# Properties

 Label 169.2.a.a.1.1 Level $169$ Weight $2$ Character 169.1 Self dual yes Analytic conductor $1.349$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(1,169)]

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([0]))

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 169.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -3.46410 q^{6} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -3.46410 q^{6} +1.73205 q^{8} +1.00000 q^{9} -3.00000 q^{10} +2.00000 q^{12} +3.46410 q^{15} -5.00000 q^{16} +3.00000 q^{17} -1.73205 q^{18} +3.46410 q^{19} +1.73205 q^{20} +6.00000 q^{23} +3.46410 q^{24} -2.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} -6.00000 q^{30} -3.46410 q^{31} +5.19615 q^{32} -5.19615 q^{34} +1.00000 q^{36} -8.66025 q^{37} -6.00000 q^{38} +3.00000 q^{40} -5.19615 q^{41} -8.00000 q^{43} +1.73205 q^{45} -10.3923 q^{46} -3.46410 q^{47} -10.0000 q^{48} -7.00000 q^{49} +3.46410 q^{50} +6.00000 q^{51} -3.00000 q^{53} +6.92820 q^{54} +6.92820 q^{57} -5.19615 q^{58} +6.92820 q^{59} +3.46410 q^{60} +1.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +3.46410 q^{67} +3.00000 q^{68} +12.0000 q^{69} -3.46410 q^{71} +1.73205 q^{72} +1.73205 q^{73} +15.0000 q^{74} -4.00000 q^{75} +3.46410 q^{76} +4.00000 q^{79} -8.66025 q^{80} -11.0000 q^{81} +9.00000 q^{82} -13.8564 q^{83} +5.19615 q^{85} +13.8564 q^{86} +6.00000 q^{87} +6.92820 q^{89} -3.00000 q^{90} +6.00000 q^{92} -6.92820 q^{93} +6.00000 q^{94} +6.00000 q^{95} +10.3923 q^{96} -6.92820 q^{97} +12.1244 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 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

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.73205 0.774597 0.387298 0.921954i $$-0.373408\pi$$
0.387298 + 0.921954i $$0.373408\pi$$
$$6$$ −3.46410 −1.41421
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.73205 0.612372
$$9$$ 1.00000 0.333333
$$10$$ −3.00000 −0.948683
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 3.46410 0.894427
$$16$$ −5.00000 −1.25000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ −1.73205 −0.408248
$$19$$ 3.46410 0.794719 0.397360 0.917663i $$-0.369927\pi$$
0.397360 + 0.917663i $$0.369927\pi$$
$$20$$ 1.73205 0.387298
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 3.46410 0.707107
$$25$$ −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
$$31$$ −3.46410 −0.622171 −0.311086 0.950382i $$-0.600693\pi$$
−0.311086 + 0.950382i $$0.600693\pi$$
$$32$$ 5.19615 0.918559
$$33$$ 0 0
$$34$$ −5.19615 −0.891133
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.66025 −1.42374 −0.711868 0.702313i $$-0.752151\pi$$
−0.711868 + 0.702313i $$0.752151\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ −5.19615 −0.811503 −0.405751 0.913984i $$-0.632990\pi$$
−0.405751 + 0.913984i $$0.632990\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 1.73205 0.258199
$$46$$ −10.3923 −1.53226
$$47$$ −3.46410 −0.505291 −0.252646 0.967559i $$-0.581301\pi$$
−0.252646 + 0.967559i $$0.581301\pi$$
$$48$$ −10.0000 −1.44338
$$49$$ −7.00000 −1.00000
$$50$$ 3.46410 0.489898
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 6.92820 0.942809
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.92820 0.917663
$$58$$ −5.19615 −0.682288
$$59$$ 6.92820 0.901975 0.450988 0.892530i $$-0.351072\pi$$
0.450988 + 0.892530i $$0.351072\pi$$
$$60$$ 3.46410 0.447214
$$61$$ 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
$$67$$ 3.46410 0.423207 0.211604 0.977356i $$-0.432131\pi$$
0.211604 + 0.977356i $$0.432131\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 12.0000 1.44463
$$70$$ 0 0
$$71$$ −3.46410 −0.411113 −0.205557 0.978645i $$-0.565900\pi$$
−0.205557 + 0.978645i $$0.565900\pi$$
$$72$$ 1.73205 0.204124
$$73$$ 1.73205 0.202721 0.101361 0.994850i $$-0.467680\pi$$
0.101361 + 0.994850i $$0.467680\pi$$
$$74$$ 15.0000 1.74371
$$75$$ −4.00000 −0.461880
$$76$$ 3.46410 0.397360
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ −8.66025 −0.968246
$$81$$ −11.0000 −1.22222
$$82$$ 9.00000 0.993884
$$83$$ −13.8564 −1.52094 −0.760469 0.649374i $$-0.775031\pi$$
−0.760469 + 0.649374i $$0.775031\pi$$
$$84$$ 0 0
$$85$$ 5.19615 0.563602
$$86$$ 13.8564 1.49417
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ 6.92820 0.734388 0.367194 0.930144i $$-0.380318\pi$$
0.367194 + 0.930144i $$0.380318\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ −6.92820 −0.718421
$$94$$ 6.00000 0.618853
$$95$$ 6.00000 0.615587
$$96$$ 10.3923 1.06066
$$97$$ −6.92820 −0.703452 −0.351726 0.936103i $$-0.614405\pi$$
−0.351726 + 0.936103i $$0.614405\pi$$
$$98$$ 12.1244 1.22474
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ −10.3923 −1.02899
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 5.19615 0.504695
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ 13.8564 1.32720 0.663602 0.748086i $$-0.269027\pi$$
0.663602 + 0.748086i $$0.269027\pi$$
$$110$$ 0 0
$$111$$ −17.3205 −1.64399
$$112$$ 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
$$115$$ 10.3923 0.969087
$$116$$ 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
$$122$$ −1.73205 −0.156813
$$123$$ −10.3923 −0.937043
$$124$$ −3.46410 −0.311086
$$125$$ −12.1244 −1.08444
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ −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
$$135$$ −6.92820 −0.596285
$$136$$ 5.19615 0.445566
$$137$$ −15.5885 −1.33181 −0.665906 0.746036i $$-0.731955\pi$$
−0.665906 + 0.746036i $$0.731955\pi$$
$$138$$ −20.7846 −1.76930
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −6.92820 −0.583460
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ −5.00000 −0.416667
$$145$$ 5.19615 0.431517
$$146$$ −3.00000 −0.248282
$$147$$ −14.0000 −1.15470
$$148$$ −8.66025 −0.711868
$$149$$ 19.0526 1.56085 0.780423 0.625252i $$-0.215004\pi$$
0.780423 + 0.625252i $$0.215004\pi$$
$$150$$ 6.92820 0.565685
$$151$$ 17.3205 1.40952 0.704761 0.709444i $$-0.251054\pi$$
0.704761 + 0.709444i $$0.251054\pi$$
$$152$$ 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 $$0.673605\pi$$
$$158$$ −6.92820 −0.551178
$$159$$ −6.00000 −0.475831
$$160$$ 9.00000 0.711512
$$161$$ 0 0
$$162$$ 19.0526 1.49691
$$163$$ 20.7846 1.62798 0.813988 0.580881i $$-0.197292\pi$$
0.813988 + 0.580881i $$0.197292\pi$$
$$164$$ −5.19615 −0.405751
$$165$$ 0 0
$$166$$ 24.0000 1.86276
$$167$$ 13.8564 1.07224 0.536120 0.844141i $$-0.319889\pi$$
0.536120 + 0.844141i $$0.319889\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −9.00000 −0.690268
$$171$$ 3.46410 0.264906
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −10.3923 −0.787839
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 13.8564 1.04151
$$178$$ −12.0000 −0.899438
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 1.73205 0.129099
$$181$$ −11.0000 −0.817624 −0.408812 0.912619i $$-0.634057\pi$$
−0.408812 + 0.912619i $$0.634057\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 10.3923 0.766131
$$185$$ −15.0000 −1.10282
$$186$$ 12.0000 0.879883
$$187$$ 0 0
$$188$$ −3.46410 −0.252646
$$189$$ 0 0
$$190$$ −10.3923 −0.753937
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 2.00000 0.144338
$$193$$ 5.19615 0.374027 0.187014 0.982357i $$-0.440119\pi$$
0.187014 + 0.982357i $$0.440119\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 13.8564 0.987228 0.493614 0.869681i $$-0.335676\pi$$
0.493614 + 0.869681i $$0.335676\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ −3.46410 −0.244949
$$201$$ 6.92820 0.488678
$$202$$ −5.19615 −0.365600
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ −9.00000 −0.628587
$$206$$ −17.3205 −1.20678
$$207$$ 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
$$213$$ −6.92820 −0.474713
$$214$$ −10.3923 −0.710403
$$215$$ −13.8564 −0.944999
$$216$$ −6.92820 −0.471405
$$217$$ 0 0
$$218$$ −24.0000 −1.62549
$$219$$ 3.46410 0.234082
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 30.0000 2.01347
$$223$$ −10.3923 −0.695920 −0.347960 0.937509i $$-0.613126\pi$$
−0.347960 + 0.937509i $$0.613126\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 25.9808 1.72821
$$227$$ −24.2487 −1.60944 −0.804722 0.593652i $$-0.797686\pi$$
−0.804722 + 0.593652i $$0.797686\pi$$
$$228$$ 6.92820 0.458831
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ −18.0000 −1.18688
$$231$$ 0 0
$$232$$ 5.19615 0.341144
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 6.92820 0.450988
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 20.7846 1.34444 0.672222 0.740349i $$-0.265340\pi$$
0.672222 + 0.740349i $$0.265340\pi$$
$$240$$ −17.3205 −1.11803
$$241$$ −1.73205 −0.111571 −0.0557856 0.998443i $$-0.517766\pi$$
−0.0557856 + 0.998443i $$0.517766\pi$$
$$242$$ 19.0526 1.22474
$$243$$ −10.0000 −0.641500
$$244$$ 1.00000 0.0640184
$$245$$ −12.1244 −0.774597
$$246$$ 18.0000 1.14764
$$247$$ 0 0
$$248$$ −6.00000 −0.381000
$$249$$ −27.7128 −1.75623
$$250$$ 21.0000 1.32816
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −3.46410 −0.217357
$$255$$ 10.3923 0.650791
$$256$$ 19.0000 1.18750
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 27.7128 1.72532
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ −31.1769 −1.92612
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ −5.19615 −0.319197
$$266$$ 0 0
$$267$$ 13.8564 0.847998
$$268$$ 3.46410 0.211604
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 12.0000 0.730297
$$271$$ −20.7846 −1.26258 −0.631288 0.775549i $$-0.717473\pi$$
−0.631288 + 0.775549i $$0.717473\pi$$
$$272$$ −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 $$-0.432558\pi$$
0.210295 + 0.977638i $$0.432558\pi$$
$$278$$ 6.92820 0.415526
$$279$$ −3.46410 −0.207390
$$280$$ 0 0
$$281$$ −22.5167 −1.34323 −0.671616 0.740900i $$-0.734399\pi$$
−0.671616 + 0.740900i $$0.734399\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −3.46410 −0.205557
$$285$$ 12.0000 0.710819
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 5.19615 0.306186
$$289$$ −8.00000 −0.470588
$$290$$ −9.00000 −0.528498
$$291$$ −13.8564 −0.812277
$$292$$ 1.73205 0.101361
$$293$$ 5.19615 0.303562 0.151781 0.988414i $$-0.451499\pi$$
0.151781 + 0.988414i $$0.451499\pi$$
$$294$$ 24.2487 1.41421
$$295$$ 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
$$304$$ −17.3205 −0.993399
$$305$$ 1.73205 0.0991769
$$306$$ −5.19615 −0.297044
$$307$$ −17.3205 −0.988534 −0.494267 0.869310i $$-0.664563\pi$$
−0.494267 + 0.869310i $$0.664563\pi$$
$$308$$ 0 0
$$309$$ 20.0000 1.13776
$$310$$ 10.3923 0.590243
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 22.5167 1.27069
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ 5.19615 0.291845 0.145922 0.989296i $$-0.453385\pi$$
0.145922 + 0.989296i $$0.453385\pi$$
$$318$$ 10.3923 0.582772
$$319$$ 0 0
$$320$$ 1.73205 0.0968246
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 10.3923 0.578243
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ −36.0000 −1.99386
$$327$$ 27.7128 1.53252
$$328$$ −9.00000 −0.496942
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 27.7128 1.52323 0.761617 0.648027i $$-0.224406\pi$$
0.761617 + 0.648027i $$0.224406\pi$$
$$332$$ −13.8564 −0.760469
$$333$$ −8.66025 −0.474579
$$334$$ −24.0000 −1.31322
$$335$$ 6.00000 0.327815
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ −30.0000 −1.62938
$$340$$ 5.19615 0.281801
$$341$$ 0 0
$$342$$ −6.00000 −0.324443
$$343$$ 0 0
$$344$$ −13.8564 −0.747087
$$345$$ 20.7846 1.11901
$$346$$ 10.3923 0.558694
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −13.8564 −0.741716 −0.370858 0.928689i $$-0.620936\pi$$
−0.370858 + 0.928689i $$0.620936\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 32.9090 1.75157 0.875784 0.482704i $$-0.160345\pi$$
0.875784 + 0.482704i $$0.160345\pi$$
$$354$$ −24.0000 −1.27559
$$355$$ −6.00000 −0.318447
$$356$$ 6.92820 0.367194
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.92820 −0.365657 −0.182828 0.983145i $$-0.558525\pi$$
−0.182828 + 0.983145i $$0.558525\pi$$
$$360$$ 3.00000 0.158114
$$361$$ −7.00000 −0.368421
$$362$$ 19.0526 1.00138
$$363$$ −22.0000 −1.15470
$$364$$ 0 0
$$365$$ 3.00000 0.157027
$$366$$ −3.46410 −0.181071
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ −30.0000 −1.56386
$$369$$ −5.19615 −0.270501
$$370$$ 25.9808 1.35068
$$371$$ 0 0
$$372$$ −6.92820 −0.359211
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ 0 0
$$375$$ −24.2487 −1.25220
$$376$$ −6.00000 −0.309426
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −24.2487 −1.24557 −0.622786 0.782392i $$-0.713999\pi$$
−0.622786 + 0.782392i $$0.713999\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 4.00000 0.204926
$$382$$ −31.1769 −1.59515
$$383$$ 20.7846 1.06204 0.531022 0.847358i $$-0.321808\pi$$
0.531022 + 0.847358i $$0.321808\pi$$
$$384$$ −24.2487 −1.23744
$$385$$ 0 0
$$386$$ −9.00000 −0.458088
$$387$$ −8.00000 −0.406663
$$388$$ −6.92820 −0.351726
$$389$$ 9.00000 0.456318 0.228159 0.973624i $$-0.426729\pi$$
0.228159 + 0.973624i $$0.426729\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ −12.1244 −0.612372
$$393$$ 36.0000 1.81596
$$394$$ −24.0000 −1.20910
$$395$$ 6.92820 0.348596
$$396$$ 0 0
$$397$$ −13.8564 −0.695433 −0.347717 0.937600i $$-0.613043\pi$$
−0.347717 + 0.937600i $$0.613043\pi$$
$$398$$ −3.46410 −0.173640
$$399$$ 0 0
$$400$$ 10.0000 0.500000
$$401$$ 1.73205 0.0864945 0.0432472 0.999064i $$-0.486230\pi$$
0.0432472 + 0.999064i $$0.486230\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ 0 0
$$404$$ 3.00000 0.149256
$$405$$ −19.0526 −0.946729
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 10.3923 0.514496
$$409$$ −15.5885 −0.770800 −0.385400 0.922750i $$-0.625936\pi$$
−0.385400 + 0.922750i $$0.625936\pi$$
$$410$$ 15.5885 0.769859
$$411$$ −31.1769 −1.53784
$$412$$ 10.0000 0.492665
$$413$$ 0 0
$$414$$ −10.3923 −0.510754
$$415$$ −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
$$421$$ −15.5885 −0.759735 −0.379867 0.925041i $$-0.624030\pi$$
−0.379867 + 0.925041i $$0.624030\pi$$
$$422$$ −17.3205 −0.843149
$$423$$ −3.46410 −0.168430
$$424$$ −5.19615 −0.252347
$$425$$ −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
$$431$$ −6.92820 −0.333720 −0.166860 0.985981i $$-0.553363\pi$$
−0.166860 + 0.985981i $$0.553363\pi$$
$$432$$ 20.0000 0.962250
$$433$$ 17.0000 0.816968 0.408484 0.912766i $$-0.366058\pi$$
0.408484 + 0.912766i $$0.366058\pi$$
$$434$$ 0 0
$$435$$ 10.3923 0.498273
$$436$$ 13.8564 0.663602
$$437$$ 20.7846 0.994263
$$438$$ −6.00000 −0.286691
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 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 $$0.592016\pi$$
$$444$$ −17.3205 −0.821995
$$445$$ 12.0000 0.568855
$$446$$ 18.0000 0.852325
$$447$$ 38.1051 1.80231
$$448$$ 0 0
$$449$$ −6.92820 −0.326962 −0.163481 0.986546i $$-0.552272\pi$$
−0.163481 + 0.986546i $$0.552272\pi$$
$$450$$ 3.46410 0.163299
$$451$$ 0 0
$$452$$ −15.0000 −0.705541
$$453$$ 34.6410 1.62758
$$454$$ 42.0000 1.97116
$$455$$ 0 0
$$456$$ 12.0000 0.561951
$$457$$ 1.73205 0.0810219 0.0405110 0.999179i $$-0.487101\pi$$
0.0405110 + 0.999179i $$0.487101\pi$$
$$458$$ 0 0
$$459$$ −12.0000 −0.560112
$$460$$ 10.3923 0.484544
$$461$$ −22.5167 −1.04871 −0.524353 0.851501i $$-0.675693\pi$$
−0.524353 + 0.851501i $$0.675693\pi$$
$$462$$ 0 0
$$463$$ 13.8564 0.643962 0.321981 0.946746i $$-0.395651\pi$$
0.321981 + 0.946746i $$0.395651\pi$$
$$464$$ −15.0000 −0.696358
$$465$$ −12.0000 −0.556487
$$466$$ 10.3923 0.481414
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 10.3923 0.479361
$$471$$ −26.0000 −1.19802
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ −13.8564 −0.636446
$$475$$ −6.92820 −0.317888
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ −36.0000 −1.64660
$$479$$ −24.2487 −1.10795 −0.553976 0.832533i $$-0.686890\pi$$
−0.553976 + 0.832533i $$0.686890\pi$$
$$480$$ 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
$$486$$ 17.3205 0.785674
$$487$$ 6.92820 0.313947 0.156973 0.987603i $$-0.449826\pi$$
0.156973 + 0.987603i $$0.449826\pi$$
$$488$$ 1.73205 0.0784063
$$489$$ 41.5692 1.87983
$$490$$ 21.0000 0.948683
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ −10.3923 −0.468521
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 17.3205 0.777714
$$497$$ 0 0
$$498$$ 48.0000 2.15093
$$499$$ −31.1769 −1.39567 −0.697835 0.716258i $$-0.745853\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ −12.1244 −0.542218
$$501$$ 27.7128 1.23812
$$502$$ −31.1769 −1.39149
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 5.19615 0.231226
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 19.0526 0.844490 0.422245 0.906482i $$-0.361242\pi$$
0.422245 + 0.906482i $$0.361242\pi$$
$$510$$ −18.0000 −0.797053
$$511$$ 0 0
$$512$$ −8.66025 −0.382733
$$513$$ −13.8564 −0.611775
$$514$$ 5.19615 0.229192
$$515$$ 17.3205 0.763233
$$516$$ −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 $$0.436832\pi$$
$$522$$ −5.19615 −0.227429
$$523$$ −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
$$526$$ −20.7846 −0.906252
$$527$$ −10.3923 −0.452696
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 9.00000 0.390935
$$531$$ 6.92820 0.300658
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −24.0000 −1.03858
$$535$$ 10.3923 0.449299
$$536$$ 6.00000 0.259161
$$537$$ 0 0
$$538$$ 10.3923 0.448044
$$539$$ 0 0
$$540$$ −6.92820 −0.298142
$$541$$ 29.4449 1.26593 0.632967 0.774179i $$-0.281837\pi$$
0.632967 + 0.774179i $$0.281837\pi$$
$$542$$ 36.0000 1.54633
$$543$$ −22.0000 −0.944110
$$544$$ 15.5885 0.668350
$$545$$ 24.0000 1.02805
$$546$$ 0 0
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ −15.5885 −0.665906
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ 10.3923 0.442727
$$552$$ 20.7846 0.884652
$$553$$ 0 0
$$554$$ −12.1244 −0.515115
$$555$$ −30.0000 −1.27343
$$556$$ −4.00000 −0.169638
$$557$$ 15.5885 0.660504 0.330252 0.943893i $$-0.392866\pi$$
0.330252 + 0.943893i $$0.392866\pi$$
$$558$$ 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 $$0.5\pi$$
$$564$$ −6.92820 −0.291730
$$565$$ −25.9808 −1.09302
$$566$$ 6.92820 0.291214
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ −20.7846 −0.870572
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ 36.0000 1.50392
$$574$$ 0 0
$$575$$ −12.0000 −0.500435
$$576$$ 1.00000 0.0416667
$$577$$ 19.0526 0.793168 0.396584 0.917998i $$-0.370195\pi$$
0.396584 + 0.917998i $$0.370195\pi$$
$$578$$ 13.8564 0.576351
$$579$$ 10.3923 0.431889
$$580$$ 5.19615 0.215758
$$581$$ 0 0
$$582$$ 24.0000 0.994832
$$583$$ 0 0
$$584$$ 3.00000 0.124141
$$585$$ 0 0
$$586$$ −9.00000 −0.371787
$$587$$ −20.7846 −0.857873 −0.428936 0.903335i $$-0.641112\pi$$
−0.428936 + 0.903335i $$0.641112\pi$$
$$588$$ −14.0000 −0.577350
$$589$$ −12.0000 −0.494451
$$590$$ −20.7846 −0.855689
$$591$$ 27.7128 1.13995
$$592$$ 43.3013 1.77967
$$593$$ −25.9808 −1.06690 −0.533451 0.845831i $$-0.679105\pi$$
−0.533451 + 0.845831i $$0.679105\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 19.0526 0.780423
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ −6.92820 −0.282843
$$601$$ 25.0000 1.01977 0.509886 0.860242i $$-0.329688\pi$$
0.509886 + 0.860242i $$0.329688\pi$$
$$602$$ 0 0
$$603$$ 3.46410 0.141069
$$604$$ 17.3205 0.704761
$$605$$ −19.0526 −0.774597
$$606$$ −10.3923 −0.422159
$$607$$ −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
$$613$$ −12.1244 −0.489698 −0.244849 0.969561i $$-0.578738\pi$$
−0.244849 + 0.969561i $$0.578738\pi$$
$$614$$ 30.0000 1.21070
$$615$$ −18.0000 −0.725830
$$616$$ 0 0
$$617$$ 22.5167 0.906487 0.453243 0.891387i $$-0.350267\pi$$
0.453243 + 0.891387i $$0.350267\pi$$
$$618$$ −34.6410 −1.39347
$$619$$ −20.7846 −0.835404 −0.417702 0.908584i $$-0.637164\pi$$
−0.417702 + 0.908584i $$0.637164\pi$$
$$620$$ −6.00000 −0.240966
$$621$$ −24.0000 −0.963087
$$622$$ −51.9615 −2.08347
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ −17.3205 −0.692267
$$627$$ 0 0
$$628$$ −13.0000 −0.518756
$$629$$ −25.9808 −1.03592
$$630$$ 0 0
$$631$$ −48.4974 −1.93065 −0.965326 0.261048i $$-0.915932\pi$$
−0.965326 + 0.261048i $$0.915932\pi$$
$$632$$ 6.92820 0.275589
$$633$$ 20.0000 0.794929
$$634$$ −9.00000 −0.357436
$$635$$ 3.46410 0.137469
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −3.46410 −0.137038
$$640$$ −21.0000 −0.830098
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ −20.7846 −0.820303
$$643$$ −13.8564 −0.546443 −0.273222 0.961951i $$-0.588089\pi$$
−0.273222 + 0.961951i $$0.588089\pi$$
$$644$$ 0 0
$$645$$ −27.7128 −1.09119
$$646$$ −18.0000 −0.708201
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ −19.0526 −0.748455
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.7846 0.813988
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ −48.0000 −1.87695
$$655$$ 31.1769 1.21818
$$656$$ 25.9808 1.01438
$$657$$ 1.73205 0.0675737
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 46.7654 1.81896 0.909481 0.415745i $$-0.136479\pi$$
0.909481 + 0.415745i $$0.136479\pi$$
$$662$$ −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
$$668$$ 13.8564 0.536120
$$669$$ −20.7846 −0.803579
$$670$$ −10.3923 −0.401490
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ −39.8372 −1.53447
$$675$$ 8.00000 0.307920
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 51.9615 1.99557
$$679$$ 0 0
$$680$$ 9.00000 0.345134
$$681$$ −48.4974 −1.85843
$$682$$ 0 0
$$683$$ 24.2487 0.927851 0.463926 0.885874i $$-0.346441\pi$$
0.463926 + 0.885874i $$0.346441\pi$$
$$684$$ 3.46410 0.132453
$$685$$ −27.0000 −1.03162
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 40.0000 1.52499
$$689$$ 0 0
$$690$$ −36.0000 −1.37050
$$691$$ 13.8564 0.527123 0.263561 0.964643i $$-0.415103\pi$$
0.263561 + 0.964643i $$0.415103\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 51.9615 1.97243
$$695$$ −6.92820 −0.262802
$$696$$ 10.3923 0.393919
$$697$$ −15.5885 −0.590455
$$698$$ 24.0000 0.908413
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ −30.0000 −1.13147
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ −57.0000 −2.14522
$$707$$ 0 0
$$708$$ 13.8564 0.520756
$$709$$ −5.19615 −0.195146 −0.0975728 0.995228i $$-0.531108\pi$$
−0.0975728 + 0.995228i $$0.531108\pi$$
$$710$$ 10.3923 0.390016
$$711$$ 4.00000 0.150012
$$712$$ 12.0000 0.449719
$$713$$ −20.7846 −0.778390
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 41.5692 1.55243
$$718$$ 12.0000 0.447836
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ −8.66025 −0.322749
$$721$$ 0 0
$$722$$ 12.1244 0.451222
$$723$$ −3.46410 −0.128831
$$724$$ −11.0000 −0.408812
$$725$$ −6.00000 −0.222834
$$726$$ 38.1051 1.41421
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ −5.19615 −0.192318
$$731$$ −24.0000 −0.887672
$$732$$ 2.00000 0.0739221
$$733$$ −12.1244 −0.447823 −0.223912 0.974609i $$-0.571883\pi$$
−0.223912 + 0.974609i $$0.571883\pi$$
$$734$$ 38.1051 1.40649
$$735$$ −24.2487 −0.894427
$$736$$ 31.1769 1.14920
$$737$$ 0 0
$$738$$ 9.00000 0.331295
$$739$$ 20.7846 0.764574 0.382287 0.924044i $$-0.375137\pi$$
0.382287 + 0.924044i $$0.375137\pi$$
$$740$$ −15.0000 −0.551411
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −34.6410 −1.27086 −0.635428 0.772160i $$-0.719176\pi$$
−0.635428 + 0.772160i $$0.719176\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ 33.0000 1.20903
$$746$$ −32.9090 −1.20488
$$747$$ −13.8564 −0.506979
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 42.0000 1.53362
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 17.3205 0.631614
$$753$$ 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
$$760$$ 10.3923 0.376969
$$761$$ 34.6410 1.25574 0.627868 0.778320i $$-0.283928\pi$$
0.627868 + 0.778320i $$0.283928\pi$$
$$762$$ −6.92820 −0.250982
$$763$$ 0 0
$$764$$ 18.0000 0.651217
$$765$$ 5.19615 0.187867
$$766$$ −36.0000 −1.30073
$$767$$ 0 0
$$768$$ 38.0000 1.37121
$$769$$ 6.92820 0.249837 0.124919 0.992167i $$-0.460133\pi$$
0.124919 + 0.992167i $$0.460133\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 5.19615 0.187014
$$773$$ 34.6410 1.24595 0.622975 0.782241i $$-0.285924\pi$$
0.622975 + 0.782241i $$0.285924\pi$$
$$774$$ 13.8564 0.498058
$$775$$ 6.92820 0.248868
$$776$$ −12.0000 −0.430775
$$777$$ 0 0
$$778$$ −15.5885 −0.558873
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −31.1769 −1.11488
$$783$$ −12.0000 −0.428845
$$784$$ 35.0000 1.25000
$$785$$ −22.5167 −0.803654
$$786$$ −62.3538 −2.22409
$$787$$ 38.1051 1.35830 0.679150 0.733999i $$-0.262348\pi$$
0.679150 + 0.733999i $$0.262348\pi$$
$$788$$ 13.8564 0.493614
$$789$$ 24.0000 0.854423
$$790$$ −12.0000 −0.426941
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 24.0000 0.851728
$$795$$ −10.3923 −0.368577
$$796$$ 2.00000 0.0708881
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ −10.3923 −0.367653
$$800$$ −10.3923 −0.367423
$$801$$ 6.92820 0.244796
$$802$$ −3.00000 −0.105934
$$803$$ 0 0
$$804$$ 6.92820 0.244339
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12.0000 −0.422420
$$808$$ 5.19615 0.182800
$$809$$ 33.0000 1.16022 0.580109 0.814539i $$-0.303010\pi$$
0.580109 + 0.814539i $$0.303010\pi$$
$$810$$ 33.0000 1.15950
$$811$$ 38.1051 1.33805 0.669026 0.743239i $$-0.266712\pi$$
0.669026 + 0.743239i $$0.266712\pi$$
$$812$$ 0 0
$$813$$ −41.5692 −1.45790
$$814$$ 0 0
$$815$$ 36.0000 1.26102
$$816$$ −30.0000 −1.05021
$$817$$ −27.7128 −0.969549
$$818$$ 27.0000 0.944033
$$819$$ 0 0
$$820$$ −9.00000 −0.314294
$$821$$ −41.5692 −1.45078 −0.725388 0.688340i $$-0.758340\pi$$
−0.725388 + 0.688340i $$0.758340\pi$$
$$822$$ 54.0000 1.88347
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 17.3205 0.603388
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.7846 0.722752 0.361376 0.932420i $$-0.382307\pi$$
0.361376 + 0.932420i $$0.382307\pi$$
$$828$$ 6.00000 0.208514
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 41.5692 1.44289
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ −21.0000 −0.727607
$$834$$ 13.8564 0.479808
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ 13.8564 0.478947
$$838$$ −31.1769 −1.07699
$$839$$ 45.0333 1.55472 0.777361 0.629054i $$-0.216558\pi$$
0.777361 + 0.629054i $$0.216558\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 27.0000 0.930481
$$843$$ −45.0333 −1.55103
$$844$$ 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
$$850$$ 10.3923 0.356453
$$851$$ −51.9615 −1.78122
$$852$$ −6.92820 −0.237356
$$853$$ 25.9808 0.889564 0.444782 0.895639i $$-0.353281\pi$$
0.444782 + 0.895639i $$0.353281\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 10.3923 0.355202
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ −13.8564 −0.472500
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ 27.7128 0.943355 0.471678 0.881771i $$-0.343649\pi$$
0.471678 + 0.881771i $$0.343649\pi$$
$$864$$ −20.7846 −0.707107
$$865$$ −10.3923 −0.353349
$$866$$ −29.4449 −1.00058
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −18.0000 −0.610257
$$871$$ 0 0
$$872$$ 24.0000 0.812743
$$873$$ −6.92820 −0.234484
$$874$$ −36.0000 −1.21772
$$875$$ 0 0
$$876$$ 3.46410 0.117041
$$877$$ −12.1244 −0.409410 −0.204705 0.978824i $$-0.565624\pi$$
−0.204705 + 0.978824i $$0.565624\pi$$
$$878$$ −48.4974 −1.63671
$$879$$ 10.3923 0.350524
$$880$$ 0 0
$$881$$ −27.0000 −0.909653 −0.454827 0.890580i $$-0.650299\pi$$
−0.454827 + 0.890580i $$0.650299\pi$$
$$882$$ 12.1244 0.408248
$$883$$ −10.0000 −0.336527 −0.168263 0.985742i $$-0.553816\pi$$
−0.168263 + 0.985742i $$0.553816\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 20.7846 0.698273
$$887$$ 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
$$890$$ −20.7846 −0.696702
$$891$$ 0 0
$$892$$ −10.3923 −0.347960
$$893$$ −12.0000 −0.401565
$$894$$ −66.0000 −2.20737
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 12.0000 0.400445
$$899$$ −10.3923 −0.346603
$$900$$ −2.00000 −0.0666667
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −25.9808 −0.864107
$$905$$ −19.0526 −0.633328
$$906$$ −60.0000 −1.99337
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ −24.2487 −0.804722
$$909$$ 3.00000 0.0995037
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −34.6410 −1.14708
$$913$$ 0 0
$$914$$ −3.00000 −0.0992312
$$915$$ 3.46410 0.114520
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 20.7846 0.685994
$$919$$ 22.0000 0.725713 0.362857 0.931845i $$-0.381802\pi$$
0.362857 + 0.931845i $$0.381802\pi$$
$$920$$ 18.0000 0.593442
$$921$$ −34.6410 −1.14146
$$922$$ 39.0000 1.28440
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 17.3205 0.569495
$$926$$ −24.0000 −0.788689
$$927$$ 10.0000 0.328443
$$928$$ 15.5885 0.511716
$$929$$ 46.7654 1.53432 0.767161 0.641455i $$-0.221669\pi$$
0.767161 + 0.641455i $$0.221669\pi$$
$$930$$ 20.7846 0.681554
$$931$$ −24.2487 −0.794719
$$932$$ −6.00000 −0.196537
$$933$$ 60.0000 1.96431
$$934$$ 20.7846 0.680093
$$935$$ 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
$$941$$ −20.7846 −0.677559 −0.338779 0.940866i $$-0.610014\pi$$
−0.338779 + 0.940866i $$0.610014\pi$$
$$942$$ 45.0333 1.46726
$$943$$ −31.1769 −1.01526
$$944$$ −34.6410 −1.12747
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.3205 0.562841 0.281420 0.959585i $$-0.409194\pi$$
0.281420 + 0.959585i $$0.409194\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ 12.0000 0.389331
$$951$$ 10.3923 0.336994
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 5.19615 0.168232
$$955$$ 31.1769 1.00886
$$956$$ 20.7846 0.672222
$$957$$ 0 0
$$958$$ 42.0000 1.35696
$$959$$ 0 0
$$960$$ 3.46410 0.111803
$$961$$ −19.0000 −0.612903
$$962$$ 0 0
$$963$$ 6.00000 0.193347
$$964$$ −1.73205 −0.0557856
$$965$$ 9.00000 0.289720
$$966$$ 0 0
$$967$$ −58.8897 −1.89377 −0.946883 0.321578i $$-0.895787\pi$$
−0.946883 + 0.321578i $$0.895787\pi$$
$$968$$ −19.0526 −0.612372
$$969$$ 20.7846 0.667698
$$970$$ 20.7846 0.667354
$$971$$ 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
$$977$$ 43.3013 1.38533 0.692665 0.721259i $$-0.256436\pi$$
0.692665 + 0.721259i $$0.256436\pi$$
$$978$$ −72.0000 −2.30231
$$979$$ 0 0
$$980$$ −12.1244 −0.387298
$$981$$ 13.8564 0.442401
$$982$$ 20.7846 0.663264
$$983$$ 51.9615 1.65732 0.828658 0.559756i $$-0.189105\pi$$
0.828658 + 0.559756i $$0.189105\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ 24.0000 0.764704
$$986$$ −15.5885 −0.496438
$$987$$ 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
$$993$$ 55.4256 1.75888
$$994$$ 0 0
$$995$$ 3.46410 0.109819
$$996$$ −27.7128 −0.878114
$$997$$ 17.0000 0.538395 0.269198 0.963085i $$-0.413241\pi$$
0.269198 + 0.963085i $$0.413241\pi$$
$$998$$ 54.0000 1.70934
$$999$$ 34.6410 1.09599
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.a.a.1.1 2
3.2 odd 2 1521.2.a.k.1.2 2
4.3 odd 2 2704.2.a.o.1.2 2
5.4 even 2 4225.2.a.v.1.2 2
7.6 odd 2 8281.2.a.q.1.1 2
13.2 odd 12 13.2.e.a.4.1 2
13.3 even 3 169.2.c.a.22.2 4
13.4 even 6 169.2.c.a.146.1 4
13.5 odd 4 169.2.b.a.168.2 2
13.6 odd 12 169.2.e.a.23.1 2
13.7 odd 12 13.2.e.a.10.1 yes 2
13.8 odd 4 169.2.b.a.168.1 2
13.9 even 3 169.2.c.a.146.2 4
13.10 even 6 169.2.c.a.22.1 4
13.11 odd 12 169.2.e.a.147.1 2
13.12 even 2 inner 169.2.a.a.1.2 2
39.2 even 12 117.2.q.c.82.1 2
39.5 even 4 1521.2.b.a.1351.1 2
39.8 even 4 1521.2.b.a.1351.2 2
39.20 even 12 117.2.q.c.10.1 2
39.38 odd 2 1521.2.a.k.1.1 2
52.7 even 12 208.2.w.b.49.1 2
52.15 even 12 208.2.w.b.17.1 2
52.31 even 4 2704.2.f.b.337.1 2
52.47 even 4 2704.2.f.b.337.2 2
52.51 odd 2 2704.2.a.o.1.1 2
65.2 even 12 325.2.m.a.199.2 4
65.7 even 12 325.2.m.a.49.1 4
65.28 even 12 325.2.m.a.199.1 4
65.33 even 12 325.2.m.a.49.2 4
65.54 odd 12 325.2.n.a.251.1 2
65.59 odd 12 325.2.n.a.101.1 2
65.64 even 2 4225.2.a.v.1.1 2
91.2 odd 12 637.2.k.a.459.1 2
91.20 even 12 637.2.q.a.491.1 2
91.33 even 12 637.2.u.b.361.1 2
91.41 even 12 637.2.q.a.589.1 2
91.46 odd 12 637.2.k.a.569.1 2
91.54 even 12 637.2.k.c.459.1 2
91.59 even 12 637.2.k.c.569.1 2
91.67 odd 12 637.2.u.c.30.1 2
91.72 odd 12 637.2.u.c.361.1 2
91.80 even 12 637.2.u.b.30.1 2
91.90 odd 2 8281.2.a.q.1.2 2
104.59 even 12 832.2.w.a.257.1 2
104.67 even 12 832.2.w.a.641.1 2
104.85 odd 12 832.2.w.d.257.1 2
104.93 odd 12 832.2.w.d.641.1 2
156.59 odd 12 1872.2.by.d.1297.1 2
156.119 odd 12 1872.2.by.d.433.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 13.2 odd 12
13.2.e.a.10.1 yes 2 13.7 odd 12
117.2.q.c.10.1 2 39.20 even 12
117.2.q.c.82.1 2 39.2 even 12
169.2.a.a.1.1 2 1.1 even 1 trivial
169.2.a.a.1.2 2 13.12 even 2 inner
169.2.b.a.168.1 2 13.8 odd 4
169.2.b.a.168.2 2 13.5 odd 4
169.2.c.a.22.1 4 13.10 even 6
169.2.c.a.22.2 4 13.3 even 3
169.2.c.a.146.1 4 13.4 even 6
169.2.c.a.146.2 4 13.9 even 3
169.2.e.a.23.1 2 13.6 odd 12
169.2.e.a.147.1 2 13.11 odd 12
208.2.w.b.17.1 2 52.15 even 12
208.2.w.b.49.1 2 52.7 even 12
325.2.m.a.49.1 4 65.7 even 12
325.2.m.a.49.2 4 65.33 even 12
325.2.m.a.199.1 4 65.28 even 12
325.2.m.a.199.2 4 65.2 even 12
325.2.n.a.101.1 2 65.59 odd 12
325.2.n.a.251.1 2 65.54 odd 12
637.2.k.a.459.1 2 91.2 odd 12
637.2.k.a.569.1 2 91.46 odd 12
637.2.k.c.459.1 2 91.54 even 12
637.2.k.c.569.1 2 91.59 even 12
637.2.q.a.491.1 2 91.20 even 12
637.2.q.a.589.1 2 91.41 even 12
637.2.u.b.30.1 2 91.80 even 12
637.2.u.b.361.1 2 91.33 even 12
637.2.u.c.30.1 2 91.67 odd 12
637.2.u.c.361.1 2 91.72 odd 12
832.2.w.a.257.1 2 104.59 even 12
832.2.w.a.641.1 2 104.67 even 12
832.2.w.d.257.1 2 104.85 odd 12
832.2.w.d.641.1 2 104.93 odd 12
1521.2.a.k.1.1 2 39.38 odd 2
1521.2.a.k.1.2 2 3.2 odd 2
1521.2.b.a.1351.1 2 39.5 even 4
1521.2.b.a.1351.2 2 39.8 even 4
1872.2.by.d.433.1 2 156.119 odd 12
1872.2.by.d.1297.1 2 156.59 odd 12
2704.2.a.o.1.1 2 52.51 odd 2
2704.2.a.o.1.2 2 4.3 odd 2
2704.2.f.b.337.1 2 52.31 even 4
2704.2.f.b.337.2 2 52.47 even 4
4225.2.a.v.1.1 2 65.64 even 2
4225.2.a.v.1.2 2 5.4 even 2
8281.2.a.q.1.1 2 7.6 odd 2
8281.2.a.q.1.2 2 91.90 odd 2