# Properties

 Label 864.3.e.e.161.8 Level $864$ Weight $3$ Character 864.161 Analytic conductor $23.542$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(161,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("864.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.e (of order $$2$$, degree $$1$$, 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: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2441150464.4 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196$$ x^8 - 14*x^6 + 77*x^4 - 188*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.8 Root $$-1.52833 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 864.161 Dual form 864.3.e.e.161.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+7.37942i q^{5} +1.26611 q^{7} +O(q^{10})$$ $$q+7.37942i q^{5} +1.26611 q^{7} +5.82843i q^{11} -10.4853 q^{13} +18.3399i q^{17} -20.8722 q^{19} -20.4853i q^{23} -29.4558 q^{25} +11.1777i q^{29} +61.3503 q^{31} +9.34315i q^{35} -38.4264 q^{37} -33.0988i q^{41} -49.3410 q^{43} +21.5980i q^{47} -47.3970 q^{49} -77.5925i q^{53} -43.0104 q^{55} +25.7157i q^{59} -55.8823 q^{61} -77.3753i q^{65} +91.0853 q^{67} -114.770i q^{71} -120.338 q^{73} +7.37942i q^{77} -8.21107 q^{79} +150.338i q^{83} -135.338 q^{85} +118.505i q^{89} -13.2755 q^{91} -154.024i q^{95} -72.8823 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 16 q^{13} - 32 q^{25} + 32 q^{37} + 96 q^{49} + 96 q^{61} - 216 q^{73} - 336 q^{85} - 40 q^{97}+O(q^{100})$$ 8 * q - 16 * q^13 - 32 * q^25 + 32 * q^37 + 96 * q^49 + 96 * q^61 - 216 * q^73 - 336 * q^85 - 40 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 7.37942i 1.47588i 0.674864 + 0.737942i $$0.264202\pi$$
−0.674864 + 0.737942i $$0.735798\pi$$
$$6$$ 0 0
$$7$$ 1.26611 0.180873 0.0904363 0.995902i $$-0.471174\pi$$
0.0904363 + 0.995902i $$0.471174\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.82843i 0.529857i 0.964268 + 0.264929i $$0.0853484\pi$$
−0.964268 + 0.264929i $$0.914652\pi$$
$$12$$ 0 0
$$13$$ −10.4853 −0.806560 −0.403280 0.915077i $$-0.632130\pi$$
−0.403280 + 0.915077i $$0.632130\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 18.3399i 1.07882i 0.842043 + 0.539410i $$0.181353\pi$$
−0.842043 + 0.539410i $$0.818647\pi$$
$$18$$ 0 0
$$19$$ −20.8722 −1.09853 −0.549267 0.835647i $$-0.685093\pi$$
−0.549267 + 0.835647i $$0.685093\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 20.4853i − 0.890664i −0.895365 0.445332i $$-0.853086\pi$$
0.895365 0.445332i $$-0.146914\pi$$
$$24$$ 0 0
$$25$$ −29.4558 −1.17823
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 11.1777i 0.385439i 0.981254 + 0.192720i $$0.0617308\pi$$
−0.981254 + 0.192720i $$0.938269\pi$$
$$30$$ 0 0
$$31$$ 61.3503 1.97904 0.989522 0.144384i $$-0.0461200\pi$$
0.989522 + 0.144384i $$0.0461200\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 9.34315i 0.266947i
$$36$$ 0 0
$$37$$ −38.4264 −1.03855 −0.519276 0.854607i $$-0.673798\pi$$
−0.519276 + 0.854607i $$0.673798\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 33.0988i − 0.807287i −0.914916 0.403644i $$-0.867744\pi$$
0.914916 0.403644i $$-0.132256\pi$$
$$42$$ 0 0
$$43$$ −49.3410 −1.14746 −0.573732 0.819043i $$-0.694505\pi$$
−0.573732 + 0.819043i $$0.694505\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 21.5980i 0.459531i 0.973246 + 0.229766i $$0.0737960\pi$$
−0.973246 + 0.229766i $$0.926204\pi$$
$$48$$ 0 0
$$49$$ −47.3970 −0.967285
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 77.5925i − 1.46401i −0.681299 0.732005i $$-0.738585\pi$$
0.681299 0.732005i $$-0.261415\pi$$
$$54$$ 0 0
$$55$$ −43.0104 −0.782008
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 25.7157i 0.435860i 0.975964 + 0.217930i $$0.0699304\pi$$
−0.975964 + 0.217930i $$0.930070\pi$$
$$60$$ 0 0
$$61$$ −55.8823 −0.916102 −0.458051 0.888926i $$-0.651452\pi$$
−0.458051 + 0.888926i $$0.651452\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 77.3753i − 1.19039i
$$66$$ 0 0
$$67$$ 91.0853 1.35948 0.679741 0.733452i $$-0.262092\pi$$
0.679741 + 0.733452i $$0.262092\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 114.770i − 1.61647i −0.588858 0.808236i $$-0.700422\pi$$
0.588858 0.808236i $$-0.299578\pi$$
$$72$$ 0 0
$$73$$ −120.338 −1.64847 −0.824234 0.566250i $$-0.808394\pi$$
−0.824234 + 0.566250i $$0.808394\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 7.37942i 0.0958366i
$$78$$ 0 0
$$79$$ −8.21107 −0.103938 −0.0519688 0.998649i $$-0.516550\pi$$
−0.0519688 + 0.998649i $$0.516550\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 150.338i 1.81130i 0.424024 + 0.905651i $$0.360617\pi$$
−0.424024 + 0.905651i $$0.639383\pi$$
$$84$$ 0 0
$$85$$ −135.338 −1.59221
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 118.505i 1.33152i 0.746166 + 0.665759i $$0.231892\pi$$
−0.746166 + 0.665759i $$0.768108\pi$$
$$90$$ 0 0
$$91$$ −13.2755 −0.145885
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 154.024i − 1.62131i
$$96$$ 0 0
$$97$$ −72.8823 −0.751363 −0.375682 0.926749i $$-0.622591\pi$$
−0.375682 + 0.926749i $$0.622591\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 113.838i 1.12711i 0.826079 + 0.563554i $$0.190566\pi$$
−0.826079 + 0.563554i $$0.809434\pi$$
$$102$$ 0 0
$$103$$ −158.766 −1.54142 −0.770709 0.637187i $$-0.780098\pi$$
−0.770709 + 0.637187i $$0.780098\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 77.7452i 0.726590i 0.931674 + 0.363295i $$0.118348\pi$$
−0.931674 + 0.363295i $$0.881652\pi$$
$$108$$ 0 0
$$109$$ −15.3970 −0.141257 −0.0706283 0.997503i $$-0.522500\pi$$
−0.0706283 + 0.997503i $$0.522500\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 76.9408i 0.680892i 0.940264 + 0.340446i $$0.110578\pi$$
−0.940264 + 0.340446i $$0.889422\pi$$
$$114$$ 0 0
$$115$$ 151.170 1.31452
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 23.2203i 0.195129i
$$120$$ 0 0
$$121$$ 87.0294 0.719252
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 32.8815i − 0.263052i
$$126$$ 0 0
$$127$$ 72.0936 0.567666 0.283833 0.958874i $$-0.408394\pi$$
0.283833 + 0.958874i $$0.408394\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 84.4214i − 0.644438i −0.946665 0.322219i $$-0.895571\pi$$
0.946665 0.322219i $$-0.104429\pi$$
$$132$$ 0 0
$$133$$ −26.4264 −0.198695
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 29.0832i 0.212286i 0.994351 + 0.106143i $$0.0338502\pi$$
−0.994351 + 0.106143i $$0.966150\pi$$
$$138$$ 0 0
$$139$$ −130.297 −0.937391 −0.468695 0.883360i $$-0.655276\pi$$
−0.468695 + 0.883360i $$0.655276\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 61.1127i − 0.427362i
$$144$$ 0 0
$$145$$ −82.4853 −0.568864
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 92.3514i 0.619808i 0.950768 + 0.309904i $$0.100297\pi$$
−0.950768 + 0.309904i $$0.899703\pi$$
$$150$$ 0 0
$$151$$ −44.9282 −0.297538 −0.148769 0.988872i $$-0.547531\pi$$
−0.148769 + 0.988872i $$0.547531\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 452.730i 2.92084i
$$156$$ 0 0
$$157$$ −131.029 −0.834582 −0.417291 0.908773i $$-0.637020\pi$$
−0.417291 + 0.908773i $$0.637020\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 25.9366i − 0.161097i
$$162$$ 0 0
$$163$$ 258.677 1.58697 0.793487 0.608587i $$-0.208263\pi$$
0.793487 + 0.608587i $$0.208263\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 304.108i − 1.82100i −0.413505 0.910502i $$-0.635696\pi$$
0.413505 0.910502i $$-0.364304\pi$$
$$168$$ 0 0
$$169$$ −59.0589 −0.349461
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0.651690i 0.00376699i 0.999998 + 0.00188350i $$0.000599536\pi$$
−0.999998 + 0.00188350i $$0.999400\pi$$
$$174$$ 0 0
$$175$$ −37.2943 −0.213110
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 126.765i − 0.708182i −0.935211 0.354091i $$-0.884790\pi$$
0.935211 0.354091i $$-0.115210\pi$$
$$180$$ 0 0
$$181$$ 304.735 1.68362 0.841810 0.539775i $$-0.181491\pi$$
0.841810 + 0.539775i $$0.181491\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 283.565i − 1.53278i
$$186$$ 0 0
$$187$$ −106.893 −0.571620
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 134.132i 0.702262i 0.936326 + 0.351131i $$0.114203\pi$$
−0.936326 + 0.351131i $$0.885797\pi$$
$$192$$ 0 0
$$193$$ 182.397 0.945062 0.472531 0.881314i $$-0.343340\pi$$
0.472531 + 0.881314i $$0.343340\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 154.533i − 0.784433i −0.919873 0.392217i $$-0.871708\pi$$
0.919873 0.392217i $$-0.128292\pi$$
$$198$$ 0 0
$$199$$ 36.7171 0.184508 0.0922541 0.995735i $$-0.470593\pi$$
0.0922541 + 0.995735i $$0.470593\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 14.1522i 0.0697155i
$$204$$ 0 0
$$205$$ 244.250 1.19146
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 121.652i − 0.582066i
$$210$$ 0 0
$$211$$ −359.891 −1.70564 −0.852822 0.522201i $$-0.825111\pi$$
−0.852822 + 0.522201i $$0.825111\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 364.108i − 1.69352i
$$216$$ 0 0
$$217$$ 77.6762 0.357955
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 192.299i − 0.870133i
$$222$$ 0 0
$$223$$ 273.870 1.22812 0.614059 0.789260i $$-0.289536\pi$$
0.614059 + 0.789260i $$0.289536\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 234.500i 1.03304i 0.856276 + 0.516519i $$0.172772\pi$$
−0.856276 + 0.516519i $$0.827228\pi$$
$$228$$ 0 0
$$229$$ 25.6468 0.111995 0.0559973 0.998431i $$-0.482166\pi$$
0.0559973 + 0.998431i $$0.482166\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 1.30338i − 0.00559390i −0.999996 0.00279695i $$-0.999110\pi$$
0.999996 0.00279695i $$-0.000890298\pi$$
$$234$$ 0 0
$$235$$ −159.381 −0.678215
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 75.1615i − 0.314483i −0.987560 0.157242i $$-0.949740\pi$$
0.987560 0.157242i $$-0.0502601\pi$$
$$240$$ 0 0
$$241$$ 252.558 1.04796 0.523980 0.851730i $$-0.324447\pi$$
0.523980 + 0.851730i $$0.324447\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 349.762i − 1.42760i
$$246$$ 0 0
$$247$$ 218.850 0.886034
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 246.000i 0.980080i 0.871700 + 0.490040i $$0.163018\pi$$
−0.871700 + 0.490040i $$0.836982\pi$$
$$252$$ 0 0
$$253$$ 119.397 0.471925
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 359.097i 1.39726i 0.715482 + 0.698632i $$0.246207\pi$$
−0.715482 + 0.698632i $$0.753793\pi$$
$$258$$ 0 0
$$259$$ −48.6520 −0.187846
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 215.897i 0.820899i 0.911883 + 0.410450i $$0.134628\pi$$
−0.911883 + 0.410450i $$0.865372\pi$$
$$264$$ 0 0
$$265$$ 572.588 2.16071
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 488.885i 1.81742i 0.417432 + 0.908708i $$0.362930\pi$$
−0.417432 + 0.908708i $$0.637070\pi$$
$$270$$ 0 0
$$271$$ 161.261 0.595059 0.297530 0.954713i $$-0.403837\pi$$
0.297530 + 0.954713i $$0.403837\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 171.681i − 0.624295i
$$276$$ 0 0
$$277$$ 248.617 0.897535 0.448768 0.893648i $$-0.351863\pi$$
0.448768 + 0.893648i $$0.351863\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 329.579i 1.17288i 0.809993 + 0.586439i $$0.199471\pi$$
−0.809993 + 0.586439i $$0.800529\pi$$
$$282$$ 0 0
$$283$$ 262.438 0.927343 0.463671 0.886007i $$-0.346532\pi$$
0.463671 + 0.886007i $$0.346532\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 41.9066i − 0.146016i
$$288$$ 0 0
$$289$$ −47.3532 −0.163852
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 335.872i 1.14632i 0.819443 + 0.573161i $$0.194283\pi$$
−0.819443 + 0.573161i $$0.805717\pi$$
$$294$$ 0 0
$$295$$ −189.767 −0.643279
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 214.794i 0.718374i
$$300$$ 0 0
$$301$$ −62.4710 −0.207545
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 412.379i − 1.35206i
$$306$$ 0 0
$$307$$ −580.045 −1.88940 −0.944698 0.327941i $$-0.893645\pi$$
−0.944698 + 0.327941i $$0.893645\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 452.132i − 1.45380i −0.686743 0.726900i $$-0.740960\pi$$
0.686743 0.726900i $$-0.259040\pi$$
$$312$$ 0 0
$$313$$ −126.632 −0.404577 −0.202288 0.979326i $$-0.564838\pi$$
−0.202288 + 0.979326i $$0.564838\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 237.767i 0.750055i 0.927014 + 0.375028i $$0.122367\pi$$
−0.927014 + 0.375028i $$0.877633\pi$$
$$318$$ 0 0
$$319$$ −65.1487 −0.204228
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 382.794i − 1.18512i
$$324$$ 0 0
$$325$$ 308.853 0.950316
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 27.3454i 0.0831167i
$$330$$ 0 0
$$331$$ 222.611 0.672542 0.336271 0.941765i $$-0.390834\pi$$
0.336271 + 0.941765i $$0.390834\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 672.156i 2.00644i
$$336$$ 0 0
$$337$$ 396.794 1.17743 0.588715 0.808341i $$-0.299634\pi$$
0.588715 + 0.808341i $$0.299634\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 357.576i 1.04861i
$$342$$ 0 0
$$343$$ −122.049 −0.355828
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 218.387i − 0.629357i −0.949198 0.314678i $$-0.898103\pi$$
0.949198 0.314678i $$-0.101897\pi$$
$$348$$ 0 0
$$349$$ 499.161 1.43026 0.715131 0.698990i $$-0.246367\pi$$
0.715131 + 0.698990i $$0.246367\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 423.991i 1.20111i 0.799585 + 0.600554i $$0.205053\pi$$
−0.799585 + 0.600554i $$0.794947\pi$$
$$354$$ 0 0
$$355$$ 846.933 2.38573
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.2254i 0.0897643i 0.998992 + 0.0448822i $$0.0142912\pi$$
−0.998992 + 0.0448822i $$0.985709\pi$$
$$360$$ 0 0
$$361$$ 74.6468 0.206778
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 888.025i − 2.43295i
$$366$$ 0 0
$$367$$ −449.710 −1.22537 −0.612684 0.790328i $$-0.709910\pi$$
−0.612684 + 0.790328i $$0.709910\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 98.2405i − 0.264799i
$$372$$ 0 0
$$373$$ −386.368 −1.03584 −0.517919 0.855430i $$-0.673293\pi$$
−0.517919 + 0.855430i $$0.673293\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 117.202i − 0.310880i
$$378$$ 0 0
$$379$$ 508.603 1.34196 0.670980 0.741475i $$-0.265874\pi$$
0.670980 + 0.741475i $$0.265874\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 308.142i 0.804549i 0.915519 + 0.402274i $$0.131780\pi$$
−0.915519 + 0.402274i $$0.868220\pi$$
$$384$$ 0 0
$$385$$ −54.4558 −0.141444
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 356.707i − 0.916985i −0.888698 0.458492i $$-0.848390\pi$$
0.888698 0.458492i $$-0.151610\pi$$
$$390$$ 0 0
$$391$$ 375.699 0.960866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 60.5929i − 0.153400i
$$396$$ 0 0
$$397$$ 524.191 1.32038 0.660190 0.751099i $$-0.270476\pi$$
0.660190 + 0.751099i $$0.270476\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 236.141i 0.588881i 0.955670 + 0.294441i $$0.0951333\pi$$
−0.955670 + 0.294441i $$0.904867\pi$$
$$402$$ 0 0
$$403$$ −643.276 −1.59622
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 223.966i − 0.550284i
$$408$$ 0 0
$$409$$ 261.235 0.638718 0.319359 0.947634i $$-0.396532\pi$$
0.319359 + 0.947634i $$0.396532\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 32.5589i 0.0788351i
$$414$$ 0 0
$$415$$ −1109.41 −2.67327
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 392.745i − 0.937339i −0.883374 0.468670i $$-0.844733\pi$$
0.883374 0.468670i $$-0.155267\pi$$
$$420$$ 0 0
$$421$$ −266.794 −0.633715 −0.316857 0.948473i $$-0.602628\pi$$
−0.316857 + 0.948473i $$0.602628\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 540.218i − 1.27110i
$$426$$ 0 0
$$427$$ −70.7530 −0.165698
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 601.029i 1.39450i 0.716828 + 0.697250i $$0.245593\pi$$
−0.716828 + 0.697250i $$0.754407\pi$$
$$432$$ 0 0
$$433$$ −605.500 −1.39838 −0.699191 0.714935i $$-0.746456\pi$$
−0.699191 + 0.714935i $$0.746456\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 427.572i 0.978425i
$$438$$ 0 0
$$439$$ 333.992 0.760801 0.380401 0.924822i $$-0.375786\pi$$
0.380401 + 0.924822i $$0.375786\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 203.823i 0.460098i 0.973179 + 0.230049i $$0.0738886\pi$$
−0.973179 + 0.230049i $$0.926111\pi$$
$$444$$ 0 0
$$445$$ −874.500 −1.96517
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 733.492i − 1.63361i −0.576912 0.816806i $$-0.695742\pi$$
0.576912 0.816806i $$-0.304258\pi$$
$$450$$ 0 0
$$451$$ 192.914 0.427747
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 97.9655i − 0.215309i
$$456$$ 0 0
$$457$$ −514.765 −1.12640 −0.563200 0.826321i $$-0.690430\pi$$
−0.563200 + 0.826321i $$0.690430\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 232.778i − 0.504941i −0.967605 0.252470i $$-0.918757\pi$$
0.967605 0.252470i $$-0.0812430\pi$$
$$462$$ 0 0
$$463$$ −72.7826 −0.157198 −0.0785989 0.996906i $$-0.525045\pi$$
−0.0785989 + 0.996906i $$0.525045\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 304.643i − 0.652340i −0.945311 0.326170i $$-0.894242\pi$$
0.945311 0.326170i $$-0.105758\pi$$
$$468$$ 0 0
$$469$$ 115.324 0.245893
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 287.580i − 0.607992i
$$474$$ 0 0
$$475$$ 614.807 1.29433
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 310.118i 0.647427i 0.946155 + 0.323714i $$0.104931\pi$$
−0.946155 + 0.323714i $$0.895069\pi$$
$$480$$ 0 0
$$481$$ 402.912 0.837654
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 537.829i − 1.10893i
$$486$$ 0 0
$$487$$ 744.415 1.52857 0.764287 0.644877i $$-0.223091\pi$$
0.764287 + 0.644877i $$0.223091\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 480.754i 0.979133i 0.871966 + 0.489567i $$0.162845\pi$$
−0.871966 + 0.489567i $$0.837155\pi$$
$$492$$ 0 0
$$493$$ −204.999 −0.415820
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 145.311i − 0.292376i
$$498$$ 0 0
$$499$$ −694.534 −1.39185 −0.695926 0.718113i $$-0.745006\pi$$
−0.695926 + 0.718113i $$0.745006\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 266.309i − 0.529441i −0.964325 0.264720i $$-0.914720\pi$$
0.964325 0.264720i $$-0.0852796\pi$$
$$504$$ 0 0
$$505$$ −840.058 −1.66348
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 205.538i 0.403807i 0.979405 + 0.201903i $$0.0647127\pi$$
−0.979405 + 0.201903i $$0.935287\pi$$
$$510$$ 0 0
$$511$$ −152.361 −0.298163
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 1171.60i − 2.27496i
$$516$$ 0 0
$$517$$ −125.882 −0.243486
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 940.550i 1.80528i 0.430398 + 0.902639i $$0.358373\pi$$
−0.430398 + 0.902639i $$0.641627\pi$$
$$522$$ 0 0
$$523$$ −94.3064 −0.180318 −0.0901591 0.995927i $$-0.528738\pi$$
−0.0901591 + 0.995927i $$0.528738\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1125.16i 2.13503i
$$528$$ 0 0
$$529$$ 109.353 0.206717
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 347.050i 0.651126i
$$534$$ 0 0
$$535$$ −573.714 −1.07236
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 276.250i − 0.512523i
$$540$$ 0 0
$$541$$ −35.7645 −0.0661081 −0.0330541 0.999454i $$-0.510523\pi$$
−0.0330541 + 0.999454i $$0.510523\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 113.621i − 0.208478i
$$546$$ 0 0
$$547$$ 443.305 0.810430 0.405215 0.914221i $$-0.367197\pi$$
0.405215 + 0.914221i $$0.367197\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 233.304i − 0.423419i
$$552$$ 0 0
$$553$$ −10.3961 −0.0187995
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 438.098i 0.786531i 0.919425 + 0.393266i $$0.128655\pi$$
−0.919425 + 0.393266i $$0.871345\pi$$
$$558$$ 0 0
$$559$$ 517.354 0.925499
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 1065.07i − 1.89178i −0.324485 0.945891i $$-0.605191\pi$$
0.324485 0.945891i $$-0.394809\pi$$
$$564$$ 0 0
$$565$$ −567.779 −1.00492
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 939.141i 1.65051i 0.564759 + 0.825256i $$0.308969\pi$$
−0.564759 + 0.825256i $$0.691031\pi$$
$$570$$ 0 0
$$571$$ −294.128 −0.515110 −0.257555 0.966264i $$-0.582917\pi$$
−0.257555 + 0.966264i $$0.582917\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 603.411i 1.04941i
$$576$$ 0 0
$$577$$ −546.500 −0.947140 −0.473570 0.880756i $$-0.657035\pi$$
−0.473570 + 0.880756i $$0.657035\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 190.344i 0.327615i
$$582$$ 0 0
$$583$$ 452.242 0.775716
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 43.7939i − 0.0746064i −0.999304 0.0373032i $$-0.988123\pi$$
0.999304 0.0373032i $$-0.0118767\pi$$
$$588$$ 0 0
$$589$$ −1280.51 −2.17405
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 823.454i 1.38862i 0.719674 + 0.694312i $$0.244291\pi$$
−0.719674 + 0.694312i $$0.755709\pi$$
$$594$$ 0 0
$$595$$ −171.353 −0.287988
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 826.244i 1.37937i 0.724109 + 0.689686i $$0.242251\pi$$
−0.724109 + 0.689686i $$0.757749\pi$$
$$600$$ 0 0
$$601$$ −378.632 −0.630004 −0.315002 0.949091i $$-0.602005\pi$$
−0.315002 + 0.949091i $$0.602005\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 642.227i 1.06153i
$$606$$ 0 0
$$607$$ −139.737 −0.230210 −0.115105 0.993353i $$-0.536720\pi$$
−0.115105 + 0.993353i $$0.536720\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 226.461i − 0.370640i
$$612$$ 0 0
$$613$$ −508.587 −0.829669 −0.414834 0.909897i $$-0.636161\pi$$
−0.414834 + 0.909897i $$0.636161\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 429.415i − 0.695973i −0.937500 0.347986i $$-0.886866\pi$$
0.937500 0.347986i $$-0.113134\pi$$
$$618$$ 0 0
$$619$$ −720.471 −1.16393 −0.581964 0.813215i $$-0.697715\pi$$
−0.581964 + 0.813215i $$0.697715\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 150.040i 0.240835i
$$624$$ 0 0
$$625$$ −493.749 −0.789999
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 704.738i − 1.12041i
$$630$$ 0 0
$$631$$ 479.482 0.759877 0.379938 0.925012i $$-0.375945\pi$$
0.379938 + 0.925012i $$0.375945\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 532.009i 0.837810i
$$636$$ 0 0
$$637$$ 496.971 0.780174
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 562.679i − 0.877815i −0.898532 0.438907i $$-0.855366\pi$$
0.898532 0.438907i $$-0.144634\pi$$
$$642$$ 0 0
$$643$$ −785.471 −1.22157 −0.610786 0.791796i $$-0.709146\pi$$
−0.610786 + 0.791796i $$0.709146\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 571.529i 0.883352i 0.897175 + 0.441676i $$0.145616\pi$$
−0.897175 + 0.441676i $$0.854384\pi$$
$$648$$ 0 0
$$649$$ −149.882 −0.230943
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 325.346i 0.498233i 0.968474 + 0.249117i $$0.0801402\pi$$
−0.968474 + 0.249117i $$0.919860\pi$$
$$654$$ 0 0
$$655$$ 622.981 0.951116
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 902.294i − 1.36919i −0.728926 0.684593i $$-0.759980\pi$$
0.728926 0.684593i $$-0.240020\pi$$
$$660$$ 0 0
$$661$$ −59.6325 −0.0902155 −0.0451078 0.998982i $$-0.514363\pi$$
−0.0451078 + 0.998982i $$0.514363\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 195.012i − 0.293250i
$$666$$ 0 0
$$667$$ 228.979 0.343297
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 325.706i − 0.485403i
$$672$$ 0 0
$$673$$ −1005.17 −1.49357 −0.746787 0.665064i $$-0.768404\pi$$
−0.746787 + 0.665064i $$0.768404\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 175.697i − 0.259523i −0.991545 0.129762i $$-0.958579\pi$$
0.991545 0.129762i $$-0.0414212\pi$$
$$678$$ 0 0
$$679$$ −92.2768 −0.135901
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 768.823i 1.12566i 0.826574 + 0.562828i $$0.190287\pi$$
−0.826574 + 0.562828i $$0.809713\pi$$
$$684$$ 0 0
$$685$$ −214.617 −0.313310
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 813.579i 1.18081i
$$690$$ 0 0
$$691$$ −590.714 −0.854868 −0.427434 0.904047i $$-0.640582\pi$$
−0.427434 + 0.904047i $$0.640582\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 961.519i − 1.38348i
$$696$$ 0 0
$$697$$ 607.029 0.870917
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 967.139i − 1.37966i −0.723974 0.689828i $$-0.757686\pi$$
0.723974 0.689828i $$-0.242314\pi$$
$$702$$ 0 0
$$703$$ 802.042 1.14088
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 144.131i 0.203863i
$$708$$ 0 0
$$709$$ 147.647 0.208246 0.104123 0.994564i $$-0.466796\pi$$
0.104123 + 0.994564i $$0.466796\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 1256.78i − 1.76266i
$$714$$ 0 0
$$715$$ 450.976 0.630736
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 627.098i 0.872181i 0.899903 + 0.436091i $$0.143637\pi$$
−0.899903 + 0.436091i $$0.856363\pi$$
$$720$$ 0 0
$$721$$ −201.015 −0.278800
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 329.250i − 0.454138i
$$726$$ 0 0
$$727$$ −150.518 −0.207040 −0.103520 0.994627i $$-0.533011\pi$$
−0.103520 + 0.994627i $$0.533011\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 904.910i − 1.23791i
$$732$$ 0 0
$$733$$ −430.073 −0.586730 −0.293365 0.956000i $$-0.594775\pi$$
−0.293365 + 0.956000i $$0.594775\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 530.884i 0.720331i
$$738$$ 0 0
$$739$$ 347.230 0.469865 0.234932 0.972012i $$-0.424513\pi$$
0.234932 + 0.972012i $$0.424513\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 48.7797i − 0.0656523i −0.999461 0.0328261i $$-0.989549\pi$$
0.999461 0.0328261i $$-0.0104508\pi$$
$$744$$ 0 0
$$745$$ −681.500 −0.914765
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 98.4338i 0.131420i
$$750$$ 0 0
$$751$$ −323.248 −0.430424 −0.215212 0.976567i $$-0.569044\pi$$
−0.215212 + 0.976567i $$0.569044\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 331.544i − 0.439131i
$$756$$ 0 0
$$757$$ 452.912 0.598298 0.299149 0.954206i $$-0.403297\pi$$
0.299149 + 0.954206i $$0.403297\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 824.757i − 1.08378i −0.840449 0.541890i $$-0.817709\pi$$
0.840449 0.541890i $$-0.182291\pi$$
$$762$$ 0 0
$$763$$ −19.4942 −0.0255495
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 269.637i − 0.351547i
$$768$$ 0 0
$$769$$ 248.955 0.323739 0.161870 0.986812i $$-0.448248\pi$$
0.161870 + 0.986812i $$0.448248\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 351.066i 0.454160i 0.973876 + 0.227080i $$0.0729179\pi$$
−0.973876 + 0.227080i $$0.927082\pi$$
$$774$$ 0 0
$$775$$ −1807.13 −2.33178
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 690.843i 0.886833i
$$780$$ 0 0
$$781$$ 668.926 0.856499
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 966.921i − 1.23175i
$$786$$ 0 0
$$787$$ 342.705 0.435458 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 97.4154i 0.123155i
$$792$$ 0 0
$$793$$ 585.941 0.738892
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 485.087i − 0.608641i −0.952570 0.304320i $$-0.901571\pi$$
0.952570 0.304320i $$-0.0984293\pi$$
$$798$$ 0 0
$$799$$ −396.106 −0.495752
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 701.382i − 0.873452i
$$804$$ 0 0
$$805$$ 191.397 0.237760
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 179.278i 0.221605i 0.993842 + 0.110802i $$0.0353421\pi$$
−0.993842 + 0.110802i $$0.964658\pi$$
$$810$$ 0 0
$$811$$ −475.144 −0.585874 −0.292937 0.956132i $$-0.594633\pi$$
−0.292937 + 0.956132i $$0.594633\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 1908.89i 2.34219i
$$816$$ 0 0
$$817$$ 1029.85 1.26053
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 598.056i 0.728448i 0.931311 + 0.364224i $$0.118666\pi$$
−0.931311 + 0.364224i $$0.881334\pi$$
$$822$$ 0 0
$$823$$ 433.288 0.526474 0.263237 0.964731i $$-0.415210\pi$$
0.263237 + 0.964731i $$0.415210\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 62.2153i − 0.0752301i −0.999292 0.0376151i $$-0.988024\pi$$
0.999292 0.0376151i $$-0.0119761\pi$$
$$828$$ 0 0
$$829$$ 3.70563 0.00447000 0.00223500 0.999998i $$-0.499289\pi$$
0.00223500 + 0.999998i $$0.499289\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 869.257i − 1.04353i
$$834$$ 0 0
$$835$$ 2244.14 2.68759
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 1382.59i − 1.64790i −0.566660 0.823952i $$-0.691765\pi$$
0.566660 0.823952i $$-0.308235\pi$$
$$840$$ 0 0
$$841$$ 716.058 0.851436
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 435.820i − 0.515764i
$$846$$ 0 0
$$847$$ 110.189 0.130093
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 787.176i 0.925001i
$$852$$ 0 0
$$853$$ −1273.53 −1.49300 −0.746500 0.665385i $$-0.768267\pi$$
−0.746500 + 0.665385i $$0.768267\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1009.57i 1.17802i 0.808125 + 0.589011i $$0.200483\pi$$
−0.808125 + 0.589011i $$0.799517\pi$$
$$858$$ 0 0
$$859$$ 719.633 0.837757 0.418878 0.908042i $$-0.362423\pi$$
0.418878 + 0.908042i $$0.362423\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 429.608i − 0.497808i −0.968528 0.248904i $$-0.919930\pi$$
0.968528 0.248904i $$-0.0800703\pi$$
$$864$$ 0 0
$$865$$ −4.80909 −0.00555964
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 47.8576i − 0.0550721i
$$870$$ 0 0
$$871$$ −955.055 −1.09650
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 41.6316i − 0.0475790i
$$876$$ 0 0
$$877$$ −164.030 −0.187036 −0.0935178 0.995618i $$-0.529811\pi$$
−0.0935178 + 0.995618i $$0.529811\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 333.265i − 0.378281i −0.981950 0.189140i $$-0.939430\pi$$
0.981950 0.189140i $$-0.0605701\pi$$
$$882$$ 0 0
$$883$$ 289.138 0.327450 0.163725 0.986506i $$-0.447649\pi$$
0.163725 + 0.986506i $$0.447649\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 483.338i − 0.544913i −0.962168 0.272457i $$-0.912164\pi$$
0.962168 0.272457i $$-0.0878361\pi$$
$$888$$ 0 0
$$889$$ 91.2784 0.102675
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 450.796i − 0.504811i
$$894$$ 0 0
$$895$$ 935.449 1.04519
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 685.759i 0.762802i
$$900$$ 0 0
$$901$$ 1423.04 1.57940
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 2248.77i 2.48483i
$$906$$ 0 0
$$907$$ −769.663 −0.848581 −0.424290 0.905526i $$-0.639476\pi$$
−0.424290 + 0.905526i $$0.639476\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 332.059i 0.364499i 0.983252 + 0.182250i $$0.0583379\pi$$
−0.983252 + 0.182250i $$0.941662\pi$$
$$912$$ 0 0
$$913$$ −876.235 −0.959731
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 106.887i − 0.116561i
$$918$$ 0 0
$$919$$ −1339.15 −1.45718 −0.728591 0.684949i $$-0.759825\pi$$
−0.728591 + 0.684949i $$0.759825\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1203.39i 1.30378i
$$924$$ 0 0
$$925$$ 1131.88 1.22366
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1194.16i 1.28543i 0.766106 + 0.642714i $$0.222192\pi$$
−0.766106 + 0.642714i $$0.777808\pi$$
$$930$$ 0 0
$$931$$ 989.277 1.06260
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 788.808i − 0.843645i
$$936$$ 0 0
$$937$$ 631.705 0.674178 0.337089 0.941473i $$-0.390558\pi$$
0.337089 + 0.941473i $$0.390558\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1616.20i 1.71753i 0.512367 + 0.858766i $$0.328769\pi$$
−0.512367 + 0.858766i $$0.671231\pi$$
$$942$$ 0 0
$$943$$ −678.038 −0.719022
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1625.50i − 1.71647i −0.513256 0.858235i $$-0.671561\pi$$
0.513256 0.858235i $$-0.328439\pi$$
$$948$$ 0 0
$$949$$ 1261.78 1.32959
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 143.033i − 0.150087i −0.997180 0.0750435i $$-0.976090\pi$$
0.997180 0.0750435i $$-0.0239096\pi$$
$$954$$ 0 0
$$955$$ −989.817 −1.03646
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 36.8225i 0.0383968i
$$960$$ 0 0
$$961$$ 2802.87 2.91661
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 1345.98i 1.39480i
$$966$$ 0 0
$$967$$ 132.178 0.136689 0.0683443 0.997662i $$-0.478228\pi$$
0.0683443 + 0.997662i $$0.478228\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1171.53i 1.20652i 0.797543 + 0.603262i $$0.206133\pi$$
−0.797543 + 0.603262i $$0.793867\pi$$
$$972$$ 0 0
$$973$$ −164.971 −0.169548
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 797.623i − 0.816400i −0.912893 0.408200i $$-0.866157\pi$$
0.912893 0.408200i $$-0.133843\pi$$
$$978$$ 0 0
$$979$$ −690.699 −0.705515
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 960.676i − 0.977290i −0.872483 0.488645i $$-0.837491\pi$$
0.872483 0.488645i $$-0.162509\pi$$
$$984$$ 0 0
$$985$$ 1140.37 1.15773
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1010.76i 1.02201i
$$990$$ 0 0
$$991$$ 474.492 0.478802 0.239401 0.970921i $$-0.423049\pi$$
0.239401 + 0.970921i $$0.423049\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 270.951i 0.272313i
$$996$$ 0 0
$$997$$ 180.219 0.180762 0.0903809 0.995907i $$-0.471192\pi$$
0.0903809 + 0.995907i $$0.471192\pi$$
$$998$$ 0 0
$$999$$ 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 864.3.e.e.161.8 yes 8
3.2 odd 2 inner 864.3.e.e.161.2 yes 8
4.3 odd 2 inner 864.3.e.e.161.7 yes 8
8.3 odd 2 1728.3.e.v.1025.1 8
8.5 even 2 1728.3.e.v.1025.2 8
12.11 even 2 inner 864.3.e.e.161.1 8
24.5 odd 2 1728.3.e.v.1025.8 8
24.11 even 2 1728.3.e.v.1025.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.e.161.1 8 12.11 even 2 inner
864.3.e.e.161.2 yes 8 3.2 odd 2 inner
864.3.e.e.161.7 yes 8 4.3 odd 2 inner
864.3.e.e.161.8 yes 8 1.1 even 1 trivial
1728.3.e.v.1025.1 8 8.3 odd 2
1728.3.e.v.1025.2 8 8.5 even 2
1728.3.e.v.1025.7 8 24.11 even 2
1728.3.e.v.1025.8 8 24.5 odd 2