# Properties

 Label 8550.2.a.cb.1.2 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.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: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +4.41421 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +4.41421 q^{7} +1.00000 q^{8} +1.41421 q^{11} +5.82843 q^{13} +4.41421 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} +1.41421 q^{22} -0.757359 q^{23} +5.82843 q^{26} +4.41421 q^{28} +0.171573 q^{29} +6.24264 q^{31} +1.00000 q^{32} -1.00000 q^{34} -8.48528 q^{37} -1.00000 q^{38} +4.24264 q^{41} +1.75736 q^{43} +1.41421 q^{44} -0.757359 q^{46} +12.4853 q^{49} +5.82843 q^{52} -5.48528 q^{53} +4.41421 q^{56} +0.171573 q^{58} -6.89949 q^{59} +14.2426 q^{61} +6.24264 q^{62} +1.00000 q^{64} +4.75736 q^{67} -1.00000 q^{68} +13.4142 q^{71} +11.4853 q^{73} -8.48528 q^{74} -1.00000 q^{76} +6.24264 q^{77} -6.48528 q^{79} +4.24264 q^{82} -14.4853 q^{83} +1.75736 q^{86} +1.41421 q^{88} -7.07107 q^{89} +25.7279 q^{91} -0.757359 q^{92} -0.343146 q^{97} +12.4853 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 6 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 6 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 6 q^{7} + 2 q^{8} + 6 q^{13} + 6 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 10 q^{23} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{38} + 12 q^{43} - 10 q^{46} + 8 q^{49} + 6 q^{52} + 6 q^{53} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 20 q^{61} + 4 q^{62} + 2 q^{64} + 18 q^{67} - 2 q^{68} + 24 q^{71} + 6 q^{73} - 2 q^{76} + 4 q^{77} + 4 q^{79} - 12 q^{83} + 12 q^{86} + 26 q^{91} - 10 q^{92} - 12 q^{97} + 8 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 6 * q^7 + 2 * q^8 + 6 * q^13 + 6 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 - 10 * q^23 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^34 - 2 * q^38 + 12 * q^43 - 10 * q^46 + 8 * q^49 + 6 * q^52 + 6 * q^53 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 20 * q^61 + 4 * q^62 + 2 * q^64 + 18 * q^67 - 2 * q^68 + 24 * q^71 + 6 * q^73 - 2 * q^76 + 4 * q^77 + 4 * q^79 - 12 * q^83 + 12 * q^86 + 26 * q^91 - 10 * q^92 - 12 * q^97 + 8 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.41421 1.66842 0.834208 0.551450i $$-0.185925\pi$$
0.834208 + 0.551450i $$0.185925\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.41421 0.426401 0.213201 0.977008i $$-0.431611\pi$$
0.213201 + 0.977008i $$0.431611\pi$$
$$12$$ 0 0
$$13$$ 5.82843 1.61651 0.808257 0.588829i $$-0.200411\pi$$
0.808257 + 0.588829i $$0.200411\pi$$
$$14$$ 4.41421 1.17975
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.41421 0.301511
$$23$$ −0.757359 −0.157920 −0.0789602 0.996878i $$-0.525160\pi$$
−0.0789602 + 0.996878i $$0.525160\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.82843 1.14305
$$27$$ 0 0
$$28$$ 4.41421 0.834208
$$29$$ 0.171573 0.0318603 0.0159301 0.999873i $$-0.494929\pi$$
0.0159301 + 0.999873i $$0.494929\pi$$
$$30$$ 0 0
$$31$$ 6.24264 1.12121 0.560606 0.828083i $$-0.310568\pi$$
0.560606 + 0.828083i $$0.310568\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.48528 −1.39497 −0.697486 0.716599i $$-0.745698\pi$$
−0.697486 + 0.716599i $$0.745698\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.24264 0.662589 0.331295 0.943527i $$-0.392515\pi$$
0.331295 + 0.943527i $$0.392515\pi$$
$$42$$ 0 0
$$43$$ 1.75736 0.267995 0.133997 0.990982i $$-0.457219\pi$$
0.133997 + 0.990982i $$0.457219\pi$$
$$44$$ 1.41421 0.213201
$$45$$ 0 0
$$46$$ −0.757359 −0.111667
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 12.4853 1.78361
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 5.82843 0.808257
$$53$$ −5.48528 −0.753461 −0.376731 0.926323i $$-0.622952\pi$$
−0.376731 + 0.926323i $$0.622952\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.41421 0.589874
$$57$$ 0 0
$$58$$ 0.171573 0.0225286
$$59$$ −6.89949 −0.898238 −0.449119 0.893472i $$-0.648262\pi$$
−0.449119 + 0.893472i $$0.648262\pi$$
$$60$$ 0 0
$$61$$ 14.2426 1.82358 0.911792 0.410653i $$-0.134699\pi$$
0.911792 + 0.410653i $$0.134699\pi$$
$$62$$ 6.24264 0.792816
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.75736 0.581204 0.290602 0.956844i $$-0.406144\pi$$
0.290602 + 0.956844i $$0.406144\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.4142 1.59197 0.795987 0.605314i $$-0.206952\pi$$
0.795987 + 0.605314i $$0.206952\pi$$
$$72$$ 0 0
$$73$$ 11.4853 1.34425 0.672125 0.740437i $$-0.265382\pi$$
0.672125 + 0.740437i $$0.265382\pi$$
$$74$$ −8.48528 −0.986394
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 6.24264 0.711415
$$78$$ 0 0
$$79$$ −6.48528 −0.729651 −0.364826 0.931076i $$-0.618871\pi$$
−0.364826 + 0.931076i $$0.618871\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 4.24264 0.468521
$$83$$ −14.4853 −1.58997 −0.794983 0.606632i $$-0.792520\pi$$
−0.794983 + 0.606632i $$0.792520\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.75736 0.189501
$$87$$ 0 0
$$88$$ 1.41421 0.150756
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 25.7279 2.69702
$$92$$ −0.757359 −0.0789602
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.343146 −0.0348412 −0.0174206 0.999848i $$-0.505545\pi$$
−0.0174206 + 0.999848i $$0.505545\pi$$
$$98$$ 12.4853 1.26120
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −13.0711 −1.30062 −0.650310 0.759669i $$-0.725361\pi$$
−0.650310 + 0.759669i $$0.725361\pi$$
$$102$$ 0 0
$$103$$ 4.24264 0.418040 0.209020 0.977911i $$-0.432973\pi$$
0.209020 + 0.977911i $$0.432973\pi$$
$$104$$ 5.82843 0.571524
$$105$$ 0 0
$$106$$ −5.48528 −0.532778
$$107$$ −19.7279 −1.90717 −0.953585 0.301124i $$-0.902638\pi$$
−0.953585 + 0.301124i $$0.902638\pi$$
$$108$$ 0 0
$$109$$ −17.9706 −1.72127 −0.860634 0.509224i $$-0.829932\pi$$
−0.860634 + 0.509224i $$0.829932\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.41421 0.417104
$$113$$ 10.2426 0.963547 0.481773 0.876296i $$-0.339993\pi$$
0.481773 + 0.876296i $$0.339993\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.171573 0.0159301
$$117$$ 0 0
$$118$$ −6.89949 −0.635150
$$119$$ −4.41421 −0.404650
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 14.2426 1.28947
$$123$$ 0 0
$$124$$ 6.24264 0.560606
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.48528 −0.220533 −0.110267 0.993902i $$-0.535170\pi$$
−0.110267 + 0.993902i $$0.535170\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.9706 1.48272 0.741362 0.671105i $$-0.234180\pi$$
0.741362 + 0.671105i $$0.234180\pi$$
$$132$$ 0 0
$$133$$ −4.41421 −0.382761
$$134$$ 4.75736 0.410973
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ −13.0000 −1.11066 −0.555332 0.831628i $$-0.687409\pi$$
−0.555332 + 0.831628i $$0.687409\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 13.4142 1.12570
$$143$$ 8.24264 0.689284
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 11.4853 0.950529
$$147$$ 0 0
$$148$$ −8.48528 −0.697486
$$149$$ −17.6569 −1.44651 −0.723253 0.690583i $$-0.757354\pi$$
−0.723253 + 0.690583i $$0.757354\pi$$
$$150$$ 0 0
$$151$$ −10.4853 −0.853280 −0.426640 0.904422i $$-0.640303\pi$$
−0.426640 + 0.904422i $$0.640303\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ 6.24264 0.503046
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.6569 0.930318 0.465159 0.885227i $$-0.345997\pi$$
0.465159 + 0.885227i $$0.345997\pi$$
$$158$$ −6.48528 −0.515941
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.34315 −0.263477
$$162$$ 0 0
$$163$$ −10.2426 −0.802266 −0.401133 0.916020i $$-0.631383\pi$$
−0.401133 + 0.916020i $$0.631383\pi$$
$$164$$ 4.24264 0.331295
$$165$$ 0 0
$$166$$ −14.4853 −1.12428
$$167$$ 18.2426 1.41166 0.705829 0.708382i $$-0.250575\pi$$
0.705829 + 0.708382i $$0.250575\pi$$
$$168$$ 0 0
$$169$$ 20.9706 1.61312
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.75736 0.133997
$$173$$ −0.485281 −0.0368953 −0.0184476 0.999830i $$-0.505872\pi$$
−0.0184476 + 0.999830i $$0.505872\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.41421 0.106600
$$177$$ 0 0
$$178$$ −7.07107 −0.529999
$$179$$ 11.6569 0.871274 0.435637 0.900122i $$-0.356523\pi$$
0.435637 + 0.900122i $$0.356523\pi$$
$$180$$ 0 0
$$181$$ −8.48528 −0.630706 −0.315353 0.948974i $$-0.602123\pi$$
−0.315353 + 0.948974i $$0.602123\pi$$
$$182$$ 25.7279 1.90708
$$183$$ 0 0
$$184$$ −0.757359 −0.0558333
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.41421 −0.103418
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.5563 −0.908546 −0.454273 0.890863i $$-0.650101\pi$$
−0.454273 + 0.890863i $$0.650101\pi$$
$$192$$ 0 0
$$193$$ −0.343146 −0.0247002 −0.0123501 0.999924i $$-0.503931\pi$$
−0.0123501 + 0.999924i $$0.503931\pi$$
$$194$$ −0.343146 −0.0246364
$$195$$ 0 0
$$196$$ 12.4853 0.891806
$$197$$ 11.7574 0.837677 0.418839 0.908061i $$-0.362437\pi$$
0.418839 + 0.908061i $$0.362437\pi$$
$$198$$ 0 0
$$199$$ 9.24264 0.655193 0.327597 0.944818i $$-0.393761\pi$$
0.327597 + 0.944818i $$0.393761\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −13.0711 −0.919677
$$203$$ 0.757359 0.0531562
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4.24264 0.295599
$$207$$ 0 0
$$208$$ 5.82843 0.404129
$$209$$ −1.41421 −0.0978232
$$210$$ 0 0
$$211$$ −19.7279 −1.35813 −0.679063 0.734080i $$-0.737614\pi$$
−0.679063 + 0.734080i $$0.737614\pi$$
$$212$$ −5.48528 −0.376731
$$213$$ 0 0
$$214$$ −19.7279 −1.34857
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 27.5563 1.87065
$$218$$ −17.9706 −1.21712
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5.82843 −0.392062
$$222$$ 0 0
$$223$$ −15.1716 −1.01596 −0.507982 0.861368i $$-0.669608\pi$$
−0.507982 + 0.861368i $$0.669608\pi$$
$$224$$ 4.41421 0.294937
$$225$$ 0 0
$$226$$ 10.2426 0.681330
$$227$$ 16.7574 1.11223 0.556113 0.831107i $$-0.312292\pi$$
0.556113 + 0.831107i $$0.312292\pi$$
$$228$$ 0 0
$$229$$ 14.9706 0.989283 0.494641 0.869097i $$-0.335299\pi$$
0.494641 + 0.869097i $$0.335299\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0.171573 0.0112643
$$233$$ 24.9706 1.63588 0.817938 0.575306i $$-0.195117\pi$$
0.817938 + 0.575306i $$0.195117\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.89949 −0.449119
$$237$$ 0 0
$$238$$ −4.41421 −0.286131
$$239$$ 6.89949 0.446291 0.223146 0.974785i $$-0.428367\pi$$
0.223146 + 0.974785i $$0.428367\pi$$
$$240$$ 0 0
$$241$$ 8.97056 0.577845 0.288922 0.957353i $$-0.406703\pi$$
0.288922 + 0.957353i $$0.406703\pi$$
$$242$$ −9.00000 −0.578542
$$243$$ 0 0
$$244$$ 14.2426 0.911792
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.82843 −0.370854
$$248$$ 6.24264 0.396408
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −3.55635 −0.224475 −0.112237 0.993681i $$-0.535802\pi$$
−0.112237 + 0.993681i $$0.535802\pi$$
$$252$$ 0 0
$$253$$ −1.07107 −0.0673375
$$254$$ −2.48528 −0.155940
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −20.7279 −1.29297 −0.646486 0.762926i $$-0.723762\pi$$
−0.646486 + 0.762926i $$0.723762\pi$$
$$258$$ 0 0
$$259$$ −37.4558 −2.32739
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 16.9706 1.04844
$$263$$ 26.9706 1.66308 0.831538 0.555468i $$-0.187461\pi$$
0.831538 + 0.555468i $$0.187461\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.41421 −0.270653
$$267$$ 0 0
$$268$$ 4.75736 0.290602
$$269$$ 16.6274 1.01379 0.506896 0.862007i $$-0.330793\pi$$
0.506896 + 0.862007i $$0.330793\pi$$
$$270$$ 0 0
$$271$$ −27.2426 −1.65487 −0.827436 0.561560i $$-0.810202\pi$$
−0.827436 + 0.561560i $$0.810202\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −13.0000 −0.785359
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4.24264 −0.253095 −0.126547 0.991961i $$-0.540390\pi$$
−0.126547 + 0.991961i $$0.540390\pi$$
$$282$$ 0 0
$$283$$ 32.1421 1.91065 0.955326 0.295555i $$-0.0955045\pi$$
0.955326 + 0.295555i $$0.0955045\pi$$
$$284$$ 13.4142 0.795987
$$285$$ 0 0
$$286$$ 8.24264 0.487398
$$287$$ 18.7279 1.10547
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.4853 0.672125
$$293$$ 5.48528 0.320454 0.160227 0.987080i $$-0.448777\pi$$
0.160227 + 0.987080i $$0.448777\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −8.48528 −0.493197
$$297$$ 0 0
$$298$$ −17.6569 −1.02283
$$299$$ −4.41421 −0.255281
$$300$$ 0 0
$$301$$ 7.75736 0.447127
$$302$$ −10.4853 −0.603360
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −17.6569 −1.00773 −0.503865 0.863782i $$-0.668089\pi$$
−0.503865 + 0.863782i $$0.668089\pi$$
$$308$$ 6.24264 0.355707
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4.75736 0.269765 0.134883 0.990862i $$-0.456934\pi$$
0.134883 + 0.990862i $$0.456934\pi$$
$$312$$ 0 0
$$313$$ −7.97056 −0.450523 −0.225261 0.974298i $$-0.572324\pi$$
−0.225261 + 0.974298i $$0.572324\pi$$
$$314$$ 11.6569 0.657834
$$315$$ 0 0
$$316$$ −6.48528 −0.364826
$$317$$ 9.48528 0.532746 0.266373 0.963870i $$-0.414175\pi$$
0.266373 + 0.963870i $$0.414175\pi$$
$$318$$ 0 0
$$319$$ 0.242641 0.0135853
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −3.34315 −0.186306
$$323$$ 1.00000 0.0556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −10.2426 −0.567287
$$327$$ 0 0
$$328$$ 4.24264 0.234261
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.2426 −1.05767 −0.528836 0.848724i $$-0.677371\pi$$
−0.528836 + 0.848724i $$0.677371\pi$$
$$332$$ −14.4853 −0.794983
$$333$$ 0 0
$$334$$ 18.2426 0.998193
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.1005 0.768103 0.384052 0.923312i $$-0.374528\pi$$
0.384052 + 0.923312i $$0.374528\pi$$
$$338$$ 20.9706 1.14065
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 8.82843 0.478086
$$342$$ 0 0
$$343$$ 24.2132 1.30739
$$344$$ 1.75736 0.0947505
$$345$$ 0 0
$$346$$ −0.485281 −0.0260889
$$347$$ 5.51472 0.296046 0.148023 0.988984i $$-0.452709\pi$$
0.148023 + 0.988984i $$0.452709\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.41421 0.0753778
$$353$$ 2.51472 0.133845 0.0669225 0.997758i $$-0.478682\pi$$
0.0669225 + 0.997758i $$0.478682\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −7.07107 −0.374766
$$357$$ 0 0
$$358$$ 11.6569 0.616084
$$359$$ 22.7574 1.20109 0.600544 0.799592i $$-0.294951\pi$$
0.600544 + 0.799592i $$0.294951\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −8.48528 −0.445976
$$363$$ 0 0
$$364$$ 25.7279 1.34851
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −25.4558 −1.32878 −0.664392 0.747384i $$-0.731309\pi$$
−0.664392 + 0.747384i $$0.731309\pi$$
$$368$$ −0.757359 −0.0394801
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.2132 −1.25709
$$372$$ 0 0
$$373$$ 9.00000 0.466002 0.233001 0.972476i $$-0.425145\pi$$
0.233001 + 0.972476i $$0.425145\pi$$
$$374$$ −1.41421 −0.0731272
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.00000 0.0515026
$$378$$ 0 0
$$379$$ 11.2426 0.577496 0.288748 0.957405i $$-0.406761\pi$$
0.288748 + 0.957405i $$0.406761\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −12.5563 −0.642439
$$383$$ −12.2426 −0.625570 −0.312785 0.949824i $$-0.601262\pi$$
−0.312785 + 0.949824i $$0.601262\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −0.343146 −0.0174657
$$387$$ 0 0
$$388$$ −0.343146 −0.0174206
$$389$$ −22.9289 −1.16254 −0.581272 0.813710i $$-0.697445\pi$$
−0.581272 + 0.813710i $$0.697445\pi$$
$$390$$ 0 0
$$391$$ 0.757359 0.0383013
$$392$$ 12.4853 0.630602
$$393$$ 0 0
$$394$$ 11.7574 0.592327
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 24.0000 1.20453 0.602263 0.798298i $$-0.294266\pi$$
0.602263 + 0.798298i $$0.294266\pi$$
$$398$$ 9.24264 0.463292
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 25.4142 1.26913 0.634563 0.772871i $$-0.281180\pi$$
0.634563 + 0.772871i $$0.281180\pi$$
$$402$$ 0 0
$$403$$ 36.3848 1.81245
$$404$$ −13.0711 −0.650310
$$405$$ 0 0
$$406$$ 0.757359 0.0375871
$$407$$ −12.0000 −0.594818
$$408$$ 0 0
$$409$$ 25.2132 1.24671 0.623356 0.781938i $$-0.285769\pi$$
0.623356 + 0.781938i $$0.285769\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.24264 0.209020
$$413$$ −30.4558 −1.49863
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 5.82843 0.285762
$$417$$ 0 0
$$418$$ −1.41421 −0.0691714
$$419$$ −19.4142 −0.948446 −0.474223 0.880405i $$-0.657271\pi$$
−0.474223 + 0.880405i $$0.657271\pi$$
$$420$$ 0 0
$$421$$ 19.4853 0.949655 0.474827 0.880079i $$-0.342511\pi$$
0.474827 + 0.880079i $$0.342511\pi$$
$$422$$ −19.7279 −0.960340
$$423$$ 0 0
$$424$$ −5.48528 −0.266389
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 62.8701 3.04250
$$428$$ −19.7279 −0.953585
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.38478 0.307544 0.153772 0.988106i $$-0.450858\pi$$
0.153772 + 0.988106i $$0.450858\pi$$
$$432$$ 0 0
$$433$$ −9.55635 −0.459249 −0.229624 0.973279i $$-0.573750\pi$$
−0.229624 + 0.973279i $$0.573750\pi$$
$$434$$ 27.5563 1.32275
$$435$$ 0 0
$$436$$ −17.9706 −0.860634
$$437$$ 0.757359 0.0362294
$$438$$ 0 0
$$439$$ −5.75736 −0.274784 −0.137392 0.990517i $$-0.543872\pi$$
−0.137392 + 0.990517i $$0.543872\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −5.82843 −0.277230
$$443$$ −4.24264 −0.201574 −0.100787 0.994908i $$-0.532136\pi$$
−0.100787 + 0.994908i $$0.532136\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −15.1716 −0.718395
$$447$$ 0 0
$$448$$ 4.41421 0.208552
$$449$$ −17.3137 −0.817084 −0.408542 0.912739i $$-0.633963\pi$$
−0.408542 + 0.912739i $$0.633963\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 10.2426 0.481773
$$453$$ 0 0
$$454$$ 16.7574 0.786462
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.00000 −0.140334 −0.0701670 0.997535i $$-0.522353\pi$$
−0.0701670 + 0.997535i $$0.522353\pi$$
$$458$$ 14.9706 0.699528
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 3.55635 0.165636 0.0828178 0.996565i $$-0.473608\pi$$
0.0828178 + 0.996565i $$0.473608\pi$$
$$462$$ 0 0
$$463$$ 14.1421 0.657241 0.328620 0.944462i $$-0.393416\pi$$
0.328620 + 0.944462i $$0.393416\pi$$
$$464$$ 0.171573 0.00796507
$$465$$ 0 0
$$466$$ 24.9706 1.15674
$$467$$ −0.727922 −0.0336842 −0.0168421 0.999858i $$-0.505361\pi$$
−0.0168421 + 0.999858i $$0.505361\pi$$
$$468$$ 0 0
$$469$$ 21.0000 0.969690
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −6.89949 −0.317575
$$473$$ 2.48528 0.114273
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.41421 −0.202325
$$477$$ 0 0
$$478$$ 6.89949 0.315576
$$479$$ 31.1127 1.42158 0.710788 0.703407i $$-0.248339\pi$$
0.710788 + 0.703407i $$0.248339\pi$$
$$480$$ 0 0
$$481$$ −49.4558 −2.25499
$$482$$ 8.97056 0.408598
$$483$$ 0 0
$$484$$ −9.00000 −0.409091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 37.7990 1.71284 0.856418 0.516283i $$-0.172685\pi$$
0.856418 + 0.516283i $$0.172685\pi$$
$$488$$ 14.2426 0.644734
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 33.5563 1.51438 0.757188 0.653197i $$-0.226572\pi$$
0.757188 + 0.653197i $$0.226572\pi$$
$$492$$ 0 0
$$493$$ −0.171573 −0.00772725
$$494$$ −5.82843 −0.262233
$$495$$ 0 0
$$496$$ 6.24264 0.280303
$$497$$ 59.2132 2.65608
$$498$$ 0 0
$$499$$ −25.7574 −1.15306 −0.576529 0.817077i $$-0.695593\pi$$
−0.576529 + 0.817077i $$0.695593\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −3.55635 −0.158728
$$503$$ −14.2721 −0.636361 −0.318180 0.948030i $$-0.603072\pi$$
−0.318180 + 0.948030i $$0.603072\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −1.07107 −0.0476148
$$507$$ 0 0
$$508$$ −2.48528 −0.110267
$$509$$ −28.9706 −1.28410 −0.642049 0.766664i $$-0.721915\pi$$
−0.642049 + 0.766664i $$0.721915\pi$$
$$510$$ 0 0
$$511$$ 50.6985 2.24277
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −20.7279 −0.914269
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −37.4558 −1.64572
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −23.3137 −1.02139 −0.510696 0.859761i $$-0.670612\pi$$
−0.510696 + 0.859761i $$0.670612\pi$$
$$522$$ 0 0
$$523$$ −2.27208 −0.0993510 −0.0496755 0.998765i $$-0.515819\pi$$
−0.0496755 + 0.998765i $$0.515819\pi$$
$$524$$ 16.9706 0.741362
$$525$$ 0 0
$$526$$ 26.9706 1.17597
$$527$$ −6.24264 −0.271934
$$528$$ 0 0
$$529$$ −22.4264 −0.975061
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.41421 −0.191380
$$533$$ 24.7279 1.07109
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.75736 0.205487
$$537$$ 0 0
$$538$$ 16.6274 0.716859
$$539$$ 17.6569 0.760535
$$540$$ 0 0
$$541$$ 9.75736 0.419502 0.209751 0.977755i $$-0.432735\pi$$
0.209751 + 0.977755i $$0.432735\pi$$
$$542$$ −27.2426 −1.17017
$$543$$ 0 0
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −17.3137 −0.740281 −0.370140 0.928976i $$-0.620690\pi$$
−0.370140 + 0.928976i $$0.620690\pi$$
$$548$$ −13.0000 −0.555332
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.171573 −0.00730925
$$552$$ 0 0
$$553$$ −28.6274 −1.21736
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ 0 0
$$559$$ 10.2426 0.433218
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −4.24264 −0.178965
$$563$$ −4.97056 −0.209484 −0.104742 0.994499i $$-0.533402\pi$$
−0.104742 + 0.994499i $$0.533402\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 32.1421 1.35103
$$567$$ 0 0
$$568$$ 13.4142 0.562848
$$569$$ 28.2843 1.18574 0.592869 0.805299i $$-0.297995\pi$$
0.592869 + 0.805299i $$0.297995\pi$$
$$570$$ 0 0
$$571$$ −2.24264 −0.0938516 −0.0469258 0.998898i $$-0.514942\pi$$
−0.0469258 + 0.998898i $$0.514942\pi$$
$$572$$ 8.24264 0.344642
$$573$$ 0 0
$$574$$ 18.7279 0.781688
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −20.3137 −0.845671 −0.422835 0.906207i $$-0.638965\pi$$
−0.422835 + 0.906207i $$0.638965\pi$$
$$578$$ −16.0000 −0.665512
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −63.9411 −2.65272
$$582$$ 0 0
$$583$$ −7.75736 −0.321277
$$584$$ 11.4853 0.475264
$$585$$ 0 0
$$586$$ 5.48528 0.226595
$$587$$ −30.2426 −1.24825 −0.624124 0.781326i $$-0.714544\pi$$
−0.624124 + 0.781326i $$0.714544\pi$$
$$588$$ 0 0
$$589$$ −6.24264 −0.257224
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −8.48528 −0.348743
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −17.6569 −0.723253
$$597$$ 0 0
$$598$$ −4.41421 −0.180511
$$599$$ 15.2132 0.621595 0.310797 0.950476i $$-0.399404\pi$$
0.310797 + 0.950476i $$0.399404\pi$$
$$600$$ 0 0
$$601$$ −20.2426 −0.825715 −0.412857 0.910796i $$-0.635469\pi$$
−0.412857 + 0.910796i $$0.635469\pi$$
$$602$$ 7.75736 0.316166
$$603$$ 0 0
$$604$$ −10.4853 −0.426640
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3.17157 0.128730 0.0643651 0.997926i $$-0.479498\pi$$
0.0643651 + 0.997926i $$0.479498\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −11.6985 −0.472497 −0.236249 0.971693i $$-0.575918\pi$$
−0.236249 + 0.971693i $$0.575918\pi$$
$$614$$ −17.6569 −0.712573
$$615$$ 0 0
$$616$$ 6.24264 0.251523
$$617$$ 4.48528 0.180571 0.0902853 0.995916i $$-0.471222\pi$$
0.0902853 + 0.995916i $$0.471222\pi$$
$$618$$ 0 0
$$619$$ 7.75736 0.311795 0.155897 0.987773i $$-0.450173\pi$$
0.155897 + 0.987773i $$0.450173\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 4.75736 0.190753
$$623$$ −31.2132 −1.25053
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −7.97056 −0.318568
$$627$$ 0 0
$$628$$ 11.6569 0.465159
$$629$$ 8.48528 0.338330
$$630$$ 0 0
$$631$$ −38.9706 −1.55139 −0.775697 0.631106i $$-0.782601\pi$$
−0.775697 + 0.631106i $$0.782601\pi$$
$$632$$ −6.48528 −0.257971
$$633$$ 0 0
$$634$$ 9.48528 0.376709
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 72.7696 2.88323
$$638$$ 0.242641 0.00960624
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0416 0.712602 0.356301 0.934371i $$-0.384038\pi$$
0.356301 + 0.934371i $$0.384038\pi$$
$$642$$ 0 0
$$643$$ −14.4853 −0.571244 −0.285622 0.958342i $$-0.592200\pi$$
−0.285622 + 0.958342i $$0.592200\pi$$
$$644$$ −3.34315 −0.131738
$$645$$ 0 0
$$646$$ 1.00000 0.0393445
$$647$$ 18.7574 0.737428 0.368714 0.929543i $$-0.379798\pi$$
0.368714 + 0.929543i $$0.379798\pi$$
$$648$$ 0 0
$$649$$ −9.75736 −0.383010
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −10.2426 −0.401133
$$653$$ 28.9706 1.13371 0.566853 0.823819i $$-0.308161\pi$$
0.566853 + 0.823819i $$0.308161\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 4.24264 0.165647
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −18.8995 −0.736220 −0.368110 0.929782i $$-0.619995\pi$$
−0.368110 + 0.929782i $$0.619995\pi$$
$$660$$ 0 0
$$661$$ −18.4558 −0.717849 −0.358925 0.933367i $$-0.616856\pi$$
−0.358925 + 0.933367i $$0.616856\pi$$
$$662$$ −19.2426 −0.747886
$$663$$ 0 0
$$664$$ −14.4853 −0.562138
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.129942 −0.00503139
$$668$$ 18.2426 0.705829
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 20.1421 0.777579
$$672$$ 0 0
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ 14.1005 0.543131
$$675$$ 0 0
$$676$$ 20.9706 0.806560
$$677$$ 40.9411 1.57350 0.786748 0.617275i $$-0.211763\pi$$
0.786748 + 0.617275i $$0.211763\pi$$
$$678$$ 0 0
$$679$$ −1.51472 −0.0581296
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 8.82843 0.338058
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 24.2132 0.924464
$$687$$ 0 0
$$688$$ 1.75736 0.0669987
$$689$$ −31.9706 −1.21798
$$690$$ 0 0
$$691$$ 22.4853 0.855380 0.427690 0.903925i $$-0.359327\pi$$
0.427690 + 0.903925i $$0.359327\pi$$
$$692$$ −0.485281 −0.0184476
$$693$$ 0 0
$$694$$ 5.51472 0.209336
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.24264 −0.160701
$$698$$ 6.00000 0.227103
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −16.9706 −0.640969 −0.320485 0.947254i $$-0.603846\pi$$
−0.320485 + 0.947254i $$0.603846\pi$$
$$702$$ 0 0
$$703$$ 8.48528 0.320028
$$704$$ 1.41421 0.0533002
$$705$$ 0 0
$$706$$ 2.51472 0.0946427
$$707$$ −57.6985 −2.16997
$$708$$ 0 0
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −7.07107 −0.264999
$$713$$ −4.72792 −0.177062
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 11.6569 0.435637
$$717$$ 0 0
$$718$$ 22.7574 0.849297
$$719$$ 5.10051 0.190217 0.0951084 0.995467i $$-0.469680\pi$$
0.0951084 + 0.995467i $$0.469680\pi$$
$$720$$ 0 0
$$721$$ 18.7279 0.697464
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −8.48528 −0.315353
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 9.72792 0.360789 0.180394 0.983594i $$-0.442263\pi$$
0.180394 + 0.983594i $$0.442263\pi$$
$$728$$ 25.7279 0.953540
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1.75736 −0.0649983
$$732$$ 0 0
$$733$$ 12.0000 0.443230 0.221615 0.975134i $$-0.428867\pi$$
0.221615 + 0.975134i $$0.428867\pi$$
$$734$$ −25.4558 −0.939592
$$735$$ 0 0
$$736$$ −0.757359 −0.0279166
$$737$$ 6.72792 0.247826
$$738$$ 0 0
$$739$$ 1.27208 0.0467941 0.0233971 0.999726i $$-0.492552\pi$$
0.0233971 + 0.999726i $$0.492552\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −24.2132 −0.888895
$$743$$ −6.72792 −0.246824 −0.123412 0.992356i $$-0.539384\pi$$
−0.123412 + 0.992356i $$0.539384\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 9.00000 0.329513
$$747$$ 0 0
$$748$$ −1.41421 −0.0517088
$$749$$ −87.0833 −3.18195
$$750$$ 0 0
$$751$$ −26.7279 −0.975316 −0.487658 0.873035i $$-0.662149\pi$$
−0.487658 + 0.873035i $$0.662149\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 1.00000 0.0364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12.3431 0.448619 0.224310 0.974518i $$-0.427987\pi$$
0.224310 + 0.974518i $$0.427987\pi$$
$$758$$ 11.2426 0.408351
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −43.9706 −1.59393 −0.796966 0.604024i $$-0.793563\pi$$
−0.796966 + 0.604024i $$0.793563\pi$$
$$762$$ 0 0
$$763$$ −79.3259 −2.87179
$$764$$ −12.5563 −0.454273
$$765$$ 0 0
$$766$$ −12.2426 −0.442345
$$767$$ −40.2132 −1.45201
$$768$$ 0 0
$$769$$ −36.4558 −1.31463 −0.657316 0.753615i $$-0.728308\pi$$
−0.657316 + 0.753615i $$0.728308\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −0.343146 −0.0123501
$$773$$ −13.9706 −0.502486 −0.251243 0.967924i $$-0.580839\pi$$
−0.251243 + 0.967924i $$0.580839\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −0.343146 −0.0123182
$$777$$ 0 0
$$778$$ −22.9289 −0.822042
$$779$$ −4.24264 −0.152008
$$780$$ 0 0
$$781$$ 18.9706 0.678820
$$782$$ 0.757359 0.0270831
$$783$$ 0 0
$$784$$ 12.4853 0.445903
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −41.1838 −1.46804 −0.734021 0.679126i $$-0.762359\pi$$
−0.734021 + 0.679126i $$0.762359\pi$$
$$788$$ 11.7574 0.418839
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 45.2132 1.60760
$$792$$ 0 0
$$793$$ 83.0122 2.94785
$$794$$ 24.0000 0.851728
$$795$$ 0 0
$$796$$ 9.24264 0.327597
$$797$$ 47.4853 1.68201 0.841007 0.541023i $$-0.181963\pi$$
0.841007 + 0.541023i $$0.181963\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 25.4142 0.897407
$$803$$ 16.2426 0.573190
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 36.3848 1.28160
$$807$$ 0 0
$$808$$ −13.0711 −0.459839
$$809$$ 28.7990 1.01252 0.506259 0.862381i $$-0.331028\pi$$
0.506259 + 0.862381i $$0.331028\pi$$
$$810$$ 0 0
$$811$$ 52.6985 1.85049 0.925247 0.379365i $$-0.123858\pi$$
0.925247 + 0.379365i $$0.123858\pi$$
$$812$$ 0.757359 0.0265781
$$813$$ 0 0
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.75736 −0.0614822
$$818$$ 25.2132 0.881559
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.34315 −0.221377 −0.110689 0.993855i $$-0.535306\pi$$
−0.110689 + 0.993855i $$0.535306\pi$$
$$822$$ 0 0
$$823$$ −15.7279 −0.548241 −0.274120 0.961695i $$-0.588387\pi$$
−0.274120 + 0.961695i $$0.588387\pi$$
$$824$$ 4.24264 0.147799
$$825$$ 0 0
$$826$$ −30.4558 −1.05969
$$827$$ 20.6985 0.719757 0.359878 0.932999i $$-0.382818\pi$$
0.359878 + 0.932999i $$0.382818\pi$$
$$828$$ 0 0
$$829$$ 10.4558 0.363146 0.181573 0.983377i $$-0.441881\pi$$
0.181573 + 0.983377i $$0.441881\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 5.82843 0.202064
$$833$$ −12.4853 −0.432589
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −1.41421 −0.0489116
$$837$$ 0 0
$$838$$ −19.4142 −0.670653
$$839$$ −22.6274 −0.781185 −0.390593 0.920564i $$-0.627730\pi$$
−0.390593 + 0.920564i $$0.627730\pi$$
$$840$$ 0 0
$$841$$ −28.9706 −0.998985
$$842$$ 19.4853 0.671507
$$843$$ 0 0
$$844$$ −19.7279 −0.679063
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −39.7279 −1.36507
$$848$$ −5.48528 −0.188365
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 6.42641 0.220294
$$852$$ 0 0
$$853$$ −27.9411 −0.956686 −0.478343 0.878173i $$-0.658762\pi$$
−0.478343 + 0.878173i $$0.658762\pi$$
$$854$$ 62.8701 2.15137
$$855$$ 0 0
$$856$$ −19.7279 −0.674286
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 4.72792 0.161315 0.0806573 0.996742i $$-0.474298\pi$$
0.0806573 + 0.996742i $$0.474298\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 6.38478 0.217466
$$863$$ −0.727922 −0.0247788 −0.0123894 0.999923i $$-0.503944\pi$$
−0.0123894 + 0.999923i $$0.503944\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −9.55635 −0.324738
$$867$$ 0 0
$$868$$ 27.5563 0.935323
$$869$$ −9.17157 −0.311124
$$870$$ 0 0
$$871$$ 27.7279 0.939525
$$872$$ −17.9706 −0.608560
$$873$$ 0 0
$$874$$ 0.757359 0.0256181
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −18.8579 −0.636785 −0.318392 0.947959i $$-0.603143\pi$$
−0.318392 + 0.947959i $$0.603143\pi$$
$$878$$ −5.75736 −0.194301
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 39.5980 1.33409 0.667045 0.745018i $$-0.267559\pi$$
0.667045 + 0.745018i $$0.267559\pi$$
$$882$$ 0 0
$$883$$ 14.4853 0.487469 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$884$$ −5.82843 −0.196031
$$885$$ 0 0
$$886$$ −4.24264 −0.142534
$$887$$ 45.2132 1.51811 0.759055 0.651026i $$-0.225661\pi$$
0.759055 + 0.651026i $$0.225661\pi$$
$$888$$ 0 0
$$889$$ −10.9706 −0.367941
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −15.1716 −0.507982
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.41421 0.147469
$$897$$ 0 0
$$898$$ −17.3137 −0.577766
$$899$$ 1.07107 0.0357221
$$900$$ 0 0
$$901$$ 5.48528 0.182741
$$902$$ 6.00000 0.199778
$$903$$ 0 0
$$904$$ 10.2426 0.340665
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38.6985 1.28496 0.642481 0.766302i $$-0.277905\pi$$
0.642481 + 0.766302i $$0.277905\pi$$
$$908$$ 16.7574 0.556113
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40.2843 −1.33468 −0.667339 0.744754i $$-0.732567\pi$$
−0.667339 + 0.744754i $$0.732567\pi$$
$$912$$ 0 0
$$913$$ −20.4853 −0.677964
$$914$$ −3.00000 −0.0992312
$$915$$ 0 0
$$916$$ 14.9706 0.494641
$$917$$ 74.9117 2.47380
$$918$$ 0 0
$$919$$ −6.21320 −0.204955 −0.102477 0.994735i $$-0.532677\pi$$
−0.102477 + 0.994735i $$0.532677\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 3.55635 0.117122
$$923$$ 78.1838 2.57345
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 14.1421 0.464739
$$927$$ 0 0
$$928$$ 0.171573 0.00563216
$$929$$ 12.1716 0.399336 0.199668 0.979864i $$-0.436014\pi$$
0.199668 + 0.979864i $$0.436014\pi$$
$$930$$ 0 0
$$931$$ −12.4853 −0.409189
$$932$$ 24.9706 0.817938
$$933$$ 0 0
$$934$$ −0.727922 −0.0238183
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 27.0000 0.882052 0.441026 0.897494i $$-0.354615\pi$$
0.441026 + 0.897494i $$0.354615\pi$$
$$938$$ 21.0000 0.685674
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −53.1421 −1.73238 −0.866192 0.499711i $$-0.833439\pi$$
−0.866192 + 0.499711i $$0.833439\pi$$
$$942$$ 0 0
$$943$$ −3.21320 −0.104636
$$944$$ −6.89949 −0.224559
$$945$$ 0 0
$$946$$ 2.48528 0.0808035
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ 66.9411 2.17300
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4.41421 −0.143065
$$953$$ −18.7279 −0.606657 −0.303328 0.952886i $$-0.598098\pi$$
−0.303328 + 0.952886i $$0.598098\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6.89949 0.223146
$$957$$ 0 0
$$958$$ 31.1127 1.00521
$$959$$ −57.3848 −1.85305
$$960$$ 0 0
$$961$$ 7.97056 0.257115
$$962$$ −49.4558 −1.59452
$$963$$ 0 0
$$964$$ 8.97056 0.288922
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43.1127 1.38641 0.693205 0.720740i $$-0.256198\pi$$
0.693205 + 0.720740i $$0.256198\pi$$
$$968$$ −9.00000 −0.289271
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −41.6569 −1.33683 −0.668416 0.743788i $$-0.733027\pi$$
−0.668416 + 0.743788i $$0.733027\pi$$
$$972$$ 0 0
$$973$$ −52.9706 −1.69816
$$974$$ 37.7990 1.21116
$$975$$ 0 0
$$976$$ 14.2426 0.455896
$$977$$ −0.242641 −0.00776276 −0.00388138 0.999992i $$-0.501235\pi$$
−0.00388138 + 0.999992i $$0.501235\pi$$
$$978$$ 0 0
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 33.5563 1.07083
$$983$$ 51.4558 1.64119 0.820593 0.571513i $$-0.193643\pi$$
0.820593 + 0.571513i $$0.193643\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −0.171573 −0.00546399
$$987$$ 0 0
$$988$$ −5.82843 −0.185427
$$989$$ −1.33095 −0.0423218
$$990$$ 0 0
$$991$$ 13.7574 0.437017 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$992$$ 6.24264 0.198204
$$993$$ 0 0
$$994$$ 59.2132 1.87813
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 48.7279 1.54323 0.771614 0.636091i $$-0.219450\pi$$
0.771614 + 0.636091i $$0.219450\pi$$
$$998$$ −25.7574 −0.815335
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.cb.1.2 2
3.2 odd 2 950.2.a.f.1.1 2
5.2 odd 4 1710.2.d.c.1369.3 4
5.3 odd 4 1710.2.d.c.1369.1 4
5.4 even 2 8550.2.a.bn.1.1 2
12.11 even 2 7600.2.a.v.1.2 2
15.2 even 4 190.2.b.a.39.2 4
15.8 even 4 190.2.b.a.39.3 yes 4
15.14 odd 2 950.2.a.g.1.2 2
60.23 odd 4 1520.2.d.e.609.3 4
60.47 odd 4 1520.2.d.e.609.2 4
60.59 even 2 7600.2.a.bg.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.2 4 15.2 even 4
190.2.b.a.39.3 yes 4 15.8 even 4
950.2.a.f.1.1 2 3.2 odd 2
950.2.a.g.1.2 2 15.14 odd 2
1520.2.d.e.609.2 4 60.47 odd 4
1520.2.d.e.609.3 4 60.23 odd 4
1710.2.d.c.1369.1 4 5.3 odd 4
1710.2.d.c.1369.3 4 5.2 odd 4
7600.2.a.v.1.2 2 12.11 even 2
7600.2.a.bg.1.1 2 60.59 even 2
8550.2.a.bn.1.1 2 5.4 even 2
8550.2.a.cb.1.2 2 1.1 even 1 trivial