# Properties

 Label 169.2.b.a.168.1 Level $169$ Weight $2$ Character 169.168 Analytic conductor $1.349$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 168.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 169.168 Dual form 169.2.b.a.168.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.73205i q^{5} -3.46410i q^{6} -1.73205i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.73205i q^{5} -3.46410i q^{6} -1.73205i q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{12} +3.46410i q^{15} -5.00000 q^{16} -3.00000 q^{17} -1.73205i q^{18} +3.46410i q^{19} -1.73205i q^{20} -6.00000 q^{23} -3.46410i q^{24} +2.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} +6.00000 q^{30} -3.46410i q^{31} +5.19615i q^{32} +5.19615i q^{34} -1.00000 q^{36} +8.66025i q^{37} +6.00000 q^{38} +3.00000 q^{40} -5.19615i q^{41} +8.00000 q^{43} +1.73205i q^{45} +10.3923i q^{46} +3.46410i q^{47} -10.0000 q^{48} +7.00000 q^{49} -3.46410i q^{50} -6.00000 q^{51} -3.00000 q^{53} +6.92820i q^{54} +6.92820i q^{57} -5.19615i q^{58} -6.92820i q^{59} -3.46410i q^{60} +1.00000 q^{61} -6.00000 q^{62} -1.00000 q^{64} +3.46410i q^{67} +3.00000 q^{68} -12.0000 q^{69} -3.46410i q^{71} -1.73205i q^{72} -1.73205i q^{73} +15.0000 q^{74} +4.00000 q^{75} -3.46410i q^{76} +4.00000 q^{79} -8.66025i q^{80} -11.0000 q^{81} -9.00000 q^{82} -13.8564i q^{83} -5.19615i q^{85} -13.8564i q^{86} +6.00000 q^{87} -6.92820i q^{89} +3.00000 q^{90} +6.00000 q^{92} -6.92820i q^{93} +6.00000 q^{94} -6.00000 q^{95} +10.3923i q^{96} -6.92820i q^{97} -12.1244i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 $$2 q + 4 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 4 q^{12} - 10 q^{16} - 6 q^{17} - 12 q^{23} + 4 q^{25} - 8 q^{27} + 6 q^{29} + 12 q^{30} - 2 q^{36} + 12 q^{38} + 6 q^{40} + 16 q^{43} - 20 q^{48} + 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} - 12 q^{62} - 2 q^{64} + 6 q^{68} - 24 q^{69} + 30 q^{74} + 8 q^{75} + 8 q^{79} - 22 q^{81} - 18 q^{82} + 12 q^{87} + 6 q^{90} + 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 + 6 * q^10 - 4 * q^12 - 10 * q^16 - 6 * q^17 - 12 * q^23 + 4 * q^25 - 8 * q^27 + 6 * q^29 + 12 * q^30 - 2 * q^36 + 12 * q^38 + 6 * q^40 + 16 * q^43 - 20 * q^48 + 14 * q^49 - 12 * q^51 - 6 * q^53 + 2 * q^61 - 12 * q^62 - 2 * q^64 + 6 * q^68 - 24 * q^69 + 30 * q^74 + 8 * q^75 + 8 * q^79 - 22 * q^81 - 18 * q^82 + 12 * q^87 + 6 * q^90 + 12 * q^92 + 12 * q^94 - 12 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.73205i − 1.22474i −0.790569 0.612372i $$-0.790215\pi$$
0.790569 0.612372i $$-0.209785\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 1.73205i 0.774597i 0.921954 + 0.387298i $$0.126592\pi$$
−0.921954 + 0.387298i $$0.873408\pi$$
$$6$$ − 3.46410i − 1.41421i
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.73205i − 0.612372i
$$9$$ 1.00000 0.333333
$$10$$ 3.00000 0.948683
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 3.46410i 0.894427i
$$16$$ −5.00000 −1.25000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ − 1.73205i − 0.408248i
$$19$$ 3.46410i 0.794719i 0.917663 + 0.397360i $$0.130073\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ − 1.73205i − 0.387298i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ − 3.46410i − 0.707107i
$$25$$ 2.00000 0.400000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 6.00000 1.09545
$$31$$ − 3.46410i − 0.622171i −0.950382 0.311086i $$-0.899307\pi$$
0.950382 0.311086i $$-0.100693\pi$$
$$32$$ 5.19615i 0.918559i
$$33$$ 0 0
$$34$$ 5.19615i 0.891133i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 8.66025i 1.42374i 0.702313 + 0.711868i $$0.252151\pi$$
−0.702313 + 0.711868i $$0.747849\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ − 5.19615i − 0.811503i −0.913984 0.405751i $$-0.867010\pi$$
0.913984 0.405751i $$-0.132990\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 1.73205i 0.258199i
$$46$$ 10.3923i 1.53226i
$$47$$ 3.46410i 0.505291i 0.967559 + 0.252646i $$0.0813007\pi$$
−0.967559 + 0.252646i $$0.918699\pi$$
$$48$$ −10.0000 −1.44338
$$49$$ 7.00000 1.00000
$$50$$ − 3.46410i − 0.489898i
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 6.92820i 0.942809i
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.92820i 0.917663i
$$58$$ − 5.19615i − 0.682288i
$$59$$ − 6.92820i − 0.901975i −0.892530 0.450988i $$-0.851072\pi$$
0.892530 0.450988i $$-0.148928\pi$$
$$60$$ − 3.46410i − 0.447214i
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.46410i 0.423207i 0.977356 + 0.211604i $$0.0678686\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ 3.00000 0.363803
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ − 3.46410i − 0.411113i −0.978645 0.205557i $$-0.934100\pi$$
0.978645 0.205557i $$-0.0659005\pi$$
$$72$$ − 1.73205i − 0.204124i
$$73$$ − 1.73205i − 0.202721i −0.994850 0.101361i $$-0.967680\pi$$
0.994850 0.101361i $$-0.0323196\pi$$
$$74$$ 15.0000 1.74371
$$75$$ 4.00000 0.461880
$$76$$ − 3.46410i − 0.397360i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ − 8.66025i − 0.968246i
$$81$$ −11.0000 −1.22222
$$82$$ −9.00000 −0.993884
$$83$$ − 13.8564i − 1.52094i −0.649374 0.760469i $$-0.724969\pi$$
0.649374 0.760469i $$-0.275031\pi$$
$$84$$ 0 0
$$85$$ − 5.19615i − 0.563602i
$$86$$ − 13.8564i − 1.49417i
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ − 6.92820i − 0.734388i −0.930144 0.367194i $$-0.880318\pi$$
0.930144 0.367194i $$-0.119682\pi$$
$$90$$ 3.00000 0.316228
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ − 6.92820i − 0.718421i
$$94$$ 6.00000 0.618853
$$95$$ −6.00000 −0.615587
$$96$$ 10.3923i 1.06066i
$$97$$ − 6.92820i − 0.703452i −0.936103 0.351726i $$-0.885595\pi$$
0.936103 0.351726i $$-0.114405\pi$$
$$98$$ − 12.1244i − 1.22474i
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 10.3923i 1.02899i
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 5.19615i 0.504695i
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 13.8564i 1.32720i 0.748086 + 0.663602i $$0.230973\pi$$
−0.748086 + 0.663602i $$0.769027\pi$$
$$110$$ 0 0
$$111$$ 17.3205i 1.64399i
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 12.0000 1.12390
$$115$$ − 10.3923i − 0.969087i
$$116$$ −3.00000 −0.278543
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ 6.00000 0.547723
$$121$$ 11.0000 1.00000
$$122$$ − 1.73205i − 0.156813i
$$123$$ − 10.3923i − 0.937043i
$$124$$ 3.46410i 0.311086i
$$125$$ 12.1244i 1.08444i
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 12.1244i 1.07165i
$$129$$ 16.0000 1.40872
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 6.00000 0.518321
$$135$$ − 6.92820i − 0.596285i
$$136$$ 5.19615i 0.445566i
$$137$$ 15.5885i 1.33181i 0.746036 + 0.665906i $$0.231955\pi$$
−0.746036 + 0.665906i $$0.768045\pi$$
$$138$$ 20.7846i 1.76930i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 6.92820i 0.583460i
$$142$$ −6.00000 −0.503509
$$143$$ 0 0
$$144$$ −5.00000 −0.416667
$$145$$ 5.19615i 0.431517i
$$146$$ −3.00000 −0.248282
$$147$$ 14.0000 1.15470
$$148$$ − 8.66025i − 0.711868i
$$149$$ 19.0526i 1.56085i 0.625252 + 0.780423i $$0.284996\pi$$
−0.625252 + 0.780423i $$0.715004\pi$$
$$150$$ − 6.92820i − 0.565685i
$$151$$ − 17.3205i − 1.40952i −0.709444 0.704761i $$-0.751054\pi$$
0.709444 0.704761i $$-0.248946\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −3.00000 −0.242536
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ − 6.92820i − 0.551178i
$$159$$ −6.00000 −0.475831
$$160$$ −9.00000 −0.711512
$$161$$ 0 0
$$162$$ 19.0526i 1.49691i
$$163$$ − 20.7846i − 1.62798i −0.580881 0.813988i $$-0.697292\pi$$
0.580881 0.813988i $$-0.302708\pi$$
$$164$$ 5.19615i 0.405751i
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ − 13.8564i − 1.07224i −0.844141 0.536120i $$-0.819889\pi$$
0.844141 0.536120i $$-0.180111\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −9.00000 −0.690268
$$171$$ 3.46410i 0.264906i
$$172$$ −8.00000 −0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ − 10.3923i − 0.787839i
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 13.8564i − 1.04151i
$$178$$ −12.0000 −0.899438
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ − 1.73205i − 0.129099i
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 10.3923i 0.766131i
$$185$$ −15.0000 −1.10282
$$186$$ −12.0000 −0.879883
$$187$$ 0 0
$$188$$ − 3.46410i − 0.252646i
$$189$$ 0 0
$$190$$ 10.3923i 0.753937i
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ − 5.19615i − 0.374027i −0.982357 0.187014i $$-0.940119\pi$$
0.982357 0.187014i $$-0.0598809\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 13.8564i 0.987228i 0.869681 + 0.493614i $$0.164324\pi$$
−0.869681 + 0.493614i $$0.835676\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ − 3.46410i − 0.244949i
$$201$$ 6.92820i 0.488678i
$$202$$ 5.19615i 0.365600i
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 9.00000 0.628587
$$206$$ 17.3205i 1.20678i
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 10.0000 0.688428 0.344214 0.938891i $$-0.388145\pi$$
0.344214 + 0.938891i $$0.388145\pi$$
$$212$$ 3.00000 0.206041
$$213$$ − 6.92820i − 0.474713i
$$214$$ − 10.3923i − 0.710403i
$$215$$ 13.8564i 0.944999i
$$216$$ 6.92820i 0.471405i
$$217$$ 0 0
$$218$$ 24.0000 1.62549
$$219$$ − 3.46410i − 0.234082i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 30.0000 2.01347
$$223$$ − 10.3923i − 0.695920i −0.937509 0.347960i $$-0.886874\pi$$
0.937509 0.347960i $$-0.113126\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 25.9808i 1.72821i
$$227$$ − 24.2487i − 1.60944i −0.593652 0.804722i $$-0.702314\pi$$
0.593652 0.804722i $$-0.297686\pi$$
$$228$$ − 6.92820i − 0.458831i
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ −18.0000 −1.18688
$$231$$ 0 0
$$232$$ − 5.19615i − 0.341144i
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 6.92820i 0.450988i
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 20.7846i 1.34444i 0.740349 + 0.672222i $$0.234660\pi$$
−0.740349 + 0.672222i $$0.765340\pi$$
$$240$$ − 17.3205i − 1.11803i
$$241$$ 1.73205i 0.111571i 0.998443 + 0.0557856i $$0.0177663\pi$$
−0.998443 + 0.0557856i $$0.982234\pi$$
$$242$$ − 19.0526i − 1.22474i
$$243$$ −10.0000 −0.641500
$$244$$ −1.00000 −0.0640184
$$245$$ 12.1244i 0.774597i
$$246$$ −18.0000 −1.14764
$$247$$ 0 0
$$248$$ −6.00000 −0.381000
$$249$$ − 27.7128i − 1.75623i
$$250$$ 21.0000 1.32816
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 3.46410i 0.217357i
$$255$$ − 10.3923i − 0.650791i
$$256$$ 19.0000 1.18750
$$257$$ 3.00000 0.187135 0.0935674 0.995613i $$-0.470173\pi$$
0.0935674 + 0.995613i $$0.470173\pi$$
$$258$$ − 27.7128i − 1.72532i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ − 31.1769i − 1.92612i
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ − 5.19615i − 0.319197i
$$266$$ 0 0
$$267$$ − 13.8564i − 0.847998i
$$268$$ − 3.46410i − 0.211604i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −12.0000 −0.730297
$$271$$ 20.7846i 1.26258i 0.775549 + 0.631288i $$0.217473\pi$$
−0.775549 + 0.631288i $$0.782527\pi$$
$$272$$ 15.0000 0.909509
$$273$$ 0 0
$$274$$ 27.0000 1.63113
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ −7.00000 −0.420589 −0.210295 0.977638i $$-0.567442\pi$$
−0.210295 + 0.977638i $$0.567442\pi$$
$$278$$ 6.92820i 0.415526i
$$279$$ − 3.46410i − 0.207390i
$$280$$ 0 0
$$281$$ 22.5167i 1.34323i 0.740900 + 0.671616i $$0.234399\pi$$
−0.740900 + 0.671616i $$0.765601\pi$$
$$282$$ 12.0000 0.714590
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 3.46410i 0.205557i
$$285$$ −12.0000 −0.710819
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 5.19615i 0.306186i
$$289$$ −8.00000 −0.470588
$$290$$ 9.00000 0.528498
$$291$$ − 13.8564i − 0.812277i
$$292$$ 1.73205i 0.101361i
$$293$$ − 5.19615i − 0.303562i −0.988414 0.151781i $$-0.951499\pi$$
0.988414 0.151781i $$-0.0485009\pi$$
$$294$$ − 24.2487i − 1.41421i
$$295$$ 12.0000 0.698667
$$296$$ 15.0000 0.871857
$$297$$ 0 0
$$298$$ 33.0000 1.91164
$$299$$ 0 0
$$300$$ −4.00000 −0.230940
$$301$$ 0 0
$$302$$ −30.0000 −1.72631
$$303$$ −6.00000 −0.344691
$$304$$ − 17.3205i − 0.993399i
$$305$$ 1.73205i 0.0991769i
$$306$$ 5.19615i 0.297044i
$$307$$ 17.3205i 0.988534i 0.869310 + 0.494267i $$0.164563\pi$$
−0.869310 + 0.494267i $$0.835437\pi$$
$$308$$ 0 0
$$309$$ −20.0000 −1.13776
$$310$$ − 10.3923i − 0.590243i
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 22.5167i 1.27069i
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 5.19615i 0.291845i 0.989296 + 0.145922i $$0.0466150\pi$$
−0.989296 + 0.145922i $$0.953385\pi$$
$$318$$ 10.3923i 0.582772i
$$319$$ 0 0
$$320$$ − 1.73205i − 0.0968246i
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 10.3923i − 0.578243i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −36.0000 −1.99386
$$327$$ 27.7128i 1.53252i
$$328$$ −9.00000 −0.496942
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 27.7128i 1.52323i 0.648027 + 0.761617i $$0.275594\pi$$
−0.648027 + 0.761617i $$0.724406\pi$$
$$332$$ 13.8564i 0.760469i
$$333$$ 8.66025i 0.474579i
$$334$$ −24.0000 −1.31322
$$335$$ −6.00000 −0.327815
$$336$$ 0 0
$$337$$ −23.0000 −1.25289 −0.626445 0.779466i $$-0.715491\pi$$
−0.626445 + 0.779466i $$0.715491\pi$$
$$338$$ 0 0
$$339$$ −30.0000 −1.62938
$$340$$ 5.19615i 0.281801i
$$341$$ 0 0
$$342$$ 6.00000 0.324443
$$343$$ 0 0
$$344$$ − 13.8564i − 0.747087i
$$345$$ − 20.7846i − 1.11901i
$$346$$ − 10.3923i − 0.558694i
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 13.8564i 0.741716i 0.928689 + 0.370858i $$0.120936\pi$$
−0.928689 + 0.370858i $$0.879064\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 32.9090i 1.75157i 0.482704 + 0.875784i $$0.339655\pi$$
−0.482704 + 0.875784i $$0.660345\pi$$
$$354$$ −24.0000 −1.27559
$$355$$ 6.00000 0.318447
$$356$$ 6.92820i 0.367194i
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 6.92820i 0.365657i 0.983145 + 0.182828i $$0.0585252\pi$$
−0.983145 + 0.182828i $$0.941475\pi$$
$$360$$ 3.00000 0.158114
$$361$$ 7.00000 0.368421
$$362$$ − 19.0526i − 1.00138i
$$363$$ 22.0000 1.15470
$$364$$ 0 0
$$365$$ 3.00000 0.157027
$$366$$ − 3.46410i − 0.181071i
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 30.0000 1.56386
$$369$$ − 5.19615i − 0.270501i
$$370$$ 25.9808i 1.35068i
$$371$$ 0 0
$$372$$ 6.92820i 0.359211i
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ 0 0
$$375$$ 24.2487i 1.25220i
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 24.2487i − 1.24557i −0.782392 0.622786i $$-0.786001\pi$$
0.782392 0.622786i $$-0.213999\pi$$
$$380$$ 6.00000 0.307794
$$381$$ −4.00000 −0.204926
$$382$$ − 31.1769i − 1.59515i
$$383$$ 20.7846i 1.06204i 0.847358 + 0.531022i $$0.178192\pi$$
−0.847358 + 0.531022i $$0.821808\pi$$
$$384$$ 24.2487i 1.23744i
$$385$$ 0 0
$$386$$ −9.00000 −0.458088
$$387$$ 8.00000 0.406663
$$388$$ 6.92820i 0.351726i
$$389$$ −9.00000 −0.456318 −0.228159 0.973624i $$-0.573271\pi$$
−0.228159 + 0.973624i $$0.573271\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ − 12.1244i − 0.612372i
$$393$$ 36.0000 1.81596
$$394$$ 24.0000 1.20910
$$395$$ 6.92820i 0.348596i
$$396$$ 0 0
$$397$$ 13.8564i 0.695433i 0.937600 + 0.347717i $$0.113043\pi$$
−0.937600 + 0.347717i $$0.886957\pi$$
$$398$$ 3.46410i 0.173640i
$$399$$ 0 0
$$400$$ −10.0000 −0.500000
$$401$$ − 1.73205i − 0.0864945i −0.999064 0.0432472i $$-0.986230\pi$$
0.999064 0.0432472i $$-0.0137703\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 0 0
$$404$$ 3.00000 0.149256
$$405$$ − 19.0526i − 0.946729i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 10.3923i 0.514496i
$$409$$ − 15.5885i − 0.770800i −0.922750 0.385400i $$-0.874064\pi$$
0.922750 0.385400i $$-0.125936\pi$$
$$410$$ − 15.5885i − 0.769859i
$$411$$ 31.1769i 1.53784i
$$412$$ 10.0000 0.492665
$$413$$ 0 0
$$414$$ 10.3923i 0.510754i
$$415$$ 24.0000 1.17811
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ 18.0000 0.879358 0.439679 0.898155i $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ 0 0
$$421$$ − 15.5885i − 0.759735i −0.925041 0.379867i $$-0.875970\pi$$
0.925041 0.379867i $$-0.124030\pi$$
$$422$$ − 17.3205i − 0.843149i
$$423$$ 3.46410i 0.168430i
$$424$$ 5.19615i 0.252347i
$$425$$ −6.00000 −0.291043
$$426$$ −12.0000 −0.581402
$$427$$ 0 0
$$428$$ −6.00000 −0.290021
$$429$$ 0 0
$$430$$ 24.0000 1.15738
$$431$$ − 6.92820i − 0.333720i −0.985981 0.166860i $$-0.946637\pi$$
0.985981 0.166860i $$-0.0533628\pi$$
$$432$$ 20.0000 0.962250
$$433$$ −17.0000 −0.816968 −0.408484 0.912766i $$-0.633942\pi$$
−0.408484 + 0.912766i $$0.633942\pi$$
$$434$$ 0 0
$$435$$ 10.3923i 0.498273i
$$436$$ − 13.8564i − 0.663602i
$$437$$ − 20.7846i − 0.994263i
$$438$$ −6.00000 −0.286691
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ − 17.3205i − 0.821995i
$$445$$ 12.0000 0.568855
$$446$$ −18.0000 −0.852325
$$447$$ 38.1051i 1.80231i
$$448$$ 0 0
$$449$$ 6.92820i 0.326962i 0.986546 + 0.163481i $$0.0522723\pi$$
−0.986546 + 0.163481i $$0.947728\pi$$
$$450$$ − 3.46410i − 0.163299i
$$451$$ 0 0
$$452$$ 15.0000 0.705541
$$453$$ − 34.6410i − 1.62758i
$$454$$ −42.0000 −1.97116
$$455$$ 0 0
$$456$$ 12.0000 0.561951
$$457$$ 1.73205i 0.0810219i 0.999179 + 0.0405110i $$0.0128986\pi$$
−0.999179 + 0.0405110i $$0.987101\pi$$
$$458$$ 0 0
$$459$$ 12.0000 0.560112
$$460$$ 10.3923i 0.484544i
$$461$$ − 22.5167i − 1.04871i −0.851501 0.524353i $$-0.824307\pi$$
0.851501 0.524353i $$-0.175693\pi$$
$$462$$ 0 0
$$463$$ − 13.8564i − 0.643962i −0.946746 0.321981i $$-0.895651\pi$$
0.946746 0.321981i $$-0.104349\pi$$
$$464$$ −15.0000 −0.696358
$$465$$ 12.0000 0.556487
$$466$$ − 10.3923i − 0.481414i
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 10.3923i 0.479361i
$$471$$ −26.0000 −1.19802
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ − 13.8564i − 0.636446i
$$475$$ 6.92820i 0.317888i
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ 36.0000 1.64660
$$479$$ 24.2487i 1.10795i 0.832533 + 0.553976i $$0.186890\pi$$
−0.832533 + 0.553976i $$0.813110\pi$$
$$480$$ −18.0000 −0.821584
$$481$$ 0 0
$$482$$ 3.00000 0.136646
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 12.0000 0.544892
$$486$$ 17.3205i 0.785674i
$$487$$ 6.92820i 0.313947i 0.987603 + 0.156973i $$0.0501737\pi$$
−0.987603 + 0.156973i $$0.949826\pi$$
$$488$$ − 1.73205i − 0.0784063i
$$489$$ − 41.5692i − 1.87983i
$$490$$ 21.0000 0.948683
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 10.3923i 0.468521i
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 17.3205i 0.777714i
$$497$$ 0 0
$$498$$ −48.0000 −2.15093
$$499$$ − 31.1769i − 1.39567i −0.716258 0.697835i $$-0.754147\pi$$
0.716258 0.697835i $$-0.245853\pi$$
$$500$$ − 12.1244i − 0.542218i
$$501$$ − 27.7128i − 1.23812i
$$502$$ 31.1769i 1.39149i
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ − 5.19615i − 0.231226i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 19.0526i 0.844490i 0.906482 + 0.422245i $$0.138758\pi$$
−0.906482 + 0.422245i $$0.861242\pi$$
$$510$$ −18.0000 −0.797053
$$511$$ 0 0
$$512$$ − 8.66025i − 0.382733i
$$513$$ − 13.8564i − 0.611775i
$$514$$ − 5.19615i − 0.229192i
$$515$$ − 17.3205i − 0.763233i
$$516$$ −16.0000 −0.704361
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 9.00000 0.394297 0.197149 0.980374i $$-0.436832\pi$$
0.197149 + 0.980374i $$0.436832\pi$$
$$522$$ − 5.19615i − 0.227429i
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −18.0000 −0.786334
$$525$$ 0 0
$$526$$ − 20.7846i − 0.906252i
$$527$$ 10.3923i 0.452696i
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −9.00000 −0.390935
$$531$$ − 6.92820i − 0.300658i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −24.0000 −1.03858
$$535$$ 10.3923i 0.449299i
$$536$$ 6.00000 0.259161
$$537$$ 0 0
$$538$$ 10.3923i 0.448044i
$$539$$ 0 0
$$540$$ 6.92820i 0.298142i
$$541$$ − 29.4449i − 1.26593i −0.774179 0.632967i $$-0.781837\pi$$
0.774179 0.632967i $$-0.218163\pi$$
$$542$$ 36.0000 1.54633
$$543$$ 22.0000 0.944110
$$544$$ − 15.5885i − 0.668350i
$$545$$ −24.0000 −1.02805
$$546$$ 0 0
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ − 15.5885i − 0.665906i
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ 10.3923i 0.442727i
$$552$$ 20.7846i 0.884652i
$$553$$ 0 0
$$554$$ 12.1244i 0.515115i
$$555$$ −30.0000 −1.27343
$$556$$ 4.00000 0.169638
$$557$$ − 15.5885i − 0.660504i −0.943893 0.330252i $$-0.892866\pi$$
0.943893 0.330252i $$-0.107134\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 39.0000 1.64512
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ − 6.92820i − 0.291730i
$$565$$ − 25.9808i − 1.09302i
$$566$$ − 6.92820i − 0.291214i
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 20.7846i 0.870572i
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ 36.0000 1.50392
$$574$$ 0 0
$$575$$ −12.0000 −0.500435
$$576$$ −1.00000 −0.0416667
$$577$$ 19.0526i 0.793168i 0.917998 + 0.396584i $$0.129805\pi$$
−0.917998 + 0.396584i $$0.870195\pi$$
$$578$$ 13.8564i 0.576351i
$$579$$ − 10.3923i − 0.431889i
$$580$$ − 5.19615i − 0.215758i
$$581$$ 0 0
$$582$$ −24.0000 −0.994832
$$583$$ 0 0
$$584$$ −3.00000 −0.124141
$$585$$ 0 0
$$586$$ −9.00000 −0.371787
$$587$$ − 20.7846i − 0.857873i −0.903335 0.428936i $$-0.858888\pi$$
0.903335 0.428936i $$-0.141112\pi$$
$$588$$ −14.0000 −0.577350
$$589$$ 12.0000 0.494451
$$590$$ − 20.7846i − 0.855689i
$$591$$ 27.7128i 1.13995i
$$592$$ − 43.3013i − 1.77967i
$$593$$ 25.9808i 1.06690i 0.845831 + 0.533451i $$0.179105\pi$$
−0.845831 + 0.533451i $$0.820895\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ − 19.0526i − 0.780423i
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ − 6.92820i − 0.282843i
$$601$$ 25.0000 1.01977 0.509886 0.860242i $$-0.329688\pi$$
0.509886 + 0.860242i $$0.329688\pi$$
$$602$$ 0 0
$$603$$ 3.46410i 0.141069i
$$604$$ 17.3205i 0.704761i
$$605$$ 19.0526i 0.774597i
$$606$$ 10.3923i 0.422159i
$$607$$ −34.0000 −1.38002 −0.690009 0.723801i $$-0.742393\pi$$
−0.690009 + 0.723801i $$0.742393\pi$$
$$608$$ −18.0000 −0.729996
$$609$$ 0 0
$$610$$ 3.00000 0.121466
$$611$$ 0 0
$$612$$ 3.00000 0.121268
$$613$$ − 12.1244i − 0.489698i −0.969561 0.244849i $$-0.921262\pi$$
0.969561 0.244849i $$-0.0787384\pi$$
$$614$$ 30.0000 1.21070
$$615$$ 18.0000 0.725830
$$616$$ 0 0
$$617$$ 22.5167i 0.906487i 0.891387 + 0.453243i $$0.149733\pi$$
−0.891387 + 0.453243i $$0.850267\pi$$
$$618$$ 34.6410i 1.39347i
$$619$$ 20.7846i 0.835404i 0.908584 + 0.417702i $$0.137164\pi$$
−0.908584 + 0.417702i $$0.862836\pi$$
$$620$$ −6.00000 −0.240966
$$621$$ 24.0000 0.963087
$$622$$ 51.9615i 2.08347i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ − 17.3205i − 0.692267i
$$627$$ 0 0
$$628$$ 13.0000 0.518756
$$629$$ − 25.9808i − 1.03592i
$$630$$ 0 0
$$631$$ 48.4974i 1.93065i 0.261048 + 0.965326i $$0.415932\pi$$
−0.261048 + 0.965326i $$0.584068\pi$$
$$632$$ − 6.92820i − 0.275589i
$$633$$ 20.0000 0.794929
$$634$$ 9.00000 0.357436
$$635$$ − 3.46410i − 0.137469i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 3.46410i − 0.137038i
$$640$$ −21.0000 −0.830098
$$641$$ 33.0000 1.30342 0.651711 0.758468i $$-0.274052\pi$$
0.651711 + 0.758468i $$0.274052\pi$$
$$642$$ − 20.7846i − 0.820303i
$$643$$ − 13.8564i − 0.546443i −0.961951 0.273222i $$-0.911911\pi$$
0.961951 0.273222i $$-0.0880892\pi$$
$$644$$ 0 0
$$645$$ 27.7128i 1.09119i
$$646$$ −18.0000 −0.708201
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 19.0526i 0.748455i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.7846i 0.813988i
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 48.0000 1.87695
$$655$$ 31.1769i 1.21818i
$$656$$ 25.9808i 1.01438i
$$657$$ − 1.73205i − 0.0675737i
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ − 46.7654i − 1.81896i −0.415745 0.909481i $$-0.636479\pi$$
0.415745 0.909481i $$-0.363521\pi$$
$$662$$ 48.0000 1.86557
$$663$$ 0 0
$$664$$ −24.0000 −0.931381
$$665$$ 0 0
$$666$$ 15.0000 0.581238
$$667$$ −18.0000 −0.696963
$$668$$ 13.8564i 0.536120i
$$669$$ − 20.7846i − 0.803579i
$$670$$ 10.3923i 0.401490i
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 39.8372i 1.53447i
$$675$$ −8.00000 −0.307920
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 51.9615i 1.99557i
$$679$$ 0 0
$$680$$ −9.00000 −0.345134
$$681$$ − 48.4974i − 1.85843i
$$682$$ 0 0
$$683$$ − 24.2487i − 0.927851i −0.885874 0.463926i $$-0.846441\pi$$
0.885874 0.463926i $$-0.153559\pi$$
$$684$$ − 3.46410i − 0.132453i
$$685$$ −27.0000 −1.03162
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −40.0000 −1.52499
$$689$$ 0 0
$$690$$ −36.0000 −1.37050
$$691$$ 13.8564i 0.527123i 0.964643 + 0.263561i $$0.0848971\pi$$
−0.964643 + 0.263561i $$0.915103\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 51.9615i 1.97243i
$$695$$ − 6.92820i − 0.262802i
$$696$$ − 10.3923i − 0.393919i
$$697$$ 15.5885i 0.590455i
$$698$$ 24.0000 0.908413
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ −30.0000 −1.13147
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 57.0000 2.14522
$$707$$ 0 0
$$708$$ 13.8564i 0.520756i
$$709$$ 5.19615i 0.195146i 0.995228 + 0.0975728i $$0.0311079\pi$$
−0.995228 + 0.0975728i $$0.968892\pi$$
$$710$$ − 10.3923i − 0.390016i
$$711$$ 4.00000 0.150012
$$712$$ −12.0000 −0.449719
$$713$$ 20.7846i 0.778390i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 41.5692i 1.55243i
$$718$$ 12.0000 0.447836
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ − 8.66025i − 0.322749i
$$721$$ 0 0
$$722$$ − 12.1244i − 0.451222i
$$723$$ 3.46410i 0.128831i
$$724$$ −11.0000 −0.408812
$$725$$ 6.00000 0.222834
$$726$$ − 38.1051i − 1.41421i
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ − 5.19615i − 0.192318i
$$731$$ −24.0000 −0.887672
$$732$$ −2.00000 −0.0739221
$$733$$ − 12.1244i − 0.447823i −0.974609 0.223912i $$-0.928117\pi$$
0.974609 0.223912i $$-0.0718827\pi$$
$$734$$ 38.1051i 1.40649i
$$735$$ 24.2487i 0.894427i
$$736$$ − 31.1769i − 1.14920i
$$737$$ 0 0
$$738$$ −9.00000 −0.331295
$$739$$ − 20.7846i − 0.764574i −0.924044 0.382287i $$-0.875137\pi$$
0.924044 0.382287i $$-0.124863\pi$$
$$740$$ 15.0000 0.551411
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 34.6410i − 1.27086i −0.772160 0.635428i $$-0.780824\pi$$
0.772160 0.635428i $$-0.219176\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ −33.0000 −1.20903
$$746$$ − 32.9090i − 1.20488i
$$747$$ − 13.8564i − 0.506979i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 42.0000 1.53362
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ − 17.3205i − 0.631614i
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ −42.0000 −1.52551
$$759$$ 0 0
$$760$$ 10.3923i 0.376969i
$$761$$ − 34.6410i − 1.25574i −0.778320 0.627868i $$-0.783928\pi$$
0.778320 0.627868i $$-0.216072\pi$$
$$762$$ 6.92820i 0.250982i
$$763$$ 0 0
$$764$$ −18.0000 −0.651217
$$765$$ − 5.19615i − 0.187867i
$$766$$ 36.0000 1.30073
$$767$$ 0 0
$$768$$ 38.0000 1.37121
$$769$$ 6.92820i 0.249837i 0.992167 + 0.124919i $$0.0398670\pi$$
−0.992167 + 0.124919i $$0.960133\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 5.19615i 0.187014i
$$773$$ 34.6410i 1.24595i 0.782241 + 0.622975i $$0.214076\pi$$
−0.782241 + 0.622975i $$0.785924\pi$$
$$774$$ − 13.8564i − 0.498058i
$$775$$ − 6.92820i − 0.248868i
$$776$$ −12.0000 −0.430775
$$777$$ 0 0
$$778$$ 15.5885i 0.558873i
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 31.1769i − 1.11488i
$$783$$ −12.0000 −0.428845
$$784$$ −35.0000 −1.25000
$$785$$ − 22.5167i − 0.803654i
$$786$$ − 62.3538i − 2.22409i
$$787$$ − 38.1051i − 1.35830i −0.733999 0.679150i $$-0.762348\pi$$
0.733999 0.679150i $$-0.237652\pi$$
$$788$$ − 13.8564i − 0.493614i
$$789$$ 24.0000 0.854423
$$790$$ 12.0000 0.426941
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 24.0000 0.851728
$$795$$ − 10.3923i − 0.368577i
$$796$$ 2.00000 0.0708881
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ − 10.3923i − 0.367653i
$$800$$ 10.3923i 0.367423i
$$801$$ − 6.92820i − 0.244796i
$$802$$ −3.00000 −0.105934
$$803$$ 0 0
$$804$$ − 6.92820i − 0.244339i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12.0000 −0.422420
$$808$$ 5.19615i 0.182800i
$$809$$ 33.0000 1.16022 0.580109 0.814539i $$-0.303010\pi$$
0.580109 + 0.814539i $$0.303010\pi$$
$$810$$ −33.0000 −1.15950
$$811$$ 38.1051i 1.33805i 0.743239 + 0.669026i $$0.233288\pi$$
−0.743239 + 0.669026i $$0.766712\pi$$
$$812$$ 0 0
$$813$$ 41.5692i 1.45790i
$$814$$ 0 0
$$815$$ 36.0000 1.26102
$$816$$ 30.0000 1.05021
$$817$$ 27.7128i 0.969549i
$$818$$ −27.0000 −0.944033
$$819$$ 0 0
$$820$$ −9.00000 −0.314294
$$821$$ − 41.5692i − 1.45078i −0.688340 0.725388i $$-0.741660\pi$$
0.688340 0.725388i $$-0.258340\pi$$
$$822$$ 54.0000 1.88347
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 17.3205i 0.603388i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 20.7846i − 0.722752i −0.932420 0.361376i $$-0.882307\pi$$
0.932420 0.361376i $$-0.117693\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 25.0000 0.868286 0.434143 0.900844i $$-0.357051\pi$$
0.434143 + 0.900844i $$0.357051\pi$$
$$830$$ − 41.5692i − 1.44289i
$$831$$ −14.0000 −0.485655
$$832$$ 0 0
$$833$$ −21.0000 −0.727607
$$834$$ 13.8564i 0.479808i
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ 13.8564i 0.478947i
$$838$$ − 31.1769i − 1.07699i
$$839$$ − 45.0333i − 1.55472i −0.629054 0.777361i $$-0.716558\pi$$
0.629054 0.777361i $$-0.283442\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −27.0000 −0.930481
$$843$$ 45.0333i 1.55103i
$$844$$ −10.0000 −0.344214
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 0 0
$$848$$ 15.0000 0.515102
$$849$$ 8.00000 0.274559
$$850$$ 10.3923i 0.356453i
$$851$$ − 51.9615i − 1.78122i
$$852$$ 6.92820i 0.237356i
$$853$$ − 25.9808i − 0.889564i −0.895639 0.444782i $$-0.853281\pi$$
0.895639 0.444782i $$-0.146719\pi$$
$$854$$ 0 0
$$855$$ −6.00000 −0.205196
$$856$$ − 10.3923i − 0.355202i
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ − 13.8564i − 0.472500i
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ 27.7128i 0.943355i 0.881771 + 0.471678i $$0.156351\pi$$
−0.881771 + 0.471678i $$0.843649\pi$$
$$864$$ − 20.7846i − 0.707107i
$$865$$ 10.3923i 0.353349i
$$866$$ 29.4449i 1.00058i
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 18.0000 0.610257
$$871$$ 0 0
$$872$$ 24.0000 0.812743
$$873$$ − 6.92820i − 0.234484i
$$874$$ −36.0000 −1.21772
$$875$$ 0 0
$$876$$ 3.46410i 0.117041i
$$877$$ − 12.1244i − 0.409410i −0.978824 0.204705i $$-0.934376\pi$$
0.978824 0.204705i $$-0.0656236\pi$$
$$878$$ 48.4974i 1.63671i
$$879$$ − 10.3923i − 0.350524i
$$880$$ 0 0
$$881$$ 27.0000 0.909653 0.454827 0.890580i $$-0.349701\pi$$
0.454827 + 0.890580i $$0.349701\pi$$
$$882$$ − 12.1244i − 0.408248i
$$883$$ 10.0000 0.336527 0.168263 0.985742i $$-0.446184\pi$$
0.168263 + 0.985742i $$0.446184\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 20.7846i 0.698273i
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 30.0000 1.00673
$$889$$ 0 0
$$890$$ − 20.7846i − 0.696702i
$$891$$ 0 0
$$892$$ 10.3923i 0.347960i
$$893$$ −12.0000 −0.401565
$$894$$ 66.0000 2.20737
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 12.0000 0.400445
$$899$$ − 10.3923i − 0.346603i
$$900$$ −2.00000 −0.0666667
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 25.9808i 0.864107i
$$905$$ 19.0526i 0.633328i
$$906$$ −60.0000 −1.99337
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 24.2487i 0.804722i
$$909$$ −3.00000 −0.0995037
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ − 34.6410i − 1.14708i
$$913$$ 0 0
$$914$$ 3.00000 0.0992312
$$915$$ 3.46410i 0.114520i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ − 20.7846i − 0.685994i
$$919$$ 22.0000 0.725713 0.362857 0.931845i $$-0.381802\pi$$
0.362857 + 0.931845i $$0.381802\pi$$
$$920$$ −18.0000 −0.593442
$$921$$ 34.6410i 1.14146i
$$922$$ −39.0000 −1.28440
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 17.3205i 0.569495i
$$926$$ −24.0000 −0.788689
$$927$$ −10.0000 −0.328443
$$928$$ 15.5885i 0.511716i
$$929$$ 46.7654i 1.53432i 0.641455 + 0.767161i $$0.278331\pi$$
−0.641455 + 0.767161i $$0.721669\pi$$
$$930$$ − 20.7846i − 0.681554i
$$931$$ 24.2487i 0.794719i
$$932$$ −6.00000 −0.196537
$$933$$ −60.0000 −1.96431
$$934$$ − 20.7846i − 0.680093i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ 6.00000 0.195698
$$941$$ − 20.7846i − 0.677559i −0.940866 0.338779i $$-0.889986\pi$$
0.940866 0.338779i $$-0.110014\pi$$
$$942$$ 45.0333i 1.46726i
$$943$$ 31.1769i 1.01526i
$$944$$ 34.6410i 1.12747i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 17.3205i − 0.562841i −0.959585 0.281420i $$-0.909194\pi$$
0.959585 0.281420i $$-0.0908056\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ 12.0000 0.389331
$$951$$ 10.3923i 0.336994i
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 5.19615i 0.168232i
$$955$$ 31.1769i 1.00886i
$$956$$ − 20.7846i − 0.672222i
$$957$$ 0 0
$$958$$ 42.0000 1.35696
$$959$$ 0 0
$$960$$ − 3.46410i − 0.111803i
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ 6.00000 0.193347
$$964$$ − 1.73205i − 0.0557856i
$$965$$ 9.00000 0.289720
$$966$$ 0 0
$$967$$ − 58.8897i − 1.89377i −0.321578 0.946883i $$-0.604213\pi$$
0.321578 0.946883i $$-0.395787\pi$$
$$968$$ − 19.0526i − 0.612372i
$$969$$ − 20.7846i − 0.667698i
$$970$$ − 20.7846i − 0.667354i
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 0 0
$$974$$ 12.0000 0.384505
$$975$$ 0 0
$$976$$ −5.00000 −0.160046
$$977$$ 43.3013i 1.38533i 0.721259 + 0.692665i $$0.243564\pi$$
−0.721259 + 0.692665i $$0.756436\pi$$
$$978$$ −72.0000 −2.30231
$$979$$ 0 0
$$980$$ − 12.1244i − 0.387298i
$$981$$ 13.8564i 0.442401i
$$982$$ − 20.7846i − 0.663264i
$$983$$ − 51.9615i − 1.65732i −0.559756 0.828658i $$-0.689105\pi$$
0.559756 0.828658i $$-0.310895\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ −24.0000 −0.764704
$$986$$ 15.5885i 0.496438i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 18.0000 0.571501
$$993$$ 55.4256i 1.75888i
$$994$$ 0 0
$$995$$ − 3.46410i − 0.109819i
$$996$$ 27.7128i 0.878114i
$$997$$ 17.0000 0.538395 0.269198 0.963085i $$-0.413241\pi$$
0.269198 + 0.963085i $$0.413241\pi$$
$$998$$ −54.0000 −1.70934
$$999$$ − 34.6410i − 1.09599i
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.a.168.1 2
3.2 odd 2 1521.2.b.a.1351.2 2
4.3 odd 2 2704.2.f.b.337.2 2
13.2 odd 12 169.2.c.a.22.2 4
13.3 even 3 169.2.e.a.147.1 2
13.4 even 6 169.2.e.a.23.1 2
13.5 odd 4 169.2.a.a.1.1 2
13.6 odd 12 169.2.c.a.146.2 4
13.7 odd 12 169.2.c.a.146.1 4
13.8 odd 4 169.2.a.a.1.2 2
13.9 even 3 13.2.e.a.10.1 yes 2
13.10 even 6 13.2.e.a.4.1 2
13.11 odd 12 169.2.c.a.22.1 4
13.12 even 2 inner 169.2.b.a.168.2 2
39.5 even 4 1521.2.a.k.1.2 2
39.8 even 4 1521.2.a.k.1.1 2
39.23 odd 6 117.2.q.c.82.1 2
39.35 odd 6 117.2.q.c.10.1 2
39.38 odd 2 1521.2.b.a.1351.1 2
52.23 odd 6 208.2.w.b.17.1 2
52.31 even 4 2704.2.a.o.1.2 2
52.35 odd 6 208.2.w.b.49.1 2
52.47 even 4 2704.2.a.o.1.1 2
52.51 odd 2 2704.2.f.b.337.1 2
65.9 even 6 325.2.n.a.101.1 2
65.22 odd 12 325.2.m.a.49.1 4
65.23 odd 12 325.2.m.a.199.1 4
65.34 odd 4 4225.2.a.v.1.1 2
65.44 odd 4 4225.2.a.v.1.2 2
65.48 odd 12 325.2.m.a.49.2 4
65.49 even 6 325.2.n.a.251.1 2
65.62 odd 12 325.2.m.a.199.2 4
91.9 even 3 637.2.u.c.361.1 2
91.10 odd 6 637.2.u.b.30.1 2
91.23 even 6 637.2.k.a.459.1 2
91.34 even 4 8281.2.a.q.1.2 2
91.48 odd 6 637.2.q.a.491.1 2
91.61 odd 6 637.2.u.b.361.1 2
91.62 odd 6 637.2.q.a.589.1 2
91.74 even 3 637.2.k.a.569.1 2
91.75 odd 6 637.2.k.c.459.1 2
91.83 even 4 8281.2.a.q.1.1 2
91.87 odd 6 637.2.k.c.569.1 2
91.88 even 6 637.2.u.c.30.1 2
104.35 odd 6 832.2.w.a.257.1 2
104.61 even 6 832.2.w.d.257.1 2
104.75 odd 6 832.2.w.a.641.1 2
104.101 even 6 832.2.w.d.641.1 2
156.23 even 6 1872.2.by.d.433.1 2
156.35 even 6 1872.2.by.d.1297.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 13.10 even 6
13.2.e.a.10.1 yes 2 13.9 even 3
117.2.q.c.10.1 2 39.35 odd 6
117.2.q.c.82.1 2 39.23 odd 6
169.2.a.a.1.1 2 13.5 odd 4
169.2.a.a.1.2 2 13.8 odd 4
169.2.b.a.168.1 2 1.1 even 1 trivial
169.2.b.a.168.2 2 13.12 even 2 inner
169.2.c.a.22.1 4 13.11 odd 12
169.2.c.a.22.2 4 13.2 odd 12
169.2.c.a.146.1 4 13.7 odd 12
169.2.c.a.146.2 4 13.6 odd 12
169.2.e.a.23.1 2 13.4 even 6
169.2.e.a.147.1 2 13.3 even 3
208.2.w.b.17.1 2 52.23 odd 6
208.2.w.b.49.1 2 52.35 odd 6
325.2.m.a.49.1 4 65.22 odd 12
325.2.m.a.49.2 4 65.48 odd 12
325.2.m.a.199.1 4 65.23 odd 12
325.2.m.a.199.2 4 65.62 odd 12
325.2.n.a.101.1 2 65.9 even 6
325.2.n.a.251.1 2 65.49 even 6
637.2.k.a.459.1 2 91.23 even 6
637.2.k.a.569.1 2 91.74 even 3
637.2.k.c.459.1 2 91.75 odd 6
637.2.k.c.569.1 2 91.87 odd 6
637.2.q.a.491.1 2 91.48 odd 6
637.2.q.a.589.1 2 91.62 odd 6
637.2.u.b.30.1 2 91.10 odd 6
637.2.u.b.361.1 2 91.61 odd 6
637.2.u.c.30.1 2 91.88 even 6
637.2.u.c.361.1 2 91.9 even 3
832.2.w.a.257.1 2 104.35 odd 6
832.2.w.a.641.1 2 104.75 odd 6
832.2.w.d.257.1 2 104.61 even 6
832.2.w.d.641.1 2 104.101 even 6
1521.2.a.k.1.1 2 39.8 even 4
1521.2.a.k.1.2 2 39.5 even 4
1521.2.b.a.1351.1 2 39.38 odd 2
1521.2.b.a.1351.2 2 3.2 odd 2
1872.2.by.d.433.1 2 156.23 even 6
1872.2.by.d.1297.1 2 156.35 even 6
2704.2.a.o.1.1 2 52.47 even 4
2704.2.a.o.1.2 2 52.31 even 4
2704.2.f.b.337.1 2 52.51 odd 2
2704.2.f.b.337.2 2 4.3 odd 2
4225.2.a.v.1.1 2 65.34 odd 4
4225.2.a.v.1.2 2 65.44 odd 4
8281.2.a.q.1.1 2 91.83 even 4
8281.2.a.q.1.2 2 91.34 even 4