# Properties

 Label 3240.2.a.n.1.2 Level $3240$ Weight $2$ Character 3240.1 Self dual yes Analytic conductor $25.872$ 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] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.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: $$25.8715302549$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 3240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} +4.27492 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} +4.27492 q^{7} -1.27492 q^{11} +6.27492 q^{13} +2.00000 q^{17} +1.00000 q^{19} +0.274917 q^{23} +1.00000 q^{25} -1.27492 q^{29} -1.27492 q^{31} +4.27492 q^{35} +4.54983 q^{37} -7.54983 q^{41} +4.00000 q^{43} +6.27492 q^{47} +11.2749 q^{49} +8.27492 q^{53} -1.27492 q^{55} -13.0000 q^{59} -6.54983 q^{61} +6.27492 q^{65} -14.5498 q^{67} -0.725083 q^{71} -15.0997 q^{73} -5.45017 q^{77} -4.54983 q^{79} +12.5498 q^{83} +2.00000 q^{85} -9.82475 q^{89} +26.8248 q^{91} +1.00000 q^{95} +16.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + q^7 $$2 q + 2 q^{5} + q^{7} + 5 q^{11} + 5 q^{13} + 4 q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 5 q^{29} + 5 q^{31} + q^{35} - 6 q^{37} + 8 q^{43} + 5 q^{47} + 15 q^{49} + 9 q^{53} + 5 q^{55} - 26 q^{59} + 2 q^{61} + 5 q^{65} - 14 q^{67} - 9 q^{71} - 26 q^{77} + 6 q^{79} + 10 q^{83} + 4 q^{85} + 3 q^{89} + 31 q^{91} + 2 q^{95} + 32 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + q^7 + 5 * q^11 + 5 * q^13 + 4 * q^17 + 2 * q^19 - 7 * q^23 + 2 * q^25 + 5 * q^29 + 5 * q^31 + q^35 - 6 * q^37 + 8 * q^43 + 5 * q^47 + 15 * q^49 + 9 * q^53 + 5 * q^55 - 26 * q^59 + 2 * q^61 + 5 * q^65 - 14 * q^67 - 9 * q^71 - 26 * q^77 + 6 * q^79 + 10 * q^83 + 4 * q^85 + 3 * q^89 + 31 * q^91 + 2 * q^95 + 32 * q^97

## 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
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.27492 1.61577 0.807883 0.589342i $$-0.200613\pi$$
0.807883 + 0.589342i $$0.200613\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.27492 −0.384402 −0.192201 0.981356i $$-0.561563\pi$$
−0.192201 + 0.981356i $$0.561563\pi$$
$$12$$ 0 0
$$13$$ 6.27492 1.74035 0.870174 0.492744i $$-0.164006\pi$$
0.870174 + 0.492744i $$0.164006\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.274917 0.0573242 0.0286621 0.999589i $$-0.490875\pi$$
0.0286621 + 0.999589i $$0.490875\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.27492 −0.236746 −0.118373 0.992969i $$-0.537768\pi$$
−0.118373 + 0.992969i $$0.537768\pi$$
$$30$$ 0 0
$$31$$ −1.27492 −0.228982 −0.114491 0.993424i $$-0.536524\pi$$
−0.114491 + 0.993424i $$0.536524\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.27492 0.722593
$$36$$ 0 0
$$37$$ 4.54983 0.747988 0.373994 0.927431i $$-0.377988\pi$$
0.373994 + 0.927431i $$0.377988\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.54983 −1.17909 −0.589543 0.807737i $$-0.700692\pi$$
−0.589543 + 0.807737i $$0.700692\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.27492 0.915291 0.457645 0.889135i $$-0.348693\pi$$
0.457645 + 0.889135i $$0.348693\pi$$
$$48$$ 0 0
$$49$$ 11.2749 1.61070
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 8.27492 1.13665 0.568324 0.822805i $$-0.307592\pi$$
0.568324 + 0.822805i $$0.307592\pi$$
$$54$$ 0 0
$$55$$ −1.27492 −0.171910
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −13.0000 −1.69246 −0.846228 0.532821i $$-0.821132\pi$$
−0.846228 + 0.532821i $$0.821132\pi$$
$$60$$ 0 0
$$61$$ −6.54983 −0.838620 −0.419310 0.907843i $$-0.637728\pi$$
−0.419310 + 0.907843i $$0.637728\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.27492 0.778308
$$66$$ 0 0
$$67$$ −14.5498 −1.77755 −0.888773 0.458348i $$-0.848441\pi$$
−0.888773 + 0.458348i $$0.848441\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −0.725083 −0.0860515 −0.0430257 0.999074i $$-0.513700\pi$$
−0.0430257 + 0.999074i $$0.513700\pi$$
$$72$$ 0 0
$$73$$ −15.0997 −1.76728 −0.883641 0.468165i $$-0.844915\pi$$
−0.883641 + 0.468165i $$0.844915\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −5.45017 −0.621104
$$78$$ 0 0
$$79$$ −4.54983 −0.511896 −0.255948 0.966691i $$-0.582388\pi$$
−0.255948 + 0.966691i $$0.582388\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.5498 1.37752 0.688762 0.724988i $$-0.258155\pi$$
0.688762 + 0.724988i $$0.258155\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.82475 −1.04142 −0.520711 0.853733i $$-0.674333\pi$$
−0.520711 + 0.853733i $$0.674333\pi$$
$$90$$ 0 0
$$91$$ 26.8248 2.81200
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 17.2749 1.71892 0.859459 0.511204i $$-0.170800\pi$$
0.859459 + 0.511204i $$0.170800\pi$$
$$102$$ 0 0
$$103$$ 16.8248 1.65779 0.828896 0.559403i $$-0.188969\pi$$
0.828896 + 0.559403i $$0.188969\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −15.0997 −1.45974 −0.729870 0.683586i $$-0.760419\pi$$
−0.729870 + 0.683586i $$0.760419\pi$$
$$108$$ 0 0
$$109$$ 1.27492 0.122115 0.0610575 0.998134i $$-0.480553\pi$$
0.0610575 + 0.998134i $$0.480553\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0.274917 0.0256362
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.54983 0.783762
$$120$$ 0 0
$$121$$ −9.37459 −0.852235
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 3.72508 0.330548 0.165274 0.986248i $$-0.447149\pi$$
0.165274 + 0.986248i $$0.447149\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −13.5498 −1.18385 −0.591927 0.805991i $$-0.701633\pi$$
−0.591927 + 0.805991i $$0.701633\pi$$
$$132$$ 0 0
$$133$$ 4.27492 0.370682
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −0.549834 −0.0469755 −0.0234878 0.999724i $$-0.507477\pi$$
−0.0234878 + 0.999724i $$0.507477\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −1.27492 −0.105876
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −4.72508 −0.384522 −0.192261 0.981344i $$-0.561582\pi$$
−0.192261 + 0.981344i $$0.561582\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1.27492 −0.102404
$$156$$ 0 0
$$157$$ 18.2749 1.45850 0.729249 0.684249i $$-0.239870\pi$$
0.729249 + 0.684249i $$0.239870\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.17525 0.0926225
$$162$$ 0 0
$$163$$ 5.45017 0.426890 0.213445 0.976955i $$-0.431532\pi$$
0.213445 + 0.976955i $$0.431532\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −22.5498 −1.74496 −0.872479 0.488651i $$-0.837489\pi$$
−0.872479 + 0.488651i $$0.837489\pi$$
$$168$$ 0 0
$$169$$ 26.3746 2.02881
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 16.2749 1.23736 0.618680 0.785643i $$-0.287668\pi$$
0.618680 + 0.785643i $$0.287668\pi$$
$$174$$ 0 0
$$175$$ 4.27492 0.323153
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3.54983 0.265327 0.132664 0.991161i $$-0.457647\pi$$
0.132664 + 0.991161i $$0.457647\pi$$
$$180$$ 0 0
$$181$$ 8.72508 0.648530 0.324265 0.945966i $$-0.394883\pi$$
0.324265 + 0.945966i $$0.394883\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.54983 0.334510
$$186$$ 0 0
$$187$$ −2.54983 −0.186462
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.3746 −1.32954 −0.664769 0.747049i $$-0.731470\pi$$
−0.664769 + 0.747049i $$0.731470\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 15.3746 1.09539 0.547697 0.836677i $$-0.315505\pi$$
0.547697 + 0.836677i $$0.315505\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −5.45017 −0.382527
$$204$$ 0 0
$$205$$ −7.54983 −0.527303
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.27492 −0.0881879
$$210$$ 0 0
$$211$$ 14.0997 0.970661 0.485331 0.874331i $$-0.338699\pi$$
0.485331 + 0.874331i $$0.338699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −5.45017 −0.369981
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.5498 0.844193
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −22.5498 −1.49669 −0.748343 0.663312i $$-0.769150\pi$$
−0.748343 + 0.663312i $$0.769150\pi$$
$$228$$ 0 0
$$229$$ −6.54983 −0.432825 −0.216413 0.976302i $$-0.569436\pi$$
−0.216413 + 0.976302i $$0.569436\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.0997 1.38229 0.691143 0.722718i $$-0.257108\pi$$
0.691143 + 0.722718i $$0.257108\pi$$
$$234$$ 0 0
$$235$$ 6.27492 0.409330
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −28.5498 −1.84674 −0.923368 0.383917i $$-0.874575\pi$$
−0.923368 + 0.383917i $$0.874575\pi$$
$$240$$ 0 0
$$241$$ −16.7251 −1.07736 −0.538679 0.842511i $$-0.681076\pi$$
−0.538679 + 0.842511i $$0.681076\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 11.2749 0.720328
$$246$$ 0 0
$$247$$ 6.27492 0.399263
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 26.2749 1.65846 0.829229 0.558909i $$-0.188780\pi$$
0.829229 + 0.558909i $$0.188780\pi$$
$$252$$ 0 0
$$253$$ −0.350497 −0.0220355
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −27.0997 −1.69043 −0.845215 0.534426i $$-0.820528\pi$$
−0.845215 + 0.534426i $$0.820528\pi$$
$$258$$ 0 0
$$259$$ 19.4502 1.20857
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −8.27492 −0.510253 −0.255127 0.966908i $$-0.582117\pi$$
−0.255127 + 0.966908i $$0.582117\pi$$
$$264$$ 0 0
$$265$$ 8.27492 0.508324
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 26.9244 1.64161 0.820805 0.571208i $$-0.193525\pi$$
0.820805 + 0.571208i $$0.193525\pi$$
$$270$$ 0 0
$$271$$ 28.5498 1.73428 0.867139 0.498065i $$-0.165956\pi$$
0.867139 + 0.498065i $$0.165956\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.27492 −0.0768804
$$276$$ 0 0
$$277$$ 17.7251 1.06500 0.532499 0.846431i $$-0.321253\pi$$
0.532499 + 0.846431i $$0.321253\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 28.8248 1.71954 0.859770 0.510681i $$-0.170607\pi$$
0.859770 + 0.510681i $$0.170607\pi$$
$$282$$ 0 0
$$283$$ −26.5498 −1.57822 −0.789112 0.614249i $$-0.789459\pi$$
−0.789112 + 0.614249i $$0.789459\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −32.2749 −1.90513
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 18.2749 1.06763 0.533816 0.845601i $$-0.320757\pi$$
0.533816 + 0.845601i $$0.320757\pi$$
$$294$$ 0 0
$$295$$ −13.0000 −0.756889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.72508 0.0997641
$$300$$ 0 0
$$301$$ 17.0997 0.985609
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −6.54983 −0.375042
$$306$$ 0 0
$$307$$ −29.0997 −1.66081 −0.830403 0.557163i $$-0.811890\pi$$
−0.830403 + 0.557163i $$0.811890\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 26.9244 1.52674 0.763372 0.645959i $$-0.223542\pi$$
0.763372 + 0.645959i $$0.223542\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.27492 −0.127772 −0.0638860 0.997957i $$-0.520349\pi$$
−0.0638860 + 0.997957i $$0.520349\pi$$
$$318$$ 0 0
$$319$$ 1.62541 0.0910057
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 6.27492 0.348070
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 26.8248 1.47890
$$330$$ 0 0
$$331$$ 3.82475 0.210227 0.105114 0.994460i $$-0.466479\pi$$
0.105114 + 0.994460i $$0.466479\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −14.5498 −0.794942
$$336$$ 0 0
$$337$$ 20.5498 1.11942 0.559710 0.828688i $$-0.310912\pi$$
0.559710 + 0.828688i $$0.310912\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.62541 0.0880211
$$342$$ 0 0
$$343$$ 18.2749 0.986753
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 14.0000 0.751559 0.375780 0.926709i $$-0.377375\pi$$
0.375780 + 0.926709i $$0.377375\pi$$
$$348$$ 0 0
$$349$$ −9.82475 −0.525907 −0.262953 0.964809i $$-0.584697\pi$$
−0.262953 + 0.964809i $$0.584697\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −8.54983 −0.455062 −0.227531 0.973771i $$-0.573065\pi$$
−0.227531 + 0.973771i $$0.573065\pi$$
$$354$$ 0 0
$$355$$ −0.725083 −0.0384834
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −21.8248 −1.15187 −0.575933 0.817497i $$-0.695361\pi$$
−0.575933 + 0.817497i $$0.695361\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.0997 −0.790353
$$366$$ 0 0
$$367$$ −2.54983 −0.133100 −0.0665501 0.997783i $$-0.521199\pi$$
−0.0665501 + 0.997783i $$0.521199\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 35.3746 1.83656
$$372$$ 0 0
$$373$$ 0.900331 0.0466174 0.0233087 0.999728i $$-0.492580\pi$$
0.0233087 + 0.999728i $$0.492580\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −8.00000 −0.412021
$$378$$ 0 0
$$379$$ 1.72508 0.0886116 0.0443058 0.999018i $$-0.485892\pi$$
0.0443058 + 0.999018i $$0.485892\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −35.3746 −1.80756 −0.903778 0.428001i $$-0.859218\pi$$
−0.903778 + 0.428001i $$0.859218\pi$$
$$384$$ 0 0
$$385$$ −5.45017 −0.277766
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −1.45017 −0.0735263 −0.0367632 0.999324i $$-0.511705\pi$$
−0.0367632 + 0.999324i $$0.511705\pi$$
$$390$$ 0 0
$$391$$ 0.549834 0.0278063
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.54983 −0.228927
$$396$$ 0 0
$$397$$ −23.0997 −1.15934 −0.579670 0.814852i $$-0.696818\pi$$
−0.579670 + 0.814852i $$0.696818\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15.7251 0.785273 0.392637 0.919694i $$-0.371563\pi$$
0.392637 + 0.919694i $$0.371563\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −5.80066 −0.287528
$$408$$ 0 0
$$409$$ −7.17525 −0.354793 −0.177397 0.984139i $$-0.556768\pi$$
−0.177397 + 0.984139i $$0.556768\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −55.5739 −2.73461
$$414$$ 0 0
$$415$$ 12.5498 0.616047
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 21.0997 1.03079 0.515393 0.856954i $$-0.327646\pi$$
0.515393 + 0.856954i $$0.327646\pi$$
$$420$$ 0 0
$$421$$ −24.7251 −1.20503 −0.602513 0.798109i $$-0.705834\pi$$
−0.602513 + 0.798109i $$0.705834\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −28.0000 −1.35501
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1.27492 0.0614106 0.0307053 0.999528i $$-0.490225\pi$$
0.0307053 + 0.999528i $$0.490225\pi$$
$$432$$ 0 0
$$433$$ −2.54983 −0.122537 −0.0612686 0.998121i $$-0.519515\pi$$
−0.0612686 + 0.998121i $$0.519515\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.274917 0.0131511
$$438$$ 0 0
$$439$$ 12.3746 0.590607 0.295303 0.955404i $$-0.404579\pi$$
0.295303 + 0.955404i $$0.404579\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −4.54983 −0.216169 −0.108085 0.994142i $$-0.534472\pi$$
−0.108085 + 0.994142i $$0.534472\pi$$
$$444$$ 0 0
$$445$$ −9.82475 −0.465738
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −13.5498 −0.639456 −0.319728 0.947509i $$-0.603592\pi$$
−0.319728 + 0.947509i $$0.603592\pi$$
$$450$$ 0 0
$$451$$ 9.62541 0.453243
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 26.8248 1.25756
$$456$$ 0 0
$$457$$ 16.0000 0.748448 0.374224 0.927338i $$-0.377909\pi$$
0.374224 + 0.927338i $$0.377909\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 14.3746 0.669491 0.334746 0.942309i $$-0.391350\pi$$
0.334746 + 0.942309i $$0.391350\pi$$
$$462$$ 0 0
$$463$$ −26.4743 −1.23036 −0.615181 0.788386i $$-0.710917\pi$$
−0.615181 + 0.788386i $$0.710917\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 20.1993 0.934714 0.467357 0.884069i $$-0.345206\pi$$
0.467357 + 0.884069i $$0.345206\pi$$
$$468$$ 0 0
$$469$$ −62.1993 −2.87210
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5.09967 −0.234483
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 3.27492 0.149635 0.0748174 0.997197i $$-0.476163\pi$$
0.0748174 + 0.997197i $$0.476163\pi$$
$$480$$ 0 0
$$481$$ 28.5498 1.30176
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 16.0000 0.726523
$$486$$ 0 0
$$487$$ −29.7251 −1.34697 −0.673486 0.739200i $$-0.735204\pi$$
−0.673486 + 0.739200i $$0.735204\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 15.0000 0.676941 0.338470 0.940977i $$-0.390091\pi$$
0.338470 + 0.940977i $$0.390091\pi$$
$$492$$ 0 0
$$493$$ −2.54983 −0.114839
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −3.09967 −0.139039
$$498$$ 0 0
$$499$$ −14.4502 −0.646878 −0.323439 0.946249i $$-0.604839\pi$$
−0.323439 + 0.946249i $$0.604839\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 17.2749 0.768724
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −32.1993 −1.42721 −0.713605 0.700548i $$-0.752939\pi$$
−0.713605 + 0.700548i $$0.752939\pi$$
$$510$$ 0 0
$$511$$ −64.5498 −2.85552
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 16.8248 0.741387
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 25.9244 1.13577 0.567885 0.823108i $$-0.307762\pi$$
0.567885 + 0.823108i $$0.307762\pi$$
$$522$$ 0 0
$$523$$ 5.64950 0.247036 0.123518 0.992342i $$-0.460582\pi$$
0.123518 + 0.992342i $$0.460582\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.54983 −0.111073
$$528$$ 0 0
$$529$$ −22.9244 −0.996714
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −47.3746 −2.05202
$$534$$ 0 0
$$535$$ −15.0997 −0.652816
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −14.3746 −0.619157
$$540$$ 0 0
$$541$$ −8.92442 −0.383691 −0.191845 0.981425i $$-0.561447\pi$$
−0.191845 + 0.981425i $$0.561447\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 1.27492 0.0546115
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.27492 −0.0543133
$$552$$ 0 0
$$553$$ −19.4502 −0.827105
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −19.9244 −0.844225 −0.422112 0.906544i $$-0.638711\pi$$
−0.422112 + 0.906544i $$0.638711\pi$$
$$558$$ 0 0
$$559$$ 25.0997 1.06160
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.90033 0.290814 0.145407 0.989372i $$-0.453551\pi$$
0.145407 + 0.989372i $$0.453551\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −41.5498 −1.74186 −0.870930 0.491407i $$-0.836483\pi$$
−0.870930 + 0.491407i $$0.836483\pi$$
$$570$$ 0 0
$$571$$ −23.8248 −0.997035 −0.498517 0.866880i $$-0.666122\pi$$
−0.498517 + 0.866880i $$0.666122\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0.274917 0.0114648
$$576$$ 0 0
$$577$$ −26.5498 −1.10528 −0.552642 0.833419i $$-0.686380\pi$$
−0.552642 + 0.833419i $$0.686380\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 53.6495 2.22576
$$582$$ 0 0
$$583$$ −10.5498 −0.436929
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −19.4502 −0.802794 −0.401397 0.915904i $$-0.631475\pi$$
−0.401397 + 0.915904i $$0.631475\pi$$
$$588$$ 0 0
$$589$$ −1.27492 −0.0525320
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 12.1993 0.500967 0.250483 0.968121i $$-0.419410\pi$$
0.250483 + 0.968121i $$0.419410\pi$$
$$594$$ 0 0
$$595$$ 8.54983 0.350509
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −23.2749 −0.950987 −0.475494 0.879719i $$-0.657731\pi$$
−0.475494 + 0.879719i $$0.657731\pi$$
$$600$$ 0 0
$$601$$ −32.0997 −1.30937 −0.654686 0.755901i $$-0.727199\pi$$
−0.654686 + 0.755901i $$0.727199\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −9.37459 −0.381131
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 39.3746 1.59293
$$612$$ 0 0
$$613$$ 37.0241 1.49539 0.747694 0.664043i $$-0.231161\pi$$
0.747694 + 0.664043i $$0.231161\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.64950 0.0664065 0.0332033 0.999449i $$-0.489429\pi$$
0.0332033 + 0.999449i $$0.489429\pi$$
$$618$$ 0 0
$$619$$ 19.3746 0.778730 0.389365 0.921083i $$-0.372694\pi$$
0.389365 + 0.921083i $$0.372694\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −42.0000 −1.68269
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 9.09967 0.362828
$$630$$ 0 0
$$631$$ −43.4743 −1.73068 −0.865341 0.501183i $$-0.832898\pi$$
−0.865341 + 0.501183i $$0.832898\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 3.72508 0.147825
$$636$$ 0 0
$$637$$ 70.7492 2.80318
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −18.9244 −0.747470 −0.373735 0.927536i $$-0.621923\pi$$
−0.373735 + 0.927536i $$0.621923\pi$$
$$642$$ 0 0
$$643$$ −2.90033 −0.114378 −0.0571889 0.998363i $$-0.518214\pi$$
−0.0571889 + 0.998363i $$0.518214\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −13.4502 −0.528781 −0.264390 0.964416i $$-0.585171\pi$$
−0.264390 + 0.964416i $$0.585171\pi$$
$$648$$ 0 0
$$649$$ 16.5739 0.650583
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 24.5498 0.960709 0.480355 0.877074i $$-0.340508\pi$$
0.480355 + 0.877074i $$0.340508\pi$$
$$654$$ 0 0
$$655$$ −13.5498 −0.529436
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9.72508 −0.378835 −0.189418 0.981897i $$-0.560660\pi$$
−0.189418 + 0.981897i $$0.560660\pi$$
$$660$$ 0 0
$$661$$ 12.9244 0.502702 0.251351 0.967896i $$-0.419125\pi$$
0.251351 + 0.967896i $$0.419125\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4.27492 0.165774
$$666$$ 0 0
$$667$$ −0.350497 −0.0135713
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.35050 0.322367
$$672$$ 0 0
$$673$$ −39.0997 −1.50718 −0.753591 0.657344i $$-0.771680\pi$$
−0.753591 + 0.657344i $$0.771680\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25.9244 0.996356 0.498178 0.867075i $$-0.334003\pi$$
0.498178 + 0.867075i $$0.334003\pi$$
$$678$$ 0 0
$$679$$ 68.3987 2.62490
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −16.5498 −0.633262 −0.316631 0.948549i $$-0.602552\pi$$
−0.316631 + 0.948549i $$0.602552\pi$$
$$684$$ 0 0
$$685$$ −0.549834 −0.0210081
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 51.9244 1.97816
$$690$$ 0 0
$$691$$ 14.8248 0.563960 0.281980 0.959420i $$-0.409009\pi$$
0.281980 + 0.959420i $$0.409009\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −9.00000 −0.341389
$$696$$ 0 0
$$697$$ −15.0997 −0.571941
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 4.37459 0.165226 0.0826129 0.996582i $$-0.473673\pi$$
0.0826129 + 0.996582i $$0.473673\pi$$
$$702$$ 0 0
$$703$$ 4.54983 0.171600
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 73.8488 2.77737
$$708$$ 0 0
$$709$$ −31.0997 −1.16797 −0.583986 0.811764i $$-0.698508\pi$$
−0.583986 + 0.811764i $$0.698508\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −0.350497 −0.0131262
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −22.1752 −0.826997 −0.413499 0.910505i $$-0.635693\pi$$
−0.413499 + 0.910505i $$0.635693\pi$$
$$720$$ 0 0
$$721$$ 71.9244 2.67861
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1.27492 −0.0473492
$$726$$ 0 0
$$727$$ 7.37459 0.273508 0.136754 0.990605i $$-0.456333\pi$$
0.136754 + 0.990605i $$0.456333\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.5498 0.683292
$$738$$ 0 0
$$739$$ −36.9244 −1.35829 −0.679143 0.734006i $$-0.737649\pi$$
−0.679143 + 0.734006i $$0.737649\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −13.4502 −0.493439 −0.246719 0.969087i $$-0.579353\pi$$
−0.246719 + 0.969087i $$0.579353\pi$$
$$744$$ 0 0
$$745$$ 2.00000 0.0732743
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −64.5498 −2.35860
$$750$$ 0 0
$$751$$ 19.4502 0.709747 0.354873 0.934914i $$-0.384524\pi$$
0.354873 + 0.934914i $$0.384524\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −4.72508 −0.171963
$$756$$ 0 0
$$757$$ −3.37459 −0.122651 −0.0613257 0.998118i $$-0.519533\pi$$
−0.0613257 + 0.998118i $$0.519533\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 34.4502 1.24882 0.624409 0.781098i $$-0.285340\pi$$
0.624409 + 0.781098i $$0.285340\pi$$
$$762$$ 0 0
$$763$$ 5.45017 0.197309
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −81.5739 −2.94546
$$768$$ 0 0
$$769$$ −8.72508 −0.314635 −0.157317 0.987548i $$-0.550285\pi$$
−0.157317 + 0.987548i $$0.550285\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −13.6495 −0.490939 −0.245469 0.969404i $$-0.578942\pi$$
−0.245469 + 0.969404i $$0.578942\pi$$
$$774$$ 0 0
$$775$$ −1.27492 −0.0457964
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −7.54983 −0.270501
$$780$$ 0 0
$$781$$ 0.924421 0.0330784
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 18.2749 0.652260
$$786$$ 0 0
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −25.6495 −0.911991
$$792$$ 0 0
$$793$$ −41.0997 −1.45949
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −40.5498 −1.43635 −0.718174 0.695863i $$-0.755022\pi$$
−0.718174 + 0.695863i $$0.755022\pi$$
$$798$$ 0 0
$$799$$ 12.5498 0.443981
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 19.2508 0.679347
$$804$$ 0 0
$$805$$ 1.17525 0.0414221
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −28.6495 −1.00726 −0.503631 0.863919i $$-0.668003\pi$$
−0.503631 + 0.863919i $$0.668003\pi$$
$$810$$ 0 0
$$811$$ 35.8248 1.25798 0.628989 0.777415i $$-0.283469\pi$$
0.628989 + 0.777415i $$0.283469\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 5.45017 0.190911
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −17.8248 −0.622088 −0.311044 0.950395i $$-0.600679\pi$$
−0.311044 + 0.950395i $$0.600679\pi$$
$$822$$ 0 0
$$823$$ 38.5498 1.34376 0.671881 0.740659i $$-0.265486\pi$$
0.671881 + 0.740659i $$0.265486\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −9.09967 −0.316426 −0.158213 0.987405i $$-0.550573\pi$$
−0.158213 + 0.987405i $$0.550573\pi$$
$$828$$ 0 0
$$829$$ −35.8248 −1.24425 −0.622123 0.782920i $$-0.713729\pi$$
−0.622123 + 0.782920i $$0.713729\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 22.5498 0.781305
$$834$$ 0 0
$$835$$ −22.5498 −0.780369
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −15.4743 −0.534231 −0.267115 0.963665i $$-0.586070\pi$$
−0.267115 + 0.963665i $$0.586070\pi$$
$$840$$ 0 0
$$841$$ −27.3746 −0.943951
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 26.3746 0.907313
$$846$$ 0 0
$$847$$ −40.0756 −1.37701
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.25083 0.0428778
$$852$$ 0 0
$$853$$ 41.2990 1.41405 0.707026 0.707188i $$-0.250037\pi$$
0.707026 + 0.707188i $$0.250037\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 22.7492 0.777097 0.388548 0.921428i $$-0.372977\pi$$
0.388548 + 0.921428i $$0.372977\pi$$
$$858$$ 0 0
$$859$$ −15.4743 −0.527975 −0.263987 0.964526i $$-0.585038\pi$$
−0.263987 + 0.964526i $$0.585038\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 15.7251 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$864$$ 0 0
$$865$$ 16.2749 0.553364
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 5.80066 0.196774
$$870$$ 0 0
$$871$$ −91.2990 −3.09355
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 4.27492 0.144519
$$876$$ 0 0
$$877$$ −5.72508 −0.193322 −0.0966612 0.995317i $$-0.530816\pi$$
−0.0966612 + 0.995317i $$0.530816\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 56.3746 1.89931 0.949654 0.313301i $$-0.101435\pi$$
0.949654 + 0.313301i $$0.101435\pi$$
$$882$$ 0 0
$$883$$ −28.1993 −0.948983 −0.474492 0.880260i $$-0.657368\pi$$
−0.474492 + 0.880260i $$0.657368\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 22.8248 0.766380 0.383190 0.923670i $$-0.374825\pi$$
0.383190 + 0.923670i $$0.374825\pi$$
$$888$$ 0 0
$$889$$ 15.9244 0.534088
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 6.27492 0.209982
$$894$$ 0 0
$$895$$ 3.54983 0.118658
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.62541 0.0542106
$$900$$ 0 0
$$901$$ 16.5498 0.551355
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 8.72508 0.290032
$$906$$ 0 0
$$907$$ 20.9003 0.693984 0.346992 0.937868i $$-0.387203\pi$$
0.346992 + 0.937868i $$0.387203\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 56.0241 1.85616 0.928080 0.372380i $$-0.121458\pi$$
0.928080 + 0.372380i $$0.121458\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −57.9244 −1.91283
$$918$$ 0 0
$$919$$ −40.9244 −1.34997 −0.674986 0.737831i $$-0.735850\pi$$
−0.674986 + 0.737831i $$0.735850\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −4.54983 −0.149760
$$924$$ 0 0
$$925$$ 4.54983 0.149598
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 53.1238 1.74293 0.871467 0.490454i $$-0.163169\pi$$
0.871467 + 0.490454i $$0.163169\pi$$
$$930$$ 0 0
$$931$$ 11.2749 0.369520
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −2.54983 −0.0833885
$$936$$ 0 0
$$937$$ 14.5498 0.475322 0.237661 0.971348i $$-0.423619\pi$$
0.237661 + 0.971348i $$0.423619\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −23.0997 −0.753028 −0.376514 0.926411i $$-0.622877\pi$$
−0.376514 + 0.926411i $$0.622877\pi$$
$$942$$ 0 0
$$943$$ −2.07558 −0.0675902
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −42.5498 −1.38268 −0.691342 0.722528i $$-0.742980\pi$$
−0.691342 + 0.722528i $$0.742980\pi$$
$$948$$ 0 0
$$949$$ −94.7492 −3.07569
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.90033 −0.0939509 −0.0469755 0.998896i $$-0.514958\pi$$
−0.0469755 + 0.998896i $$0.514958\pi$$
$$954$$ 0 0
$$955$$ −18.3746 −0.594588
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2.35050 −0.0759015
$$960$$ 0 0
$$961$$ −29.3746 −0.947567
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 55.6495 1.78957 0.894784 0.446500i $$-0.147330\pi$$
0.894784 + 0.446500i $$0.147330\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −8.64950 −0.277576 −0.138788 0.990322i $$-0.544321\pi$$
−0.138788 + 0.990322i $$0.544321\pi$$
$$972$$ 0 0
$$973$$ −38.4743 −1.23343
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −41.6495 −1.33249 −0.666243 0.745735i $$-0.732099\pi$$
−0.666243 + 0.745735i $$0.732099\pi$$
$$978$$ 0 0
$$979$$ 12.5257 0.400325
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 28.7492 0.916956 0.458478 0.888706i $$-0.348395\pi$$
0.458478 + 0.888706i $$0.348395\pi$$
$$984$$ 0 0
$$985$$ 15.3746 0.489875
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.09967 0.0349674
$$990$$ 0 0
$$991$$ −9.27492 −0.294627 −0.147314 0.989090i $$-0.547063\pi$$
−0.147314 + 0.989090i $$0.547063\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 8.00000 0.253617
$$996$$ 0 0
$$997$$ −29.3746 −0.930302 −0.465151 0.885231i $$-0.654000\pi$$
−0.465151 + 0.885231i $$0.654000\pi$$
$$998$$ 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 3240.2.a.n.1.2 yes 2
3.2 odd 2 3240.2.a.h.1.2 2
4.3 odd 2 6480.2.a.bm.1.1 2
9.2 odd 6 3240.2.q.be.1081.1 4
9.4 even 3 3240.2.q.y.2161.1 4
9.5 odd 6 3240.2.q.be.2161.1 4
9.7 even 3 3240.2.q.y.1081.1 4
12.11 even 2 6480.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3240.2.a.h.1.2 2 3.2 odd 2
3240.2.a.n.1.2 yes 2 1.1 even 1 trivial
3240.2.q.y.1081.1 4 9.7 even 3
3240.2.q.y.2161.1 4 9.4 even 3
3240.2.q.be.1081.1 4 9.2 odd 6
3240.2.q.be.2161.1 4 9.5 odd 6
6480.2.a.bf.1.1 2 12.11 even 2
6480.2.a.bm.1.1 2 4.3 odd 2