$$ −14.7360 −1.77400
$$70$$ 0 0
$$71$$ −2.30219 −0.273219 −0.136610 0.990625i $$-0.543621\pi$$
−0.136610 + 0.990625i $$0.543621\pi$$
$$72$$ 5.57608i 0.657147i
$$73$$ 3.19887i 0.374400i 0.982322 + 0.187200i $$0.0599412\pi$$
−0.982322 + 0.187200i $$0.940059\pi$$
$$74$$ 9.21942 1.07174
$$75$$ 0 0
$$76$$ 0.726109 0.0832905
$$77$$ 0.519485i 0.0592008i
$$78$$ 11.0750i 1.25400i
$$79$$ 6.71836 0.755874 0.377937 0.925831i $$-0.376634\pi$$
0.377937 + 0.925831i $$0.376634\pi$$
$$80$$ 0 0
$$81$$ −9.92498 −1.10278
$$82$$ − 14.0360i − 1.55002i
$$83$$ − 18.2165i − 1.99952i −0.0218996 0.999760i $$-0.506971\pi$$
0.0218996 0.999760i $$-0.493029\pi$$
$$84$$ 0.651093 0.0710401
$$85$$ 0 0
$$86$$ 20.0643 2.16359
$$87$$ − 8.02830i − 0.860724i
$$88$$ 2.89669i 0.308788i
$$89$$ −1.50106 −0.159112 −0.0795561 0.996830i $$-0.525350\pi$$
−0.0795561 + 0.996830i $$0.525350\pi$$
$$90$$ 0 0
$$91$$ −1.06434 −0.111573
$$92$$ 4.50106i 0.469268i
$$93$$ − 5.89669i − 0.611458i
$$94$$ −11.3567 −1.17135
$$95$$ 0 0
$$96$$ 9.33048 0.952288
$$97$$ − 11.7827i − 1.19635i −0.801365 0.598176i $$-0.795892\pi$$
0.801365 0.598176i $$-0.204108\pi$$
$$98$$ − 11.3227i − 1.14377i
$$99$$ 3.65109 0.366949
$$100$$ 0 0
$$101$$ −9.18820 −0.914260 −0.457130 0.889400i $$-0.651123\pi$$
−0.457130 + 0.889400i $$0.651123\pi$$
$$102$$ − 25.0304i − 2.47838i
$$103$$ 9.47277i 0.933379i 0.884421 + 0.466690i $$0.154553\pi$$
−0.884421 + 0.466690i $$0.845447\pi$$
$$104$$ −5.93486 −0.581961
$$105$$ 0 0
$$106$$ −19.0665 −1.85190
$$107$$ − 15.3510i − 1.48404i −0.670378 0.742020i $$-0.733868\pi$$
0.670378 0.742020i $$-0.266132\pi$$
$$108$$ 0.602251i 0.0579516i
$$109$$ 16.7643 1.60573 0.802863 0.596163i $$-0.203309\pi$$
0.802863 + 0.596163i $$0.203309\pi$$
$$110$$ 0 0
$$111$$ 13.2739 1.25990
$$112$$ 1.85772i 0.175538i
$$113$$ 13.3305i 1.25403i 0.779009 + 0.627013i $$0.215723\pi$$
−0.779009 + 0.627013i $$0.784277\pi$$
$$114$$ 3.92498 0.367608
$$115$$ 0 0
$$116$$ −2.45222 −0.227683
$$117$$ 7.48052i 0.691574i
$$118$$ 10.0000i 0.920575i
$$119$$ 2.40550 0.220512
$$120$$ 0 0
$$121$$ −9.10331 −0.827574
$$122$$ − 8.30219i − 0.751645i
$$123$$ − 20.2087i − 1.82216i
$$124$$ −1.80113 −0.161746
$$125$$ 0 0
$$126$$ 1.65109 0.147091
$$127$$ − 3.04672i − 0.270353i −0.990822 0.135176i $$-0.956840\pi$$
0.990822 0.135176i $$-0.0431601\pi$$
$$128$$ 13.4132i 1.18557i
$$129$$ 28.8881 2.54346
$$130$$ 0 0
$$131$$ −8.70769 −0.760794 −0.380397 0.924823i $$-0.624213\pi$$
−0.380397 + 0.924823i $$0.624213\pi$$
$$132$$ − 2.37720i − 0.206909i
$$133$$ 0.377203i 0.0327076i
$$134$$ 5.32836 0.460300
$$135$$ 0 0
$$136$$ 13.4132 1.15018
$$137$$ 6.59450i 0.563406i 0.959502 + 0.281703i $$0.0908993\pi$$
−0.959502 + 0.281703i $$0.909101\pi$$
$$138$$ 24.3305i 2.07115i
$$139$$ 10.1677 0.862409 0.431205 0.902254i $$-0.358089\pi$$
0.431205 + 0.902254i $$0.358089\pi$$
$$140$$ 0 0
$$141$$ −16.3510 −1.37701
$$142$$ 3.80113i 0.318983i
$$143$$ 3.88601i 0.324965i
$$144$$ 13.0566 1.08805
$$145$$ 0 0
$$146$$ 5.28164 0.437112
$$147$$ − 16.3022i − 1.34458i
$$148$$ − 4.05447i − 0.333275i
$$149$$ 5.54003 0.453857 0.226929 0.973911i $$-0.427132\pi$$
0.226929 + 0.973911i $$0.427132\pi$$
$$150$$ 0 0
$$151$$ 5.12386 0.416974 0.208487 0.978025i $$-0.433146\pi$$
0.208487 + 0.978025i $$0.433146\pi$$
$$152$$ 2.10331i 0.170601i
$$153$$ − 16.9066i − 1.36681i
$$154$$ 0.857718 0.0691169
$$155$$ 0 0
$$156$$ 4.87051 0.389953
$$157$$ 19.7643i 1.57736i 0.614803 + 0.788681i $$0.289235\pi$$
−0.614803 + 0.788681i $$0.710765\pi$$
$$158$$ − 11.0926i − 0.882483i
$$159$$ −27.4514 −2.17704
$$160$$ 0 0
$$161$$ −2.33823 −0.184279
$$162$$ 16.3871i 1.28749i
$$163$$ − 13.5195i − 1.05893i −0.848332 0.529464i $$-0.822393\pi$$
0.848332 0.529464i $$-0.177607\pi$$
$$164$$ −6.17270 −0.482007
$$165$$ 0 0
$$166$$ −30.0771 −2.33444
$$167$$ 13.1054i 1.01413i 0.861908 + 0.507065i $$0.169269\pi$$
−0.861908 + 0.507065i $$0.830731\pi$$
$$168$$ 1.88601i 0.145509i
$$169$$ 5.03817 0.387551
$$170$$ 0 0
$$171$$ 2.65109 0.202734
$$172$$ − 8.82379i − 0.672808i
$$173$$ − 1.48827i − 0.113151i −0.998398 0.0565754i $$-0.981982\pi$$
0.998398 0.0565754i $$-0.0180181\pi$$
$$174$$ −13.2555 −1.00489
$$175$$ 0 0
$$176$$ 6.78270 0.511265
$$177$$ 14.3977i 1.08220i
$$178$$ 2.47839i 0.185763i
$$179$$ 17.1132 1.27910 0.639550 0.768750i $$-0.279121\pi$$
0.639550 + 0.768750i $$0.279121\pi$$
$$180$$ 0 0
$$181$$ −5.73891 −0.426569 −0.213285 0.976990i $$-0.568416\pi$$
−0.213285 + 0.976990i $$0.568416\pi$$
$$182$$ 1.75733i 0.130262i
$$183$$ − 11.9533i − 0.883612i
$$184$$ −13.0382 −0.961187
$$185$$ 0 0
$$186$$ −9.73598 −0.713877
$$187$$ − 8.78270i − 0.642255i
$$188$$ 4.99437i 0.364252i
$$189$$ −0.312860 −0.0227572
$$190$$ 0 0
$$191$$ 22.3948 1.62043 0.810216 0.586131i $$-0.199350\pi$$
0.810216 + 0.586131i $$0.199350\pi$$
$$192$$ 8.00987i 0.578063i
$$193$$ 21.3687i 1.53815i 0.639159 + 0.769075i $$0.279283\pi$$
−0.639159 + 0.769075i $$0.720717\pi$$
$$194$$ −19.4543 −1.39674
$$195$$ 0 0
$$196$$ −4.97945 −0.355675
$$197$$ 1.11399i 0.0793682i 0.999212 + 0.0396841i $$0.0126352\pi$$
−0.999212 + 0.0396841i $$0.987365\pi$$
$$198$$ − 6.02830i − 0.428412i
$$199$$ 2.22505 0.157729 0.0788647 0.996885i $$-0.474870\pi$$
0.0788647 + 0.996885i $$0.474870\pi$$
$$200$$ 0 0
$$201$$ 7.67164 0.541116
$$202$$ 15.1706i 1.06740i
$$203$$ − 1.27389i − 0.0894096i
$$204$$ −11.0078 −0.770697
$$205$$ 0 0
$$206$$ 15.6404 1.08972
$$207$$ 16.4338i 1.14223i
$$208$$ 13.8967i 0.963562i
$$209$$ 1.37720 0.0952631
$$210$$ 0 0
$$211$$ −9.75441 −0.671521 −0.335760 0.941947i $$-0.608993\pi$$
−0.335760 + 0.941947i $$0.608993\pi$$
$$212$$ 8.38495i 0.575881i
$$213$$ 5.47277i 0.374988i
$$214$$ −25.3460 −1.73262
$$215$$ 0 0
$$216$$ −1.74453 −0.118700
$$217$$ − 0.935657i − 0.0635166i
$$218$$ − 27.6794i − 1.87468i
$$219$$ 7.60437 0.513856
$$220$$ 0 0
$$221$$ 17.9944 1.21043
$$222$$ − 21.9164i − 1.47093i
$$223$$ 18.6433i 1.24845i 0.781244 + 0.624225i $$0.214585\pi$$
−0.781244 + 0.624225i $$0.785415\pi$$
$$224$$ 1.48052 0.0989211
$$225$$ 0 0
$$226$$ 22.0099 1.46407
$$227$$ − 8.04672i − 0.534080i −0.963686 0.267040i $$-0.913954\pi$$
0.963686 0.267040i $$-0.0860455\pi$$
$$228$$ − 1.72611i − 0.114314i
$$229$$ −20.1415 −1.33099 −0.665493 0.746404i $$-0.731779\pi$$
−0.665493 + 0.746404i $$0.731779\pi$$
$$230$$ 0 0
$$231$$ 1.23492 0.0812518
$$232$$ − 7.10331i − 0.466355i
$$233$$ 1.73386i 0.113589i 0.998386 + 0.0567945i $$0.0180880\pi$$
−0.998386 + 0.0567945i $$0.981912\pi$$
$$234$$ 12.3510 0.807412
$$235$$ 0 0
$$236$$ 4.39775 0.286269
$$237$$ − 15.9709i − 1.03742i
$$238$$ − 3.97170i − 0.257447i
$$239$$ 18.3150 1.18470 0.592349 0.805682i $$-0.298201\pi$$
0.592349 + 0.805682i $$0.298201\pi$$
$$240$$ 0 0
$$241$$ −2.19675 −0.141505 −0.0707526 0.997494i $$-0.522540\pi$$
−0.0707526 + 0.997494i $$0.522540\pi$$
$$242$$ 15.0304i 0.966192i
$$243$$ 21.1054i 1.35391i
$$244$$ −3.65109 −0.233737
$$245$$ 0 0
$$246$$ −33.3665 −2.12737
$$247$$ 2.82167i 0.179539i
$$248$$ − 5.21730i − 0.331299i
$$249$$ −43.3043 −2.74430
$$250$$ 0 0
$$251$$ −13.6716 −0.862946 −0.431473 0.902126i $$-0.642006\pi$$
−0.431473 + 0.902126i $$0.642006\pi$$
$$252$$ − 0.726109i − 0.0457406i
$$253$$ 8.53711i 0.536723i
$$254$$ −5.03042 −0.315637
$$255$$ 0 0
$$256$$ 15.4076 0.962976
$$257$$ 28.4904i 1.77718i 0.458701 + 0.888591i $$0.348315\pi$$
−0.458701 + 0.888591i $$0.651685\pi$$
$$258$$ − 47.6970i − 2.96949i
$$259$$ 2.10624 0.130875
$$260$$ 0 0
$$261$$ −8.95328 −0.554194
$$262$$ 14.3772i 0.888227i
$$263$$ − 5.36945i − 0.331095i −0.986202 0.165547i $$-0.947061\pi$$
0.986202 0.165547i $$-0.0529391\pi$$
$$264$$ 6.88601 0.423805
$$265$$ 0 0
$$266$$ 0.622797 0.0381861
$$267$$ 3.56833i 0.218378i
$$268$$ − 2.34328i − 0.143139i
$$269$$ −7.06727 −0.430899 −0.215449 0.976515i $$-0.569122\pi$$
−0.215449 + 0.976515i $$0.569122\pi$$
$$270$$ 0 0
$$271$$ 21.6970 1.31800 0.659000 0.752143i $$-0.270980\pi$$
0.659000 + 0.752143i $$0.270980\pi$$
$$272$$ − 31.4076i − 1.90437i
$$273$$ 2.53016i 0.153132i
$$274$$ 10.8881 0.657776
$$275$$ 0 0
$$276$$ 10.6999 0.644060
$$277$$ 25.1132i 1.50891i 0.656355 + 0.754453i $$0.272098\pi$$
−0.656355 + 0.754453i $$0.727902\pi$$
$$278$$ − 16.7877i − 1.00686i
$$279$$ −6.57608 −0.393699
$$280$$ 0 0
$$281$$ −10.0411 −0.599001 −0.299501 0.954096i $$-0.596820\pi$$
−0.299501 + 0.954096i $$0.596820\pi$$
$$282$$ 26.9971i 1.60765i
$$283$$ 20.9143i 1.24323i 0.783324 + 0.621613i $$0.213522\pi$$
−0.783324 + 0.621613i $$0.786478\pi$$
$$284$$ 1.67164 0.0991936
$$285$$ 0 0
$$286$$ 6.41617 0.379396
$$287$$ − 3.20662i − 0.189281i
$$288$$ − 10.4055i − 0.613150i
$$289$$ −23.6687 −1.39228
$$290$$ 0 0
$$291$$ −28.0099 −1.64197
$$292$$ − 2.32273i − 0.135928i
$$293$$ 0.818748i 0.0478318i 0.999714 + 0.0239159i $$0.00761339\pi$$
−0.999714 + 0.0239159i $$0.992387\pi$$
$$294$$ −26.9164 −1.56980
$$295$$ 0 0
$$296$$ 11.7445 0.682637
$$297$$ 1.14228i 0.0662819i
$$298$$ − 9.14711i − 0.529878i
$$299$$ −17.4912 −1.01154
$$300$$ 0 0
$$301$$ 4.58383 0.264207
$$302$$ − 8.45997i − 0.486817i
$$303$$ 21.8422i 1.25480i
$$304$$ 4.92498 0.282467
$$305$$ 0 0
$$306$$ −27.9143 −1.59575
$$307$$ − 7.44447i − 0.424878i −0.977174 0.212439i $$-0.931859\pi$$
0.977174 0.212439i $$-0.0681407\pi$$
$$308$$ − 0.377203i − 0.0214931i
$$309$$ 22.5187 1.28104
$$310$$ 0 0
$$311$$ −0.956204 −0.0542213 −0.0271107 0.999632i $$-0.508631\pi$$
−0.0271107 + 0.999632i $$0.508631\pi$$
$$312$$ 14.1084i 0.798729i
$$313$$ − 5.98158i − 0.338099i −0.985608 0.169049i $$-0.945930\pi$$
0.985608 0.169049i $$-0.0540697\pi$$
$$314$$ 32.6327 1.84157
$$315$$ 0 0
$$316$$ −4.87826 −0.274424
$$317$$ 22.6511i 1.27221i 0.771602 + 0.636106i $$0.219456\pi$$
−0.771602 + 0.636106i $$0.780544\pi$$
$$318$$ 45.3249i 2.54169i
$$319$$ −4.65109 −0.260411
$$320$$ 0 0
$$321$$ −36.4925 −2.03681
$$322$$ 3.86064i 0.215145i
$$323$$ − 6.37720i − 0.354837i
$$324$$ 7.20662 0.400368
$$325$$ 0 0
$$326$$ −22.3219 −1.23630
$$327$$ − 39.8521i − 2.20383i
$$328$$ − 17.8804i − 0.987279i
$$329$$ −2.59450 −0.143039
$$330$$ 0 0
$$331$$ 4.16283 0.228810 0.114405 0.993434i $$-0.463504\pi$$
0.114405 + 0.993434i $$0.463504\pi$$
$$332$$ 13.2272i 0.725935i
$$333$$ − 14.8032i − 0.811213i
$$334$$ 21.6383 1.18399
$$335$$ 0 0
$$336$$ 4.41617 0.240922
$$337$$ − 18.0304i − 0.982180i −0.871109 0.491090i $$-0.836599\pi$$
0.871109 0.491090i $$-0.163401\pi$$
$$338$$ − 8.31849i − 0.452466i
$$339$$ 31.6893 1.72112
$$340$$ 0 0
$$341$$ −3.41617 −0.184996
$$342$$ − 4.37720i − 0.236692i
$$343$$ − 5.22717i − 0.282241i
$$344$$ 25.5598 1.37809
$$345$$ 0 0
$$346$$ −2.45726 −0.132103
$$347$$ 7.43380i 0.399067i 0.979891 + 0.199534i $$0.0639427\pi$$
−0.979891 + 0.199534i $$0.936057\pi$$
$$348$$ 5.82942i 0.312490i
$$349$$ −21.9194 −1.17332 −0.586658 0.809835i $$-0.699557\pi$$
−0.586658 + 0.809835i $$0.699557\pi$$
$$350$$ 0 0
$$351$$ −2.34036 −0.124919
$$352$$ − 5.40550i − 0.288114i
$$353$$ − 10.3695i − 0.551910i −0.961171 0.275955i $$-0.911006\pi$$
0.961171 0.275955i $$-0.0889941\pi$$
$$354$$ 23.7720 1.26347
$$355$$ 0 0
$$356$$ 1.08993 0.0577664
$$357$$ − 5.71836i − 0.302648i
$$358$$ − 28.2555i − 1.49335i
$$359$$ 23.9893 1.26611 0.633054 0.774108i $$-0.281801\pi$$
0.633054 + 0.774108i $$0.281801\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 9.47547i 0.498020i
$$363$$ 21.6404i 1.13583i
$$364$$ 0.772829 0.0405073
$$365$$ 0 0
$$366$$ −19.7360 −1.03162
$$367$$ 35.0275i 1.82842i 0.405240 + 0.914210i $$0.367188\pi$$
−0.405240 + 0.914210i $$0.632812\pi$$
$$368$$ 30.5294i 1.59145i
$$369$$ −22.5371 −1.17323
$$370$$ 0 0
$$371$$ −4.35586 −0.226145
$$372$$ 4.28164i 0.221993i
$$373$$ − 30.8003i − 1.59478i −0.603464 0.797390i $$-0.706213\pi$$
0.603464 0.797390i $$-0.293787\pi$$
$$374$$ −14.5011 −0.749832
$$375$$ 0 0
$$376$$ −14.4671 −0.746086
$$377$$ − 9.52936i − 0.490787i
$$378$$ 0.516561i 0.0265691i
$$379$$ −0.671640 −0.0344998 −0.0172499 0.999851i $$-0.505491\pi$$
−0.0172499 + 0.999851i $$0.505491\pi$$
$$380$$ 0 0
$$381$$ −7.24267 −0.371053
$$382$$ − 36.9759i − 1.89185i
$$383$$ 30.1805i 1.54215i 0.636745 + 0.771075i $$0.280280\pi$$
−0.636745 + 0.771075i $$0.719720\pi$$
$$384$$ 31.8860 1.62718
$$385$$ 0 0
$$386$$ 35.2816 1.79579
$$387$$ − 32.2165i − 1.63766i
$$388$$ 8.55553i 0.434341i
$$389$$ 31.5109 1.59767 0.798834 0.601552i $$-0.205451\pi$$
0.798834 + 0.601552i $$0.205451\pi$$
$$390$$ 0 0
$$391$$ 39.5315 1.99919
$$392$$ − 14.4239i − 0.728518i
$$393$$ 20.6999i 1.04417i
$$394$$ 1.83929 0.0926623
$$395$$ 0 0
$$396$$ −2.65109 −0.133222
$$397$$ − 12.5761i − 0.631175i −0.948896 0.315588i $$-0.897798\pi$$
0.948896 0.315588i $$-0.102202\pi$$
$$398$$ − 3.67376i − 0.184149i
$$399$$ 0.896688 0.0448905
$$400$$ 0 0
$$401$$ 24.6036 1.22864 0.614322 0.789056i $$-0.289430\pi$$
0.614322 + 0.789056i $$0.289430\pi$$
$$402$$ − 12.6666i − 0.631752i
$$403$$ − 6.99920i − 0.348655i
$$404$$ 6.67164 0.331926
$$405$$ 0 0
$$406$$ −2.10331 −0.104386
$$407$$ − 7.69006i − 0.381182i
$$408$$ − 31.8860i − 1.57859i
$$409$$ −13.6666 −0.675770 −0.337885 0.941187i $$-0.609711\pi$$
−0.337885 + 0.941187i $$0.609711\pi$$
$$410$$ 0 0
$$411$$ 15.6765 0.773263
$$412$$ − 6.87826i − 0.338868i
$$413$$ 2.28456i 0.112416i
$$414$$ 27.1337 1.33355
$$415$$ 0 0
$$416$$ 11.0750 0.542997
$$417$$ − 24.1706i − 1.18364i
$$418$$ − 2.27389i − 0.111220i
$$419$$ −12.6893 −0.619911 −0.309956 0.950751i $$-0.600314\pi$$
−0.309956 + 0.950751i $$0.600314\pi$$
$$420$$ 0 0
$$421$$ 29.1826 1.42227 0.711136 0.703055i $$-0.248181\pi$$
0.711136 + 0.703055i $$0.248181\pi$$
$$422$$ 16.1054i 0.784000i
$$423$$ 18.2349i 0.886612i
$$424$$ −24.2886 −1.17956
$$425$$ 0 0
$$426$$ 9.03605 0.437798
$$427$$ − 1.89669i − 0.0917872i
$$428$$ 11.1465i 0.538788i
$$429$$ 9.23784 0.446007
$$430$$ 0 0
$$431$$ −29.9816 −1.44416 −0.722081 0.691809i $$-0.756814\pi$$
−0.722081 + 0.691809i $$0.756814\pi$$
$$432$$ 4.08489i 0.196534i
$$433$$ 4.76216i 0.228855i 0.993432 + 0.114427i $$0.0365033\pi$$
−0.993432 + 0.114427i $$0.963497\pi$$
$$434$$ −1.54486 −0.0741555
$$435$$ 0 0
$$436$$ −12.1727 −0.582967
$$437$$ 6.19887i 0.296532i
$$438$$ − 12.5555i − 0.599926i
$$439$$ −6.88601 −0.328652 −0.164326 0.986406i $$-0.552545\pi$$
−0.164326 + 0.986406i $$0.552545\pi$$
$$440$$ 0 0
$$441$$ −18.1805 −0.865736
$$442$$ − 29.7104i − 1.41318i
$$443$$ − 8.65109i − 0.411026i −0.978654 0.205513i $$-0.934114\pi$$
0.978654 0.205513i $$-0.0658863\pi$$
$$444$$ −9.63830 −0.457413
$$445$$ 0 0
$$446$$ 30.7819 1.45757
$$447$$ − 13.1698i − 0.622909i
$$448$$ 1.27097i 0.0600476i
$$449$$ −8.63055 −0.407301 −0.203650 0.979044i $$-0.565281\pi$$
−0.203650 + 0.979044i $$0.565281\pi$$
$$450$$ 0 0
$$451$$ −11.7077 −0.551293
$$452$$ − 9.67939i − 0.455280i
$$453$$ − 12.1805i − 0.572288i
$$454$$ −13.2859 −0.623538
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ − 28.6092i − 1.33828i −0.743135 0.669141i $$-0.766662\pi$$
0.743135 0.669141i $$-0.233338\pi$$
$$458$$ 33.2555i 1.55393i
$$459$$ 5.28939 0.246888
$$460$$ 0 0
$$461$$ −39.1025 −1.82119 −0.910593 0.413305i $$-0.864374\pi$$
−0.910593 + 0.413305i $$0.864374\pi$$
$$462$$ − 2.03897i − 0.0948615i
$$463$$ 8.04380i 0.373827i 0.982376 + 0.186913i $$0.0598484\pi$$
−0.982376 + 0.186913i $$0.940152\pi$$
$$464$$ −16.6327 −0.772152
$$465$$ 0 0
$$466$$ 2.86276 0.132615
$$467$$ 17.0488i 0.788926i 0.918912 + 0.394463i $$0.129069\pi$$
−0.918912 + 0.394463i $$0.870931\pi$$
$$468$$ − 5.43167i − 0.251079i
$$469$$ 1.21730 0.0562096
$$470$$ 0 0
$$471$$ 46.9837 2.16489
$$472$$ 12.7389i 0.586356i
$$473$$ − 16.7360i − 0.769521i
$$474$$ −26.3695 −1.21119
$$475$$ 0 0
$$476$$ −1.74666 −0.0800578
$$477$$ 30.6142i 1.40173i
$$478$$ − 30.2397i − 1.38313i
$$479$$ −22.5478 −1.03023 −0.515117 0.857120i $$-0.672252\pi$$
−0.515117 + 0.857120i $$0.672252\pi$$
$$480$$ 0 0
$$481$$ 15.7557 0.718399
$$482$$ 3.62704i 0.165207i
$$483$$ 5.55845i 0.252918i
$$484$$ 6.61000 0.300455
$$485$$ 0 0
$$486$$ 34.8470 1.58069
$$487$$ − 24.5577i − 1.11281i −0.830910 0.556407i $$-0.812180\pi$$
0.830910 0.556407i $$-0.187820\pi$$
$$488$$ − 10.5761i − 0.478757i
$$489$$ −32.1386 −1.45336
$$490$$ 0 0
$$491$$ −16.9298 −0.764032 −0.382016 0.924156i $$-0.624770\pi$$
−0.382016 + 0.924156i $$0.624770\pi$$
$$492$$ 14.6738i 0.661544i
$$493$$ 21.5371i 0.969983i
$$494$$ 4.65884 0.209611
$$495$$ 0 0
$$496$$ −12.2165 −0.548537
$$497$$ 0.868391i 0.0389527i
$$498$$ 71.4995i 3.20397i
$$499$$ −0.418295 −0.0187255 −0.00936273 0.999956i $$-0.502980\pi$$
−0.00936273 + 0.999956i $$0.502980\pi$$
$$500$$ 0 0
$$501$$ 31.1543 1.39187
$$502$$ 22.5732i 1.00749i
$$503$$ 12.5993i 0.561776i 0.959741 + 0.280888i $$0.0906290\pi$$
−0.959741 + 0.280888i $$0.909371\pi$$
$$504$$ 2.10331 0.0936890
$$505$$ 0 0
$$506$$ 14.0956 0.626624
$$507$$ − 11.9767i − 0.531906i
$$508$$ 2.21225i 0.0981528i
$$509$$ −12.8209 −0.568275 −0.284138 0.958784i $$-0.591707\pi$$
−0.284138 + 0.958784i $$0.591707\pi$$
$$510$$ 0 0
$$511$$ 1.20662 0.0533779
$$512$$ 1.38708i 0.0613007i
$$513$$ 0.829422i 0.0366199i
$$514$$ 47.0403 2.07486
$$515$$ 0 0
$$516$$ −20.9759 −0.923415
$$517$$ 9.47277i 0.416612i
$$518$$ − 3.47759i − 0.152797i
$$519$$ −3.53791 −0.155297
$$520$$ 0 0
$$521$$ 11.3743 0.498316 0.249158 0.968463i $$-0.419846\pi$$
0.249158 + 0.968463i $$0.419846\pi$$
$$522$$ 14.7827i 0.647021i
$$523$$ − 33.0948i − 1.44713i −0.690255 0.723566i $$-0.742502\pi$$
0.690255 0.723566i $$-0.257498\pi$$
$$524$$ 6.32273 0.276210
$$525$$ 0 0
$$526$$ −8.86547 −0.386553
$$527$$ 15.8187i 0.689075i
$$528$$ − 16.1239i − 0.701701i
$$529$$ −15.4260 −0.670698
$$530$$ 0 0
$$531$$ 16.0566 0.696797
$$532$$ − 0.273891i − 0.0118747i
$$533$$ − 23.9872i − 1.03900i
$$534$$ 5.89164 0.254956
$$535$$ 0 0
$$536$$ 6.78775 0.293186
$$537$$ − 40.6815i − 1.75554i
$$538$$ 11.6687i 0.503074i
$$539$$ −9.44447 −0.406802
$$540$$ 0 0
$$541$$ 31.4338 1.35144 0.675722 0.737156i $$-0.263832\pi$$
0.675722 + 0.737156i $$0.263832\pi$$
$$542$$ − 35.8238i − 1.53876i
$$543$$ 13.6425i 0.585458i
$$544$$ −25.0304 −1.07317
$$545$$ 0 0
$$546$$ 4.17753 0.178782
$$547$$ − 23.3460i − 0.998202i −0.866544 0.499101i $$-0.833664\pi$$
0.866544 0.499101i $$-0.166336\pi$$
$$548$$ − 4.78833i − 0.204547i
$$549$$ −13.3305 −0.568931
$$550$$ 0 0
$$551$$ −3.37720 −0.143874
$$552$$ 30.9944i 1.31921i
$$553$$ − 2.53418i − 0.107764i
$$554$$ 41.4642 1.76165
$$555$$ 0 0
$$556$$ −7.38283 −0.313102
$$557$$ 15.9229i 0.674673i 0.941384 + 0.337337i $$0.109526\pi$$
−0.941384 + 0.337337i $$0.890474\pi$$
$$558$$ 10.8577i 0.459644i
$$559$$ 34.2894 1.45029
$$560$$ 0 0
$$561$$ −20.8783 −0.881481
$$562$$ 16.5788i 0.699334i
$$563$$ 29.6404i 1.24919i 0.780947 + 0.624597i $$0.214737\pi$$
−0.780947 + 0.624597i $$0.785263\pi$$
$$564$$ 11.8726 0.499928
$$565$$ 0 0
$$566$$ 34.5315 1.45147
$$567$$ 3.74373i 0.157222i
$$568$$ 4.84222i 0.203175i
$$569$$ 7.64334 0.320426 0.160213 0.987082i $$-0.448782\pi$$
0.160213 + 0.987082i $$0.448782\pi$$
$$570$$ 0 0
$$571$$ −10.5577 −0.441824 −0.220912 0.975294i $$-0.570903\pi$$
−0.220912 + 0.975294i $$0.570903\pi$$
$$572$$ − 2.82167i − 0.117980i
$$573$$ − 53.2370i − 2.22401i
$$574$$ −5.29444 −0.220986
$$575$$ 0 0
$$576$$ 8.93273 0.372197
$$577$$ 38.9447i 1.62129i 0.585538 + 0.810645i $$0.300883\pi$$
−0.585538 + 0.810645i $$0.699117\pi$$
$$578$$ 39.0793i 1.62548i
$$579$$ 50.7976 2.11108
$$580$$ 0 0
$$581$$ −6.87131 −0.285070
$$582$$ 46.2469i 1.91700i
$$583$$ 15.9036i 0.658661i
$$584$$ 6.72823 0.278416
$$585$$ 0 0
$$586$$ 1.35183 0.0558436
$$587$$ − 10.9992i − 0.453986i −0.973896 0.226993i $$-0.927111\pi$$
0.973896 0.226993i $$-0.0728894\pi$$
$$588$$ 11.8372i 0.488157i
$$589$$ −2.48052 −0.102208
$$590$$ 0 0
$$591$$ 2.64817 0.108931
$$592$$ − 27.5003i − 1.13025i
$$593$$ − 26.7848i − 1.09992i −0.835191 0.549960i $$-0.814643\pi$$
0.835191 0.549960i $$-0.185357\pi$$
$$594$$ 1.88601 0.0773841
$$595$$ 0 0
$$596$$ −4.02267 −0.164775
$$597$$ − 5.28939i − 0.216480i
$$598$$ 28.8796i 1.18097i
$$599$$ 3.44930 0.140934 0.0704672 0.997514i $$-0.477551\pi$$
0.0704672 + 0.997514i $$0.477551\pi$$
$$600$$ 0 0
$$601$$ −37.0686 −1.51206 −0.756030 0.654537i $$-0.772863\pi$$
−0.756030 + 0.654537i $$0.772863\pi$$
$$602$$ − 7.56833i − 0.308462i
$$603$$ − 8.55553i − 0.348408i
$$604$$ −3.72048 −0.151384
$$605$$ 0 0
$$606$$ 36.0635 1.46498
$$607$$ − 14.8732i − 0.603685i −0.953358 0.301843i $$-0.902398\pi$$
0.953358 0.301843i $$-0.0976017\pi$$
$$608$$ − 3.92498i − 0.159179i
$$609$$ −3.02830 −0.122713
$$610$$ 0 0
$$611$$ −19.4082 −0.785172
$$612$$ 12.2760i 0.496228i
$$613$$ 40.4671i 1.63445i 0.576317 + 0.817226i $$0.304489\pi$$
−0.576317 + 0.817226i $$0.695511\pi$$
$$614$$ −12.2915 −0.496045
$$615$$ 0 0
$$616$$ 1.09264 0.0440237
$$617$$ 8.76991i 0.353063i 0.984295 + 0.176532i $$0.0564878\pi$$
−0.984295 + 0.176532i $$0.943512\pi$$
$$618$$ − 37.1805i − 1.49562i
$$619$$ −32.0510 −1.28824 −0.644119 0.764926i $$-0.722776\pi$$
−0.644119 + 0.764926i $$0.722776\pi$$
$$620$$ 0 0
$$621$$ −5.14148 −0.206321
$$622$$ 1.57878i 0.0633034i
$$623$$ 0.566205i 0.0226845i
$$624$$ 33.0352 1.32247
$$625$$ 0 0
$$626$$ −9.87614 −0.394730
$$627$$ − 3.27389i − 0.130747i
$$628$$ − 14.3510i − 0.572668i
$$629$$ −35.6092 −1.41983
$$630$$ 0 0
$$631$$ −18.9709 −0.755220 −0.377610 0.925965i $$-0.623254\pi$$
−0.377610 + 0.925965i $$0.623254\pi$$
$$632$$ − 14.1308i − 0.562093i
$$633$$ 23.1882i 0.921648i
$$634$$ 37.3991 1.48531
$$635$$ 0 0
$$636$$ 19.9327 0.790384
$$637$$ − 19.3502i − 0.766684i
$$638$$ 7.67939i 0.304030i
$$639$$ 6.10331 0.241443
$$640$$ 0 0
$$641$$ −44.3433 −1.75145 −0.875727 0.482806i $$-0.839617\pi$$
−0.875727 + 0.482806i $$0.839617\pi$$
$$642$$ 60.2525i 2.37798i
$$643$$ − 42.9397i − 1.69338i −0.532090 0.846688i $$-0.678593\pi$$
0.532090 0.846688i $$-0.321407\pi$$
$$644$$ 1.69781 0.0669032
$$645$$ 0 0
$$646$$ −10.5294 −0.414272
$$647$$ 17.9864i 0.707118i 0.935412 + 0.353559i $$0.115029\pi$$
−0.935412 + 0.353559i $$0.884971\pi$$
$$648$$ 20.8753i 0.820061i
$$649$$ 8.34116 0.327419
$$650$$ 0 0
$$651$$ −2.22425 −0.0871751
$$652$$ 9.81663i 0.384449i
$$653$$ 21.7274i 0.850260i 0.905132 + 0.425130i $$0.139772\pi$$
−0.905132 + 0.425130i $$0.860228\pi$$
$$654$$ −65.7995 −2.57297
$$655$$ 0 0
$$656$$ −41.8676 −1.63465
$$657$$ − 8.48052i − 0.330856i
$$658$$ 4.28376i 0.166998i
$$659$$ 40.8590 1.59164 0.795821 0.605532i $$-0.207040\pi$$
0.795821 + 0.605532i $$0.207040\pi$$
$$660$$ 0 0
$$661$$ 33.7282 1.31188 0.655938 0.754815i $$-0.272273\pi$$
0.655938 + 0.754815i $$0.272273\pi$$
$$662$$ − 6.87322i − 0.267135i
$$663$$ − 42.7763i − 1.66129i
$$664$$ −38.3150 −1.48691
$$665$$ 0 0
$$666$$ −24.4415 −0.947091
$$667$$ − 20.9349i − 0.810601i
$$668$$ − 9.51598i − 0.368184i
$$669$$ 44.3190 1.71347
$$670$$ 0 0
$$671$$ −6.92498 −0.267336
$$672$$ − 3.51948i − 0.135767i
$$673$$ 11.0622i 0.426417i 0.977007 + 0.213209i $$0.0683914\pi$$
−0.977007 + 0.213209i $$0.931609\pi$$
$$674$$ −29.7699 −1.14669
$$675$$ 0 0
$$676$$ −3.65826 −0.140702
$$677$$ − 6.14711i − 0.236253i −0.992999 0.118126i $$-0.962311\pi$$
0.992999 0.118126i $$-0.0376888\pi$$
$$678$$ − 52.3219i − 2.00941i
$$679$$ −4.44447 −0.170563
$$680$$ 0 0
$$681$$ −19.1287 −0.733013
$$682$$ 5.64042i 0.215983i
$$683$$ 13.6073i 0.520669i 0.965519 + 0.260334i $$0.0838328\pi$$
−0.965519 + 0.260334i $$0.916167\pi$$
$$684$$ −1.92498 −0.0736036
$$685$$ 0 0
$$686$$ −8.63055 −0.329516
$$687$$ 47.8804i 1.82675i
$$688$$ − 59.8492i − 2.28173i
$$689$$ −32.5840 −1.24135
$$690$$ 0 0
$$691$$ −40.3043 −1.53325 −0.766624 0.642096i $$-0.778065\pi$$
−0.766624 + 0.642096i $$0.778065\pi$$
$$692$$ 1.08064i 0.0410799i
$$693$$ − 1.37720i − 0.0523156i
$$694$$ 12.2739 0.465911
$$695$$ 0 0
$$696$$ −16.8860 −0.640063
$$697$$ 54.2130i 2.05346i
$$698$$ 36.1909i 1.36985i
$$699$$ 4.12174 0.155898
$$700$$ 0 0
$$701$$ −3.23704 −0.122261 −0.0611307 0.998130i $$-0.519471\pi$$
−0.0611307 + 0.998130i $$0.519471\pi$$
$$702$$ 3.86415i 0.145843i
$$703$$ − 5.58383i − 0.210598i
$$704$$ 4.64042 0.174892
$$705$$ 0 0
$$706$$ −17.1209 −0.644355
$$707$$ 3.46582i 0.130345i
$$708$$ − 10.4543i − 0.392898i
$$709$$ −20.2370 −0.760018 −0.380009 0.924983i $$-0.624079\pi$$
−0.380009 + 0.924983i $$0.624079\pi$$
$$710$$ 0 0
$$711$$ −17.8110 −0.667965
$$712$$ 3.15720i 0.118321i
$$713$$ − 15.3764i − 0.575851i
$$714$$ −9.44155 −0.353341
$$715$$ 0 0
$$716$$ −12.4260 −0.464383
$$717$$ − 43.5384i − 1.62597i
$$718$$ − 39.6086i − 1.47818i
$$719$$ −47.8804 −1.78564 −0.892819 0.450416i $$-0.851276\pi$$
−0.892819 + 0.450416i $$0.851276\pi$$
$$720$$ 0 0
$$721$$ 3.57315 0.133071
$$722$$ − 1.65109i − 0.0614473i
$$723$$ 5.22212i 0.194213i
$$724$$ 4.16707 0.154868
$$725$$ 0 0
$$726$$ 35.7304 1.32608
$$727$$ 17.7926i 0.659890i 0.944000 + 0.329945i $$0.107030\pi$$
−0.944000 + 0.329945i $$0.892970\pi$$
$$728$$ 2.23864i 0.0829697i
$$729$$ 20.3969 0.755443
$$730$$ 0 0
$$731$$ −77.4968 −2.86632
$$732$$ 8.67939i 0.320799i
$$733$$ 5.66097i 0.209093i 0.994520 + 0.104546i $$0.0333390\pi$$
−0.994520 + 0.104546i $$0.966661\pi$$
$$734$$ 57.8337 2.13468
$$735$$ 0 0
$$736$$ 24.3305 0.896834
$$737$$ − 4.44447i − 0.163714i
$$738$$ 37.2109i 1.36975i
$$739$$ 5.59955 0.205983 0.102991 0.994682i $$-0.467159\pi$$
0.102991 + 0.994682i $$0.467159\pi$$
$$740$$ 0 0
$$741$$ 6.70769 0.246413
$$742$$ 7.19193i 0.264024i
$$743$$ − 5.81663i − 0.213391i −0.994292 0.106696i $$-0.965973\pi$$
0.994292 0.106696i $$-0.0340270\pi$$
$$744$$ −12.4026 −0.454700
$$745$$ 0 0
$$746$$ −50.8542 −1.86191
$$747$$ 48.2936i 1.76697i
$$748$$ 6.37720i 0.233174i
$$749$$ −5.79045 −0.211579
$$750$$ 0 0
$$751$$ −41.1797 −1.50267 −0.751333 0.659923i $$-0.770589\pi$$
−0.751333 + 0.659923i $$0.770589\pi$$
$$752$$ 33.8753i 1.23531i
$$753$$ 32.5003i 1.18438i
$$754$$ −15.7339 −0.572993
$$755$$ 0 0
$$756$$ 0.227171 0.00826212
$$757$$ − 33.5208i − 1.21833i −0.793042 0.609167i $$-0.791504\pi$$
0.793042 0.609167i $$-0.208496\pi$$
$$758$$ 1.10894i 0.0402785i
$$759$$ 20.2944 0.736641
$$760$$ 0 0
$$761$$ 31.5032 1.14199 0.570995 0.820954i $$-0.306558\pi$$
0.570995 + 0.820954i $$0.306558\pi$$
$$762$$ 11.9583i 0.433204i
$$763$$ − 6.32353i − 0.228927i
$$764$$ −16.2611 −0.588306
$$765$$ 0 0
$$766$$ 49.8307 1.80046
$$767$$ 17.0897i 0.617074i
$$768$$ − 36.6270i − 1.32166i
$$769$$ 1.22425 0.0441475 0.0220737 0.999756i $$-0.492973\pi$$
0.0220737 + 0.999756i $$0.492973\pi$$
$$770$$ 0 0
$$771$$ 67.7274 2.43914
$$772$$ − 15.5160i − 0.558432i
$$773$$ − 40.6687i − 1.46275i −0.681974 0.731376i $$-0.738878\pi$$
0.681974 0.731376i $$-0.261122\pi$$
$$774$$ −53.1924 −1.91196
$$775$$ 0 0
$$776$$ −24.7827 −0.889647
$$777$$ − 5.00695i − 0.179623i
$$778$$ − 52.0275i − 1.86528i
$$779$$ −8.50106 −0.304582
$$780$$ 0 0
$$781$$ 3.17058 0.113452
$$782$$ − 65.2702i − 2.33406i
$$783$$ − 2.80113i − 0.100104i
$$784$$ −33.7742 −1.20622
$$785$$ 0 0
$$786$$ 34.1775 1.21907
$$787$$ − 39.9525i − 1.42415i −0.702102 0.712076i $$-0.747755\pi$$
0.702102 0.712076i $$-0.252245\pi$$
$$788$$ − 0.808876i − 0.0288150i
$$789$$ −12.7643 −0.454420
$$790$$ 0 0
$$791$$ 5.02830 0.178786
$$792$$ − 7.67939i − 0.272875i
$$793$$ − 14.1882i − 0.503838i
$$794$$ −20.7643 −0.736897
$$795$$ 0 0
$$796$$ −1.61563 −0.0572644
$$797$$ − 4.53791i − 0.160741i −0.996765 0.0803705i $$-0.974390\pi$$
0.996765 0.0803705i $$-0.0256103\pi$$
$$798$$ − 1.48052i − 0.0524097i
$$799$$ 43.8641 1.55180
$$800$$ 0 0
$$801$$ 3.97945 0.140607
$$802$$ − 40.6228i − 1.43444i
$$803$$ − 4.40550i − 0.155467i
$$804$$ −5.57045 −0.196455
$$805$$ 0 0
$$806$$ −11.5563 −0.407054
$$807$$ 16.8003i 0.591399i
$$808$$ 19.3257i 0.679874i
$$809$$ −55.5958 −1.95465 −0.977323 0.211756i $$-0.932082\pi$$
−0.977323 + 0.211756i $$0.932082\pi$$
$$810$$ 0 0
$$811$$ 42.3326 1.48650 0.743249 0.669014i $$-0.233284\pi$$
0.743249 + 0.669014i $$0.233284\pi$$
$$812$$ 0.924984i 0.0324606i
$$813$$ − 51.5782i − 1.80893i
$$814$$ −12.6970 −0.445030
$$815$$ 0 0
$$816$$ −74.6623 −2.61370
$$817$$ − 12.1522i − 0.425150i
$$818$$ 22.5648i 0.788961i
$$819$$ 2.82167 0.0985972
$$820$$ 0 0
$$821$$ −3.10543 −0.108380 −0.0541902 0.998531i $$-0.517258\pi$$
−0.0541902 + 0.998531i $$0.517258\pi$$
$$822$$ − 25.8833i − 0.902784i
$$823$$ 45.2002i 1.57558i 0.615944 + 0.787790i $$0.288775\pi$$
−0.615944 + 0.787790i $$0.711225\pi$$
$$824$$ 19.9242 0.694092
$$825$$ 0 0
$$826$$ 3.77203 0.131246
$$827$$ − 35.6503i − 1.23968i −0.784727 0.619841i $$-0.787197\pi$$
0.784727 0.619841i $$-0.212803\pi$$
$$828$$ − 11.9327i − 0.414691i
$$829$$ 18.4330 0.640204 0.320102 0.947383i $$-0.396283\pi$$
0.320102 + 0.947383i $$0.396283\pi$$
$$830$$ 0 0
$$831$$ 59.6991 2.07094
$$832$$ 9.50749i 0.329613i
$$833$$ 43.7331i 1.51526i
$$834$$ −39.9079 −1.38190
$$835$$ 0 0
$$836$$ −1.00000 −0.0345857
$$837$$ − 2.05739i − 0.0711139i
$$838$$ 20.9512i 0.723746i
$$839$$ 14.1805 0.489564 0.244782 0.969578i $$-0.421284\pi$$
0.244782 + 0.969578i $$0.421284\pi$$
$$840$$ 0 0
$$841$$ −17.5945 −0.606707
$$842$$ − 48.1832i − 1.66050i
$$843$$ 23.8697i 0.822117i
$$844$$ 7.08277 0.243799
$$845$$ 0 0
$$846$$ 30.1076 1.03512
$$847$$ 3.43380i 0.117987i
$$848$$ 56.8726i 1.95301i
$$849$$ 49.7176 1.70630
$$850$$ 0 0
$$851$$ 34.6134 1.18653
$$852$$ − 3.97383i − 0.136141i
$$853$$ − 42.6639i − 1.46078i −0.683028 0.730392i $$-0.739337\pi$$
0.683028 0.730392i $$-0.260663\pi$$
$$854$$ −3.13161 −0.107161
$$855$$ 0 0
$$856$$ −32.2880 −1.10358
$$857$$ − 1.42392i − 0.0486403i −0.999704 0.0243201i $$-0.992258\pi$$
0.999704 0.0243201i $$-0.00774210\pi$$
$$858$$ − 15.2525i − 0.520713i
$$859$$ 8.27894 0.282474 0.141237 0.989976i $$-0.454892\pi$$
0.141237 + 0.989976i $$0.454892\pi$$
$$860$$ 0 0
$$861$$ −7.62280 −0.259784
$$862$$ 49.5024i 1.68606i
$$863$$ − 30.4154i − 1.03535i −0.855577 0.517676i $$-0.826797\pi$$
0.855577 0.517676i $$-0.173203\pi$$
$$864$$ 3.25547 0.110753
$$865$$ 0 0
$$866$$ 7.86276 0.267188
$$867$$ 56.2653i 1.91087i
$$868$$ 0.679390i 0.0230600i
$$869$$ −9.25254 −0.313871
$$870$$ 0 0
$$871$$ 9.10602 0.308546
$$872$$ − 35.2605i − 1.19407i
$$873$$ 31.2370i 1.05721i
$$874$$ 10.2349 0.346201
$$875$$ 0 0
$$876$$ −5.52161 −0.186558
$$877$$ 57.8484i 1.95340i 0.214608 + 0.976700i $$0.431153\pi$$
−0.214608 + 0.976700i $$0.568847\pi$$
$$878$$ 11.3695i 0.383700i
$$879$$ 1.94633 0.0656481
$$880$$ 0 0
$$881$$ 18.4055 0.620097 0.310049 0.950721i $$-0.399655\pi$$
0.310049 + 0.950721i $$0.399655\pi$$
$$882$$ 30.0176i 1.01075i
$$883$$ 18.7947i 0.632492i 0.948677 + 0.316246i $$0.102422\pi$$
−0.948677 + 0.316246i $$0.897578\pi$$
$$884$$ −13.0659 −0.439453
$$885$$ 0 0
$$886$$ −14.2838 −0.479872
$$887$$ − 11.1471i − 0.374283i −0.982333 0.187142i $$-0.940078\pi$$
0.982333 0.187142i $$-0.0599223\pi$$
$$888$$ − 27.9191i − 0.936905i
$$889$$ −1.14923 −0.0385440
$$890$$ 0 0
$$891$$ 13.6687 0.457919
$$892$$ − 13.5371i − 0.453256i
$$893$$ 6.87826i 0.230172i
$$894$$ −21.7445 −0.727246
$$895$$ 0 0
$$896$$ 5.05952 0.169027
$$897$$ 41.5801i 1.38832i
$$898$$ 14.2498i 0.475523i
$$899$$ 8.37720 0.279395
$$900$$ 0 0
$$901$$ 73.6425 2.45339
$$902$$ 19.3305i 0.643635i
$$903$$ − 10.8967i − 0.362619i
$$904$$ 28.0382 0.932536
$$905$$ 0 0
$$906$$ −20.1111 −0.668145
$$907$$ 38.5598i 1.28036i 0.768226 + 0.640178i $$0.221139\pi$$
−0.768226 + 0.640178i $$0.778861\pi$$
$$908$$ 5.84280i 0.193900i
$$909$$ 24.3588 0.807930
$$910$$ 0 0
$$911$$ −1.79820 −0.0595771 −0.0297885 0.999556i $$-0.509483\pi$$
−0.0297885 + 0.999556i $$0.509483\pi$$
$$912$$ − 11.7077i − 0.387680i
$$913$$ 25.0878i 0.830285i
$$914$$ −47.2365 −1.56244
$$915$$ 0 0
$$916$$ 14.6249 0.483221
$$917$$ 3.28456i 0.108466i
$$918$$ − 8.73328i − 0.288241i
$$919$$ 32.4458 1.07029 0.535144 0.844761i $$-0.320257\pi$$
0.535144 + 0.844761i $$0.320257\pi$$
$$920$$ 0 0
$$921$$ −17.6970 −0.583136
$$922$$ 64.5619i 2.12623i
$$923$$ 6.49602i 0.213819i
$$924$$ −0.896688 −0.0294989
$$925$$ 0 0
$$926$$ 13.2811 0.436443
$$927$$ − 25.1132i − 0.824825i
$$928$$ 13.2555i 0.435132i
$$929$$ 18.4231 0.604443 0.302222 0.953238i $$-0.402272\pi$$
0.302222 + 0.953238i $$0.402272\pi$$
$$930$$ 0 0
$$931$$ −6.85772 −0.224753
$$932$$ − 1.25897i − 0.0412390i
$$933$$ 2.27309i 0.0744176i
$$934$$ 28.1492 0.921071
$$935$$ 0 0
$$936$$ 15.7339 0.514277
$$937$$ 8.29926i 0.271125i 0.990769 + 0.135563i $$0.0432842\pi$$
−0.990769 + 0.135563i $$0.956716\pi$$
$$938$$ − 2.00987i − 0.0656247i
$$939$$ −14.2194 −0.464033
$$940$$ 0 0
$$941$$ 33.6687 1.09757 0.548784 0.835964i $$-0.315091\pi$$
0.548784 + 0.835964i $$0.315091\pi$$
$$942$$ − 77.5745i − 2.52751i
$$943$$ − 52.6970i − 1.71605i
$$944$$ 29.8286 0.970839
$$945$$ 0 0
$$946$$ −27.6327 −0.898416
$$947$$ 1.35103i 0.0439026i 0.999759 + 0.0219513i $$0.00698787\pi$$
−0.999759 + 0.0219513i $$0.993012\pi$$
$$948$$ 11.5966i 0.376641i
$$949$$ 9.02617 0.293002
$$950$$ 0 0
$$951$$ 53.8462 1.74608
$$952$$ − 5.05952i − 0.163980i
$$953$$ 6.70769i 0.217283i 0.994081 + 0.108642i $$0.0346501\pi$$
−0.994081 + 0.108642i $$0.965350\pi$$
$$954$$ 50.5470 1.63652
$$955$$ 0 0
$$956$$ −13.2987 −0.430110
$$957$$ 11.0566i 0.357409i
$$958$$ 37.2285i 1.20280i
$$959$$ 2.48746 0.0803244
$$960$$ 0 0
$$961$$ −24.8470 −0.801518
$$962$$ − 26.0142i − 0.838731i
$$963$$ 40.6970i 1.31144i
$$964$$ 1.59508 0.0513741
$$965$$ 0 0
$$966$$ 9.17753 0.295282
$$967$$ 19.5174i 0.627636i 0.949483 + 0.313818i $$0.101608\pi$$
−0.949483 + 0.313818i $$0.898392\pi$$
$$968$$ 19.1471i 0.615411i
$$969$$ −15.1599 −0.487006
$$970$$ 0 0
$$971$$ −8.50669 −0.272993 −0.136496 0.990641i $$-0.543584\pi$$
−0.136496 + 0.990641i $$0.543584\pi$$
$$972$$ − 15.3249i − 0.491545i
$$973$$ − 3.83527i − 0.122953i
$$974$$ −40.5470 −1.29921
$$975$$ 0 0
$$976$$ −24.7643 −0.792685
$$977$$ 19.7048i 0.630411i 0.949023 + 0.315206i $$0.102073\pi$$
−0.949023 + 0.315206i $$0.897927\pi$$
$$978$$ 53.0638i 1.69679i
$$979$$ 2.06727 0.0660701
$$980$$ 0 0
$$981$$ −44.4437 −1.41898
$$982$$ 27.9527i 0.892006i
$$983$$ 12.4677i 0.397658i 0.980034 + 0.198829i $$0.0637139\pi$$
−0.980034 + 0.198829i $$0.936286\pi$$
$$984$$ −42.5053 −1.35502
$$985$$ 0 0
$$986$$ 35.5598 1.13245
$$987$$ 6.16765i 0.196319i
$$988$$ − 2.04884i − 0.0651824i
$$989$$ 75.3297 2.39534
$$990$$ 0 0
$$991$$ 25.0841 0.796822 0.398411 0.917207i $$-0.369562\pi$$
0.398411 + 0.917207i $$0.369562\pi$$
$$992$$ 9.73598i 0.309118i
$$993$$ − 9.89589i − 0.314036i
$$994$$ 1.43380 0.0454772
$$995$$ 0 0
$$996$$ 31.4437 0.996331
$$997$$ 11.8775i 0.376163i 0.982153 + 0.188082i $$0.0602269\pi$$
−0.982153 + 0.188082i $$0.939773\pi$$
$$998$$ 0.690644i 0.0218620i
$$999$$ 4.63135 0.146529
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 475.2.b.b.324.2 6
5.2 odd 4 475.2.a.e.1.3 3
5.3 odd 4 475.2.a.g.1.1 yes 3
5.4 even 2 inner 475.2.b.b.324.5 6
15.2 even 4 4275.2.a.bm.1.1 3
15.8 even 4 4275.2.a.ba.1.3 3
20.3 even 4 7600.2.a.bh.1.1 3
20.7 even 4 7600.2.a.cc.1.3 3
95.18 even 4 9025.2.a.y.1.3 3
95.37 even 4 9025.2.a.bc.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.3 3 5.2 odd 4
475.2.a.g.1.1 yes 3 5.3 odd 4
475.2.b.b.324.2 6 1.1 even 1 trivial
475.2.b.b.324.5 6 5.4 even 2 inner
4275.2.a.ba.1.3 3 15.8 even 4
4275.2.a.bm.1.1 3 15.2 even 4
7600.2.a.bh.1.1 3 20.3 even 4
7600.2.a.cc.1.3 3 20.7 even 4
9025.2.a.y.1.3 3 95.18 even 4
9025.2.a.bc.1.1 3 95.37 even 4