# Properties

 Label 1520.2.a.p.1.3 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.70928 q^{3} +1.00000 q^{5} -1.07838 q^{7} -0.0783777 q^{9} +O(q^{10})$$ $$q+1.70928 q^{3} +1.00000 q^{5} -1.07838 q^{7} -0.0783777 q^{9} +6.34017 q^{11} +1.36910 q^{13} +1.70928 q^{15} +3.26180 q^{17} +1.00000 q^{19} -1.84324 q^{21} -2.34017 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.41855 q^{29} -8.68035 q^{31} +10.8371 q^{33} -1.07838 q^{35} +5.36910 q^{37} +2.34017 q^{39} -3.26180 q^{41} +11.9155 q^{43} -0.0783777 q^{45} -1.07838 q^{47} -5.83710 q^{49} +5.57531 q^{51} +6.63090 q^{53} +6.34017 q^{55} +1.70928 q^{57} +11.4186 q^{59} +5.60197 q^{61} +0.0845208 q^{63} +1.36910 q^{65} -10.3896 q^{67} -4.00000 q^{69} +10.8371 q^{71} +5.41855 q^{73} +1.70928 q^{75} -6.83710 q^{77} -14.2557 q^{79} -8.75872 q^{81} +14.3402 q^{83} +3.26180 q^{85} +2.42469 q^{87} +7.57531 q^{89} -1.47641 q^{91} -14.8371 q^{93} +1.00000 q^{95} -8.88655 q^{97} -0.496928 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 + 8 * q^11 + 8 * q^13 - 2 * q^15 + 2 * q^17 + 3 * q^19 - 12 * q^21 + 4 * q^23 + 3 * q^25 - 8 * q^27 - 10 * q^29 - 4 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 11 * q^49 - 4 * q^51 + 16 * q^53 + 8 * q^55 - 2 * q^57 + 20 * q^59 - 2 * q^61 + 32 * q^63 + 8 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^77 - q^81 + 32 * q^83 + 2 * q^85 + 28 * q^87 + 2 * q^89 - 20 * q^91 - 16 * q^93 + 3 * q^95 + 20 * q^97 + 16 * q^99

## 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
$$3$$ 1.70928 0.986851 0.493425 0.869788i $$-0.335745\pi$$
0.493425 + 0.869788i $$0.335745\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.07838 −0.407588 −0.203794 0.979014i $$-0.565327\pi$$
−0.203794 + 0.979014i $$0.565327\pi$$
$$8$$ 0 0
$$9$$ −0.0783777 −0.0261259
$$10$$ 0 0
$$11$$ 6.34017 1.91163 0.955817 0.293962i $$-0.0949740\pi$$
0.955817 + 0.293962i $$0.0949740\pi$$
$$12$$ 0 0
$$13$$ 1.36910 0.379721 0.189860 0.981811i $$-0.439196\pi$$
0.189860 + 0.981811i $$0.439196\pi$$
$$14$$ 0 0
$$15$$ 1.70928 0.441333
$$16$$ 0 0
$$17$$ 3.26180 0.791102 0.395551 0.918444i $$-0.370554\pi$$
0.395551 + 0.918444i $$0.370554\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −1.84324 −0.402229
$$22$$ 0 0
$$23$$ −2.34017 −0.487960 −0.243980 0.969780i $$-0.578453\pi$$
−0.243980 + 0.969780i $$0.578453\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.26180 −1.01263
$$28$$ 0 0
$$29$$ 1.41855 0.263418 0.131709 0.991288i $$-0.457954\pi$$
0.131709 + 0.991288i $$0.457954\pi$$
$$30$$ 0 0
$$31$$ −8.68035 −1.55904 −0.779518 0.626380i $$-0.784536\pi$$
−0.779518 + 0.626380i $$0.784536\pi$$
$$32$$ 0 0
$$33$$ 10.8371 1.88650
$$34$$ 0 0
$$35$$ −1.07838 −0.182279
$$36$$ 0 0
$$37$$ 5.36910 0.882675 0.441337 0.897341i $$-0.354504\pi$$
0.441337 + 0.897341i $$0.354504\pi$$
$$38$$ 0 0
$$39$$ 2.34017 0.374728
$$40$$ 0 0
$$41$$ −3.26180 −0.509407 −0.254703 0.967019i $$-0.581978\pi$$
−0.254703 + 0.967019i $$0.581978\pi$$
$$42$$ 0 0
$$43$$ 11.9155 1.81709 0.908547 0.417783i $$-0.137193\pi$$
0.908547 + 0.417783i $$0.137193\pi$$
$$44$$ 0 0
$$45$$ −0.0783777 −0.0116839
$$46$$ 0 0
$$47$$ −1.07838 −0.157298 −0.0786488 0.996902i $$-0.525061\pi$$
−0.0786488 + 0.996902i $$0.525061\pi$$
$$48$$ 0 0
$$49$$ −5.83710 −0.833872
$$50$$ 0 0
$$51$$ 5.57531 0.780699
$$52$$ 0 0
$$53$$ 6.63090 0.910824 0.455412 0.890281i $$-0.349492\pi$$
0.455412 + 0.890281i $$0.349492\pi$$
$$54$$ 0 0
$$55$$ 6.34017 0.854909
$$56$$ 0 0
$$57$$ 1.70928 0.226399
$$58$$ 0 0
$$59$$ 11.4186 1.48657 0.743284 0.668976i $$-0.233267\pi$$
0.743284 + 0.668976i $$0.233267\pi$$
$$60$$ 0 0
$$61$$ 5.60197 0.717259 0.358629 0.933480i $$-0.383244\pi$$
0.358629 + 0.933480i $$0.383244\pi$$
$$62$$ 0 0
$$63$$ 0.0845208 0.0106486
$$64$$ 0 0
$$65$$ 1.36910 0.169816
$$66$$ 0 0
$$67$$ −10.3896 −1.26929 −0.634647 0.772802i $$-0.718855\pi$$
−0.634647 + 0.772802i $$0.718855\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 10.8371 1.28613 0.643064 0.765813i $$-0.277663\pi$$
0.643064 + 0.765813i $$0.277663\pi$$
$$72$$ 0 0
$$73$$ 5.41855 0.634193 0.317097 0.948393i $$-0.397292\pi$$
0.317097 + 0.948393i $$0.397292\pi$$
$$74$$ 0 0
$$75$$ 1.70928 0.197370
$$76$$ 0 0
$$77$$ −6.83710 −0.779160
$$78$$ 0 0
$$79$$ −14.2557 −1.60389 −0.801943 0.597400i $$-0.796200\pi$$
−0.801943 + 0.597400i $$0.796200\pi$$
$$80$$ 0 0
$$81$$ −8.75872 −0.973192
$$82$$ 0 0
$$83$$ 14.3402 1.57404 0.787019 0.616928i $$-0.211623\pi$$
0.787019 + 0.616928i $$0.211623\pi$$
$$84$$ 0 0
$$85$$ 3.26180 0.353791
$$86$$ 0 0
$$87$$ 2.42469 0.259954
$$88$$ 0 0
$$89$$ 7.57531 0.802981 0.401490 0.915863i $$-0.368492\pi$$
0.401490 + 0.915863i $$0.368492\pi$$
$$90$$ 0 0
$$91$$ −1.47641 −0.154770
$$92$$ 0 0
$$93$$ −14.8371 −1.53854
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −8.88655 −0.902292 −0.451146 0.892450i $$-0.648985\pi$$
−0.451146 + 0.892450i $$0.648985\pi$$
$$98$$ 0 0
$$99$$ −0.496928 −0.0499432
$$100$$ 0 0
$$101$$ −4.92162 −0.489720 −0.244860 0.969558i $$-0.578742\pi$$
−0.244860 + 0.969558i $$0.578742\pi$$
$$102$$ 0 0
$$103$$ −6.38962 −0.629588 −0.314794 0.949160i $$-0.601935\pi$$
−0.314794 + 0.949160i $$0.601935\pi$$
$$104$$ 0 0
$$105$$ −1.84324 −0.179882
$$106$$ 0 0
$$107$$ −2.29072 −0.221453 −0.110726 0.993851i $$-0.535318\pi$$
−0.110726 + 0.993851i $$0.535318\pi$$
$$108$$ 0 0
$$109$$ −12.8371 −1.22957 −0.614786 0.788694i $$-0.710757\pi$$
−0.614786 + 0.788694i $$0.710757\pi$$
$$110$$ 0 0
$$111$$ 9.17727 0.871068
$$112$$ 0 0
$$113$$ −12.8865 −1.21226 −0.606132 0.795364i $$-0.707280\pi$$
−0.606132 + 0.795364i $$0.707280\pi$$
$$114$$ 0 0
$$115$$ −2.34017 −0.218222
$$116$$ 0 0
$$117$$ −0.107307 −0.00992055
$$118$$ 0 0
$$119$$ −3.51745 −0.322444
$$120$$ 0 0
$$121$$ 29.1978 2.65434
$$122$$ 0 0
$$123$$ −5.57531 −0.502708
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 8.23287 0.730549 0.365274 0.930900i $$-0.380975\pi$$
0.365274 + 0.930900i $$0.380975\pi$$
$$128$$ 0 0
$$129$$ 20.3668 1.79320
$$130$$ 0 0
$$131$$ −1.47641 −0.128995 −0.0644973 0.997918i $$-0.520544\pi$$
−0.0644973 + 0.997918i $$0.520544\pi$$
$$132$$ 0 0
$$133$$ −1.07838 −0.0935072
$$134$$ 0 0
$$135$$ −5.26180 −0.452863
$$136$$ 0 0
$$137$$ −3.94214 −0.336800 −0.168400 0.985719i $$-0.553860\pi$$
−0.168400 + 0.985719i $$0.553860\pi$$
$$138$$ 0 0
$$139$$ −8.86376 −0.751815 −0.375907 0.926657i $$-0.622669\pi$$
−0.375907 + 0.926657i $$0.622669\pi$$
$$140$$ 0 0
$$141$$ −1.84324 −0.155229
$$142$$ 0 0
$$143$$ 8.68035 0.725887
$$144$$ 0 0
$$145$$ 1.41855 0.117804
$$146$$ 0 0
$$147$$ −9.97721 −0.822907
$$148$$ 0 0
$$149$$ −19.7587 −1.61870 −0.809349 0.587328i $$-0.800180\pi$$
−0.809349 + 0.587328i $$0.800180\pi$$
$$150$$ 0 0
$$151$$ −3.41855 −0.278198 −0.139099 0.990279i $$-0.544421\pi$$
−0.139099 + 0.990279i $$0.544421\pi$$
$$152$$ 0 0
$$153$$ −0.255652 −0.0206683
$$154$$ 0 0
$$155$$ −8.68035 −0.697222
$$156$$ 0 0
$$157$$ 9.41855 0.751682 0.375841 0.926684i $$-0.377354\pi$$
0.375841 + 0.926684i $$0.377354\pi$$
$$158$$ 0 0
$$159$$ 11.3340 0.898847
$$160$$ 0 0
$$161$$ 2.52359 0.198887
$$162$$ 0 0
$$163$$ 2.92162 0.228839 0.114420 0.993433i $$-0.463499\pi$$
0.114420 + 0.993433i $$0.463499\pi$$
$$164$$ 0 0
$$165$$ 10.8371 0.843667
$$166$$ 0 0
$$167$$ −20.9132 −1.61831 −0.809156 0.587593i $$-0.800076\pi$$
−0.809156 + 0.587593i $$0.800076\pi$$
$$168$$ 0 0
$$169$$ −11.1256 −0.855812
$$170$$ 0 0
$$171$$ −0.0783777 −0.00599370
$$172$$ 0 0
$$173$$ −1.05559 −0.0802551 −0.0401276 0.999195i $$-0.512776\pi$$
−0.0401276 + 0.999195i $$0.512776\pi$$
$$174$$ 0 0
$$175$$ −1.07838 −0.0815177
$$176$$ 0 0
$$177$$ 19.5174 1.46702
$$178$$ 0 0
$$179$$ −0.894960 −0.0668925 −0.0334462 0.999441i $$-0.510648\pi$$
−0.0334462 + 0.999441i $$0.510648\pi$$
$$180$$ 0 0
$$181$$ −0.837101 −0.0622213 −0.0311106 0.999516i $$-0.509904\pi$$
−0.0311106 + 0.999516i $$0.509904\pi$$
$$182$$ 0 0
$$183$$ 9.57531 0.707827
$$184$$ 0 0
$$185$$ 5.36910 0.394744
$$186$$ 0 0
$$187$$ 20.6803 1.51230
$$188$$ 0 0
$$189$$ 5.67420 0.412738
$$190$$ 0 0
$$191$$ −22.0410 −1.59483 −0.797417 0.603429i $$-0.793801\pi$$
−0.797417 + 0.603429i $$0.793801\pi$$
$$192$$ 0 0
$$193$$ 12.7877 0.920475 0.460238 0.887796i $$-0.347764\pi$$
0.460238 + 0.887796i $$0.347764\pi$$
$$194$$ 0 0
$$195$$ 2.34017 0.167583
$$196$$ 0 0
$$197$$ −9.20394 −0.655753 −0.327877 0.944721i $$-0.606333\pi$$
−0.327877 + 0.944721i $$0.606333\pi$$
$$198$$ 0 0
$$199$$ 16.1978 1.14823 0.574116 0.818774i $$-0.305346\pi$$
0.574116 + 0.818774i $$0.305346\pi$$
$$200$$ 0 0
$$201$$ −17.7587 −1.25260
$$202$$ 0 0
$$203$$ −1.52973 −0.107366
$$204$$ 0 0
$$205$$ −3.26180 −0.227814
$$206$$ 0 0
$$207$$ 0.183417 0.0127484
$$208$$ 0 0
$$209$$ 6.34017 0.438559
$$210$$ 0 0
$$211$$ −7.78539 −0.535968 −0.267984 0.963423i $$-0.586357\pi$$
−0.267984 + 0.963423i $$0.586357\pi$$
$$212$$ 0 0
$$213$$ 18.5236 1.26922
$$214$$ 0 0
$$215$$ 11.9155 0.812629
$$216$$ 0 0
$$217$$ 9.36069 0.635445
$$218$$ 0 0
$$219$$ 9.26180 0.625854
$$220$$ 0 0
$$221$$ 4.46573 0.300398
$$222$$ 0 0
$$223$$ −12.5464 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$224$$ 0 0
$$225$$ −0.0783777 −0.00522518
$$226$$ 0 0
$$227$$ −2.29072 −0.152041 −0.0760204 0.997106i $$-0.524221\pi$$
−0.0760204 + 0.997106i $$0.524221\pi$$
$$228$$ 0 0
$$229$$ −5.91548 −0.390906 −0.195453 0.980713i $$-0.562618\pi$$
−0.195453 + 0.980713i $$0.562618\pi$$
$$230$$ 0 0
$$231$$ −11.6865 −0.768915
$$232$$ 0 0
$$233$$ 13.5174 0.885557 0.442779 0.896631i $$-0.353993\pi$$
0.442779 + 0.896631i $$0.353993\pi$$
$$234$$ 0 0
$$235$$ −1.07838 −0.0703456
$$236$$ 0 0
$$237$$ −24.3668 −1.58280
$$238$$ 0 0
$$239$$ −13.8432 −0.895445 −0.447723 0.894173i $$-0.647765\pi$$
−0.447723 + 0.894173i $$0.647765\pi$$
$$240$$ 0 0
$$241$$ 7.26180 0.467773 0.233887 0.972264i $$-0.424856\pi$$
0.233887 + 0.972264i $$0.424856\pi$$
$$242$$ 0 0
$$243$$ 0.814315 0.0522383
$$244$$ 0 0
$$245$$ −5.83710 −0.372919
$$246$$ 0 0
$$247$$ 1.36910 0.0871139
$$248$$ 0 0
$$249$$ 24.5113 1.55334
$$250$$ 0 0
$$251$$ −10.4703 −0.660877 −0.330439 0.943827i $$-0.607197\pi$$
−0.330439 + 0.943827i $$0.607197\pi$$
$$252$$ 0 0
$$253$$ −14.8371 −0.932801
$$254$$ 0 0
$$255$$ 5.57531 0.349139
$$256$$ 0 0
$$257$$ −23.6248 −1.47367 −0.736836 0.676072i $$-0.763681\pi$$
−0.736836 + 0.676072i $$0.763681\pi$$
$$258$$ 0 0
$$259$$ −5.78992 −0.359768
$$260$$ 0 0
$$261$$ −0.111183 −0.00688204
$$262$$ 0 0
$$263$$ −5.65983 −0.349000 −0.174500 0.984657i $$-0.555831\pi$$
−0.174500 + 0.984657i $$0.555831\pi$$
$$264$$ 0 0
$$265$$ 6.63090 0.407333
$$266$$ 0 0
$$267$$ 12.9483 0.792422
$$268$$ 0 0
$$269$$ −2.31351 −0.141057 −0.0705286 0.997510i $$-0.522469\pi$$
−0.0705286 + 0.997510i $$0.522469\pi$$
$$270$$ 0 0
$$271$$ 19.7009 1.19674 0.598371 0.801219i $$-0.295815\pi$$
0.598371 + 0.801219i $$0.295815\pi$$
$$272$$ 0 0
$$273$$ −2.52359 −0.152735
$$274$$ 0 0
$$275$$ 6.34017 0.382327
$$276$$ 0 0
$$277$$ −25.7321 −1.54609 −0.773045 0.634351i $$-0.781267\pi$$
−0.773045 + 0.634351i $$0.781267\pi$$
$$278$$ 0 0
$$279$$ 0.680346 0.0407312
$$280$$ 0 0
$$281$$ −6.58145 −0.392616 −0.196308 0.980542i $$-0.562895\pi$$
−0.196308 + 0.980542i $$0.562895\pi$$
$$282$$ 0 0
$$283$$ 0.496928 0.0295393 0.0147697 0.999891i $$-0.495298\pi$$
0.0147697 + 0.999891i $$0.495298\pi$$
$$284$$ 0 0
$$285$$ 1.70928 0.101249
$$286$$ 0 0
$$287$$ 3.51745 0.207628
$$288$$ 0 0
$$289$$ −6.36069 −0.374158
$$290$$ 0 0
$$291$$ −15.1896 −0.890428
$$292$$ 0 0
$$293$$ 6.63090 0.387381 0.193691 0.981063i $$-0.437954\pi$$
0.193691 + 0.981063i $$0.437954\pi$$
$$294$$ 0 0
$$295$$ 11.4186 0.664814
$$296$$ 0 0
$$297$$ −33.3607 −1.93578
$$298$$ 0 0
$$299$$ −3.20394 −0.185288
$$300$$ 0 0
$$301$$ −12.8494 −0.740626
$$302$$ 0 0
$$303$$ −8.41241 −0.483280
$$304$$ 0 0
$$305$$ 5.60197 0.320768
$$306$$ 0 0
$$307$$ −14.6042 −0.833508 −0.416754 0.909019i $$-0.636832\pi$$
−0.416754 + 0.909019i $$0.636832\pi$$
$$308$$ 0 0
$$309$$ −10.9216 −0.621309
$$310$$ 0 0
$$311$$ −19.3340 −1.09633 −0.548166 0.836369i $$-0.684674\pi$$
−0.548166 + 0.836369i $$0.684674\pi$$
$$312$$ 0 0
$$313$$ 30.6803 1.73416 0.867078 0.498173i $$-0.165995\pi$$
0.867078 + 0.498173i $$0.165995\pi$$
$$314$$ 0 0
$$315$$ 0.0845208 0.00476221
$$316$$ 0 0
$$317$$ 18.7298 1.05197 0.525985 0.850494i $$-0.323697\pi$$
0.525985 + 0.850494i $$0.323697\pi$$
$$318$$ 0 0
$$319$$ 8.99386 0.503559
$$320$$ 0 0
$$321$$ −3.91548 −0.218541
$$322$$ 0 0
$$323$$ 3.26180 0.181491
$$324$$ 0 0
$$325$$ 1.36910 0.0759441
$$326$$ 0 0
$$327$$ −21.9421 −1.21340
$$328$$ 0 0
$$329$$ 1.16290 0.0641127
$$330$$ 0 0
$$331$$ 2.73820 0.150505 0.0752527 0.997164i $$-0.476024\pi$$
0.0752527 + 0.997164i $$0.476024\pi$$
$$332$$ 0 0
$$333$$ −0.420818 −0.0230607
$$334$$ 0 0
$$335$$ −10.3896 −0.567646
$$336$$ 0 0
$$337$$ −6.04945 −0.329534 −0.164767 0.986332i $$-0.552687\pi$$
−0.164767 + 0.986332i $$0.552687\pi$$
$$338$$ 0 0
$$339$$ −22.0267 −1.19632
$$340$$ 0 0
$$341$$ −55.0349 −2.98031
$$342$$ 0 0
$$343$$ 13.8432 0.747465
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ 5.97334 0.320666 0.160333 0.987063i $$-0.448743\pi$$
0.160333 + 0.987063i $$0.448743\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −7.20394 −0.384518
$$352$$ 0 0
$$353$$ −14.0989 −0.750409 −0.375204 0.926942i $$-0.622427\pi$$
−0.375204 + 0.926942i $$0.622427\pi$$
$$354$$ 0 0
$$355$$ 10.8371 0.575174
$$356$$ 0 0
$$357$$ −6.01229 −0.318204
$$358$$ 0 0
$$359$$ −6.02666 −0.318075 −0.159038 0.987273i $$-0.550839\pi$$
−0.159038 + 0.987273i $$0.550839\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 49.9071 2.61944
$$364$$ 0 0
$$365$$ 5.41855 0.283620
$$366$$ 0 0
$$367$$ −1.07838 −0.0562909 −0.0281454 0.999604i $$-0.508960\pi$$
−0.0281454 + 0.999604i $$0.508960\pi$$
$$368$$ 0 0
$$369$$ 0.255652 0.0133087
$$370$$ 0 0
$$371$$ −7.15061 −0.371241
$$372$$ 0 0
$$373$$ 12.3051 0.637134 0.318567 0.947900i $$-0.396798\pi$$
0.318567 + 0.947900i $$0.396798\pi$$
$$374$$ 0 0
$$375$$ 1.70928 0.0882666
$$376$$ 0 0
$$377$$ 1.94214 0.100025
$$378$$ 0 0
$$379$$ −1.04718 −0.0537901 −0.0268950 0.999638i $$-0.508562\pi$$
−0.0268950 + 0.999638i $$0.508562\pi$$
$$380$$ 0 0
$$381$$ 14.0722 0.720942
$$382$$ 0 0
$$383$$ −0.0806452 −0.00412078 −0.00206039 0.999998i $$-0.500656\pi$$
−0.00206039 + 0.999998i $$0.500656\pi$$
$$384$$ 0 0
$$385$$ −6.83710 −0.348451
$$386$$ 0 0
$$387$$ −0.933908 −0.0474732
$$388$$ 0 0
$$389$$ −20.5236 −1.04059 −0.520294 0.853987i $$-0.674178\pi$$
−0.520294 + 0.853987i $$0.674178\pi$$
$$390$$ 0 0
$$391$$ −7.63317 −0.386026
$$392$$ 0 0
$$393$$ −2.52359 −0.127298
$$394$$ 0 0
$$395$$ −14.2557 −0.717280
$$396$$ 0 0
$$397$$ 39.4596 1.98042 0.990210 0.139586i $$-0.0445771\pi$$
0.990210 + 0.139586i $$0.0445771\pi$$
$$398$$ 0 0
$$399$$ −1.84324 −0.0922776
$$400$$ 0 0
$$401$$ 0.470266 0.0234840 0.0117420 0.999931i $$-0.496262\pi$$
0.0117420 + 0.999931i $$0.496262\pi$$
$$402$$ 0 0
$$403$$ −11.8843 −0.591998
$$404$$ 0 0
$$405$$ −8.75872 −0.435224
$$406$$ 0 0
$$407$$ 34.0410 1.68735
$$408$$ 0 0
$$409$$ 10.4826 0.518329 0.259164 0.965833i $$-0.416553\pi$$
0.259164 + 0.965833i $$0.416553\pi$$
$$410$$ 0 0
$$411$$ −6.73820 −0.332371
$$412$$ 0 0
$$413$$ −12.3135 −0.605908
$$414$$ 0 0
$$415$$ 14.3402 0.703931
$$416$$ 0 0
$$417$$ −15.1506 −0.741929
$$418$$ 0 0
$$419$$ −34.6681 −1.69365 −0.846823 0.531875i $$-0.821488\pi$$
−0.846823 + 0.531875i $$0.821488\pi$$
$$420$$ 0 0
$$421$$ −34.6102 −1.68680 −0.843399 0.537288i $$-0.819449\pi$$
−0.843399 + 0.537288i $$0.819449\pi$$
$$422$$ 0 0
$$423$$ 0.0845208 0.00410954
$$424$$ 0 0
$$425$$ 3.26180 0.158220
$$426$$ 0 0
$$427$$ −6.04104 −0.292346
$$428$$ 0 0
$$429$$ 14.8371 0.716342
$$430$$ 0 0
$$431$$ −6.73820 −0.324568 −0.162284 0.986744i $$-0.551886\pi$$
−0.162284 + 0.986744i $$0.551886\pi$$
$$432$$ 0 0
$$433$$ 20.4741 0.983924 0.491962 0.870617i $$-0.336280\pi$$
0.491962 + 0.870617i $$0.336280\pi$$
$$434$$ 0 0
$$435$$ 2.42469 0.116255
$$436$$ 0 0
$$437$$ −2.34017 −0.111946
$$438$$ 0 0
$$439$$ 21.4596 1.02421 0.512105 0.858923i $$-0.328866\pi$$
0.512105 + 0.858923i $$0.328866\pi$$
$$440$$ 0 0
$$441$$ 0.457499 0.0217857
$$442$$ 0 0
$$443$$ 21.5441 1.02359 0.511796 0.859107i $$-0.328980\pi$$
0.511796 + 0.859107i $$0.328980\pi$$
$$444$$ 0 0
$$445$$ 7.57531 0.359104
$$446$$ 0 0
$$447$$ −33.7731 −1.59741
$$448$$ 0 0
$$449$$ −8.47027 −0.399737 −0.199868 0.979823i $$-0.564051\pi$$
−0.199868 + 0.979823i $$0.564051\pi$$
$$450$$ 0 0
$$451$$ −20.6803 −0.973799
$$452$$ 0 0
$$453$$ −5.84324 −0.274540
$$454$$ 0 0
$$455$$ −1.47641 −0.0692151
$$456$$ 0 0
$$457$$ 11.3607 0.531431 0.265715 0.964052i $$-0.414392\pi$$
0.265715 + 0.964052i $$0.414392\pi$$
$$458$$ 0 0
$$459$$ −17.1629 −0.801096
$$460$$ 0 0
$$461$$ 3.04718 0.141921 0.0709607 0.997479i $$-0.477394\pi$$
0.0709607 + 0.997479i $$0.477394\pi$$
$$462$$ 0 0
$$463$$ 9.97334 0.463500 0.231750 0.972775i $$-0.425555\pi$$
0.231750 + 0.972775i $$0.425555\pi$$
$$464$$ 0 0
$$465$$ −14.8371 −0.688054
$$466$$ 0 0
$$467$$ −1.49079 −0.0689853 −0.0344927 0.999405i $$-0.510982\pi$$
−0.0344927 + 0.999405i $$0.510982\pi$$
$$468$$ 0 0
$$469$$ 11.2039 0.517350
$$470$$ 0 0
$$471$$ 16.0989 0.741798
$$472$$ 0 0
$$473$$ 75.5462 3.47362
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −0.519715 −0.0237961
$$478$$ 0 0
$$479$$ 10.1711 0.464731 0.232365 0.972629i $$-0.425353\pi$$
0.232365 + 0.972629i $$0.425353\pi$$
$$480$$ 0 0
$$481$$ 7.35085 0.335170
$$482$$ 0 0
$$483$$ 4.31351 0.196272
$$484$$ 0 0
$$485$$ −8.88655 −0.403517
$$486$$ 0 0
$$487$$ 24.2784 1.10016 0.550081 0.835112i $$-0.314597\pi$$
0.550081 + 0.835112i $$0.314597\pi$$
$$488$$ 0 0
$$489$$ 4.99386 0.225830
$$490$$ 0 0
$$491$$ −19.2039 −0.866662 −0.433331 0.901235i $$-0.642662\pi$$
−0.433331 + 0.901235i $$0.642662\pi$$
$$492$$ 0 0
$$493$$ 4.62702 0.208391
$$494$$ 0 0
$$495$$ −0.496928 −0.0223353
$$496$$ 0 0
$$497$$ −11.6865 −0.524211
$$498$$ 0 0
$$499$$ −7.33403 −0.328316 −0.164158 0.986434i $$-0.552491\pi$$
−0.164158 + 0.986434i $$0.552491\pi$$
$$500$$ 0 0
$$501$$ −35.7464 −1.59703
$$502$$ 0 0
$$503$$ −29.0616 −1.29579 −0.647895 0.761729i $$-0.724351\pi$$
−0.647895 + 0.761729i $$0.724351\pi$$
$$504$$ 0 0
$$505$$ −4.92162 −0.219009
$$506$$ 0 0
$$507$$ −19.0166 −0.844559
$$508$$ 0 0
$$509$$ 26.0456 1.15445 0.577225 0.816585i $$-0.304136\pi$$
0.577225 + 0.816585i $$0.304136\pi$$
$$510$$ 0 0
$$511$$ −5.84324 −0.258490
$$512$$ 0 0
$$513$$ −5.26180 −0.232314
$$514$$ 0 0
$$515$$ −6.38962 −0.281560
$$516$$ 0 0
$$517$$ −6.83710 −0.300695
$$518$$ 0 0
$$519$$ −1.80430 −0.0791998
$$520$$ 0 0
$$521$$ −38.8248 −1.70095 −0.850473 0.526019i $$-0.823684\pi$$
−0.850473 + 0.526019i $$0.823684\pi$$
$$522$$ 0 0
$$523$$ −4.59970 −0.201131 −0.100565 0.994930i $$-0.532065\pi$$
−0.100565 + 0.994930i $$0.532065\pi$$
$$524$$ 0 0
$$525$$ −1.84324 −0.0804458
$$526$$ 0 0
$$527$$ −28.3135 −1.23336
$$528$$ 0 0
$$529$$ −17.5236 −0.761895
$$530$$ 0 0
$$531$$ −0.894960 −0.0388380
$$532$$ 0 0
$$533$$ −4.46573 −0.193432
$$534$$ 0 0
$$535$$ −2.29072 −0.0990367
$$536$$ 0 0
$$537$$ −1.52973 −0.0660129
$$538$$ 0 0
$$539$$ −37.0082 −1.59406
$$540$$ 0 0
$$541$$ −12.1256 −0.521318 −0.260659 0.965431i $$-0.583940\pi$$
−0.260659 + 0.965431i $$0.583940\pi$$
$$542$$ 0 0
$$543$$ −1.43084 −0.0614031
$$544$$ 0 0
$$545$$ −12.8371 −0.549881
$$546$$ 0 0
$$547$$ 9.54023 0.407911 0.203955 0.978980i $$-0.434620\pi$$
0.203955 + 0.978980i $$0.434620\pi$$
$$548$$ 0 0
$$549$$ −0.439070 −0.0187390
$$550$$ 0 0
$$551$$ 1.41855 0.0604323
$$552$$ 0 0
$$553$$ 15.3730 0.653726
$$554$$ 0 0
$$555$$ 9.17727 0.389554
$$556$$ 0 0
$$557$$ 19.9421 0.844976 0.422488 0.906369i $$-0.361157\pi$$
0.422488 + 0.906369i $$0.361157\pi$$
$$558$$ 0 0
$$559$$ 16.3135 0.689988
$$560$$ 0 0
$$561$$ 35.3484 1.49241
$$562$$ 0 0
$$563$$ −23.5525 −0.992620 −0.496310 0.868145i $$-0.665312\pi$$
−0.496310 + 0.868145i $$0.665312\pi$$
$$564$$ 0 0
$$565$$ −12.8865 −0.542141
$$566$$ 0 0
$$567$$ 9.44521 0.396662
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 23.5031 0.983573 0.491786 0.870716i $$-0.336344\pi$$
0.491786 + 0.870716i $$0.336344\pi$$
$$572$$ 0 0
$$573$$ −37.6742 −1.57386
$$574$$ 0 0
$$575$$ −2.34017 −0.0975920
$$576$$ 0 0
$$577$$ 16.4703 0.685666 0.342833 0.939396i $$-0.388613\pi$$
0.342833 + 0.939396i $$0.388613\pi$$
$$578$$ 0 0
$$579$$ 21.8576 0.908372
$$580$$ 0 0
$$581$$ −15.4641 −0.641560
$$582$$ 0 0
$$583$$ 42.0410 1.74116
$$584$$ 0 0
$$585$$ −0.107307 −0.00443660
$$586$$ 0 0
$$587$$ 28.8104 1.18913 0.594567 0.804046i $$-0.297323\pi$$
0.594567 + 0.804046i $$0.297323\pi$$
$$588$$ 0 0
$$589$$ −8.68035 −0.357667
$$590$$ 0 0
$$591$$ −15.7321 −0.647131
$$592$$ 0 0
$$593$$ −43.2450 −1.77586 −0.887929 0.459980i $$-0.847856\pi$$
−0.887929 + 0.459980i $$0.847856\pi$$
$$594$$ 0 0
$$595$$ −3.51745 −0.144201
$$596$$ 0 0
$$597$$ 27.6865 1.13313
$$598$$ 0 0
$$599$$ 44.2967 1.80991 0.904957 0.425503i $$-0.139903\pi$$
0.904957 + 0.425503i $$0.139903\pi$$
$$600$$ 0 0
$$601$$ −24.3090 −0.991584 −0.495792 0.868441i $$-0.665122\pi$$
−0.495792 + 0.868441i $$0.665122\pi$$
$$602$$ 0 0
$$603$$ 0.814315 0.0331615
$$604$$ 0 0
$$605$$ 29.1978 1.18706
$$606$$ 0 0
$$607$$ 6.29072 0.255333 0.127666 0.991817i $$-0.459251\pi$$
0.127666 + 0.991817i $$0.459251\pi$$
$$608$$ 0 0
$$609$$ −2.61474 −0.105954
$$610$$ 0 0
$$611$$ −1.47641 −0.0597291
$$612$$ 0 0
$$613$$ −12.7915 −0.516645 −0.258322 0.966059i $$-0.583170\pi$$
−0.258322 + 0.966059i $$0.583170\pi$$
$$614$$ 0 0
$$615$$ −5.57531 −0.224818
$$616$$ 0 0
$$617$$ 17.9299 0.721829 0.360914 0.932599i $$-0.382465\pi$$
0.360914 + 0.932599i $$0.382465\pi$$
$$618$$ 0 0
$$619$$ −26.8515 −1.07925 −0.539626 0.841905i $$-0.681434\pi$$
−0.539626 + 0.841905i $$0.681434\pi$$
$$620$$ 0 0
$$621$$ 12.3135 0.494124
$$622$$ 0 0
$$623$$ −8.16904 −0.327286
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 10.8371 0.432792
$$628$$ 0 0
$$629$$ 17.5129 0.698286
$$630$$ 0 0
$$631$$ −35.5318 −1.41450 −0.707250 0.706964i $$-0.750064\pi$$
−0.707250 + 0.706964i $$0.750064\pi$$
$$632$$ 0 0
$$633$$ −13.3074 −0.528920
$$634$$ 0 0
$$635$$ 8.23287 0.326711
$$636$$ 0 0
$$637$$ −7.99159 −0.316638
$$638$$ 0 0
$$639$$ −0.849388 −0.0336013
$$640$$ 0 0
$$641$$ 12.9360 0.510941 0.255471 0.966817i $$-0.417770\pi$$
0.255471 + 0.966817i $$0.417770\pi$$
$$642$$ 0 0
$$643$$ −8.49693 −0.335086 −0.167543 0.985865i $$-0.553583\pi$$
−0.167543 + 0.985865i $$0.553583\pi$$
$$644$$ 0 0
$$645$$ 20.3668 0.801943
$$646$$ 0 0
$$647$$ 45.4908 1.78843 0.894214 0.447640i $$-0.147736\pi$$
0.894214 + 0.447640i $$0.147736\pi$$
$$648$$ 0 0
$$649$$ 72.3956 2.84178
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 0 0
$$653$$ 35.9421 1.40652 0.703262 0.710930i $$-0.251726\pi$$
0.703262 + 0.710930i $$0.251726\pi$$
$$654$$ 0 0
$$655$$ −1.47641 −0.0576881
$$656$$ 0 0
$$657$$ −0.424694 −0.0165689
$$658$$ 0 0
$$659$$ −30.9360 −1.20510 −0.602548 0.798083i $$-0.705848\pi$$
−0.602548 + 0.798083i $$0.705848\pi$$
$$660$$ 0 0
$$661$$ 10.0989 0.392802 0.196401 0.980524i $$-0.437075\pi$$
0.196401 + 0.980524i $$0.437075\pi$$
$$662$$ 0 0
$$663$$ 7.63317 0.296448
$$664$$ 0 0
$$665$$ −1.07838 −0.0418177
$$666$$ 0 0
$$667$$ −3.31965 −0.128538
$$668$$ 0 0
$$669$$ −21.4452 −0.829120
$$670$$ 0 0
$$671$$ 35.5174 1.37114
$$672$$ 0 0
$$673$$ −8.67194 −0.334279 −0.167139 0.985933i $$-0.553453\pi$$
−0.167139 + 0.985933i $$0.553453\pi$$
$$674$$ 0 0
$$675$$ −5.26180 −0.202527
$$676$$ 0 0
$$677$$ 41.9793 1.61340 0.806698 0.590964i $$-0.201253\pi$$
0.806698 + 0.590964i $$0.201253\pi$$
$$678$$ 0 0
$$679$$ 9.58306 0.367764
$$680$$ 0 0
$$681$$ −3.91548 −0.150041
$$682$$ 0 0
$$683$$ −46.3896 −1.77505 −0.887525 0.460760i $$-0.847577\pi$$
−0.887525 + 0.460760i $$0.847577\pi$$
$$684$$ 0 0
$$685$$ −3.94214 −0.150621
$$686$$ 0 0
$$687$$ −10.1112 −0.385766
$$688$$ 0 0
$$689$$ 9.07838 0.345859
$$690$$ 0 0
$$691$$ 34.8515 1.32581 0.662906 0.748702i $$-0.269323\pi$$
0.662906 + 0.748702i $$0.269323\pi$$
$$692$$ 0 0
$$693$$ 0.535877 0.0203563
$$694$$ 0 0
$$695$$ −8.86376 −0.336222
$$696$$ 0 0
$$697$$ −10.6393 −0.402993
$$698$$ 0 0
$$699$$ 23.1050 0.873913
$$700$$ 0 0
$$701$$ 35.6430 1.34622 0.673109 0.739543i $$-0.264959\pi$$
0.673109 + 0.739543i $$0.264959\pi$$
$$702$$ 0 0
$$703$$ 5.36910 0.202500
$$704$$ 0 0
$$705$$ −1.84324 −0.0694206
$$706$$ 0 0
$$707$$ 5.30737 0.199604
$$708$$ 0 0
$$709$$ 16.7214 0.627985 0.313992 0.949426i $$-0.398333\pi$$
0.313992 + 0.949426i $$0.398333\pi$$
$$710$$ 0 0
$$711$$ 1.11733 0.0419030
$$712$$ 0 0
$$713$$ 20.3135 0.760747
$$714$$ 0 0
$$715$$ 8.68035 0.324627
$$716$$ 0 0
$$717$$ −23.6619 −0.883670
$$718$$ 0 0
$$719$$ 6.85148 0.255517 0.127758 0.991805i $$-0.459222\pi$$
0.127758 + 0.991805i $$0.459222\pi$$
$$720$$ 0 0
$$721$$ 6.89043 0.256613
$$722$$ 0 0
$$723$$ 12.4124 0.461622
$$724$$ 0 0
$$725$$ 1.41855 0.0526837
$$726$$ 0 0
$$727$$ −34.4391 −1.27727 −0.638637 0.769508i $$-0.720502\pi$$
−0.638637 + 0.769508i $$0.720502\pi$$
$$728$$ 0 0
$$729$$ 27.6681 1.02474
$$730$$ 0 0
$$731$$ 38.8659 1.43751
$$732$$ 0 0
$$733$$ −19.3607 −0.715103 −0.357552 0.933893i $$-0.616388\pi$$
−0.357552 + 0.933893i $$0.616388\pi$$
$$734$$ 0 0
$$735$$ −9.97721 −0.368015
$$736$$ 0 0
$$737$$ −65.8720 −2.42643
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 2.34017 0.0859684
$$742$$ 0 0
$$743$$ 12.7154 0.466483 0.233242 0.972419i $$-0.425067\pi$$
0.233242 + 0.972419i $$0.425067\pi$$
$$744$$ 0 0
$$745$$ −19.7587 −0.723904
$$746$$ 0 0
$$747$$ −1.12395 −0.0411232
$$748$$ 0 0
$$749$$ 2.47027 0.0902616
$$750$$ 0 0
$$751$$ −3.15836 −0.115250 −0.0576252 0.998338i $$-0.518353\pi$$
−0.0576252 + 0.998338i $$0.518353\pi$$
$$752$$ 0 0
$$753$$ −17.8966 −0.652187
$$754$$ 0 0
$$755$$ −3.41855 −0.124414
$$756$$ 0 0
$$757$$ 28.0410 1.01917 0.509584 0.860421i $$-0.329799\pi$$
0.509584 + 0.860421i $$0.329799\pi$$
$$758$$ 0 0
$$759$$ −25.3607 −0.920535
$$760$$ 0 0
$$761$$ −16.4924 −0.597849 −0.298924 0.954277i $$-0.596628\pi$$
−0.298924 + 0.954277i $$0.596628\pi$$
$$762$$ 0 0
$$763$$ 13.8432 0.501159
$$764$$ 0 0
$$765$$ −0.255652 −0.00924312
$$766$$ 0 0
$$767$$ 15.6332 0.564481
$$768$$ 0 0
$$769$$ 26.9627 0.972298 0.486149 0.873876i $$-0.338401\pi$$
0.486149 + 0.873876i $$0.338401\pi$$
$$770$$ 0 0
$$771$$ −40.3812 −1.45429
$$772$$ 0 0
$$773$$ 42.1939 1.51761 0.758805 0.651318i $$-0.225784\pi$$
0.758805 + 0.651318i $$0.225784\pi$$
$$774$$ 0 0
$$775$$ −8.68035 −0.311807
$$776$$ 0 0
$$777$$ −9.89657 −0.355037
$$778$$ 0 0
$$779$$ −3.26180 −0.116866
$$780$$ 0 0
$$781$$ 68.7091 2.45860
$$782$$ 0 0
$$783$$ −7.46412 −0.266746
$$784$$ 0 0
$$785$$ 9.41855 0.336162
$$786$$ 0 0
$$787$$ 35.4368 1.26319 0.631593 0.775300i $$-0.282401\pi$$
0.631593 + 0.775300i $$0.282401\pi$$
$$788$$ 0 0
$$789$$ −9.67420 −0.344411
$$790$$ 0 0
$$791$$ 13.8966 0.494105
$$792$$ 0 0
$$793$$ 7.66967 0.272358
$$794$$ 0 0
$$795$$ 11.3340 0.401977
$$796$$ 0 0
$$797$$ 15.2579 0.540463 0.270232 0.962795i $$-0.412900\pi$$
0.270232 + 0.962795i $$0.412900\pi$$
$$798$$ 0 0
$$799$$ −3.51745 −0.124438
$$800$$ 0 0
$$801$$ −0.593735 −0.0209786
$$802$$ 0 0
$$803$$ 34.3545 1.21235
$$804$$ 0 0
$$805$$ 2.52359 0.0889449
$$806$$ 0 0
$$807$$ −3.95443 −0.139202
$$808$$ 0 0
$$809$$ −7.16290 −0.251834 −0.125917 0.992041i $$-0.540187\pi$$
−0.125917 + 0.992041i $$0.540187\pi$$
$$810$$ 0 0
$$811$$ 50.3545 1.76819 0.884094 0.467310i $$-0.154777\pi$$
0.884094 + 0.467310i $$0.154777\pi$$
$$812$$ 0 0
$$813$$ 33.6742 1.18101
$$814$$ 0 0
$$815$$ 2.92162 0.102340
$$816$$ 0 0
$$817$$ 11.9155 0.416870
$$818$$ 0 0
$$819$$ 0.115718 0.00404350
$$820$$ 0 0
$$821$$ 0.952819 0.0332536 0.0166268 0.999862i $$-0.494707\pi$$
0.0166268 + 0.999862i $$0.494707\pi$$
$$822$$ 0 0
$$823$$ 44.2290 1.54173 0.770863 0.637001i $$-0.219825\pi$$
0.770863 + 0.637001i $$0.219825\pi$$
$$824$$ 0 0
$$825$$ 10.8371 0.377299
$$826$$ 0 0
$$827$$ 37.8615 1.31657 0.658287 0.752767i $$-0.271281\pi$$
0.658287 + 0.752767i $$0.271281\pi$$
$$828$$ 0 0
$$829$$ −56.7214 −1.97002 −0.985008 0.172511i $$-0.944812\pi$$
−0.985008 + 0.172511i $$0.944812\pi$$
$$830$$ 0 0
$$831$$ −43.9832 −1.52576
$$832$$ 0 0
$$833$$ −19.0394 −0.659677
$$834$$ 0 0
$$835$$ −20.9132 −0.723732
$$836$$ 0 0
$$837$$ 45.6742 1.57873
$$838$$ 0 0
$$839$$ 28.3591 0.979064 0.489532 0.871985i $$-0.337168\pi$$
0.489532 + 0.871985i $$0.337168\pi$$
$$840$$ 0 0
$$841$$ −26.9877 −0.930611
$$842$$ 0 0
$$843$$ −11.2495 −0.387454
$$844$$ 0 0
$$845$$ −11.1256 −0.382731
$$846$$ 0 0
$$847$$ −31.4863 −1.08188
$$848$$ 0 0
$$849$$ 0.849388 0.0291509
$$850$$ 0 0
$$851$$ −12.5646 −0.430710
$$852$$ 0 0
$$853$$ −17.0061 −0.582279 −0.291140 0.956681i $$-0.594034\pi$$
−0.291140 + 0.956681i $$0.594034\pi$$
$$854$$ 0 0
$$855$$ −0.0783777 −0.00268046
$$856$$ 0 0
$$857$$ 15.4101 0.526400 0.263200 0.964741i $$-0.415222\pi$$
0.263200 + 0.964741i $$0.415222\pi$$
$$858$$ 0 0
$$859$$ 37.7275 1.28725 0.643623 0.765342i $$-0.277430\pi$$
0.643623 + 0.765342i $$0.277430\pi$$
$$860$$ 0 0
$$861$$ 6.01229 0.204898
$$862$$ 0 0
$$863$$ −32.1340 −1.09385 −0.546927 0.837181i $$-0.684202\pi$$
−0.546927 + 0.837181i $$0.684202\pi$$
$$864$$ 0 0
$$865$$ −1.05559 −0.0358912
$$866$$ 0 0
$$867$$ −10.8722 −0.369238
$$868$$ 0 0
$$869$$ −90.3833 −3.06604
$$870$$ 0 0
$$871$$ −14.2245 −0.481977
$$872$$ 0 0
$$873$$ 0.696508 0.0235732
$$874$$ 0 0
$$875$$ −1.07838 −0.0364558
$$876$$ 0 0
$$877$$ −19.8927 −0.671729 −0.335864 0.941910i $$-0.609028\pi$$
−0.335864 + 0.941910i $$0.609028\pi$$
$$878$$ 0 0
$$879$$ 11.3340 0.382287
$$880$$ 0 0
$$881$$ 24.0722 0.811014 0.405507 0.914092i $$-0.367095\pi$$
0.405507 + 0.914092i $$0.367095\pi$$
$$882$$ 0 0
$$883$$ 31.1727 1.04905 0.524523 0.851396i $$-0.324244\pi$$
0.524523 + 0.851396i $$0.324244\pi$$
$$884$$ 0 0
$$885$$ 19.5174 0.656072
$$886$$ 0 0
$$887$$ −4.86764 −0.163439 −0.0817197 0.996655i $$-0.526041\pi$$
−0.0817197 + 0.996655i $$0.526041\pi$$
$$888$$ 0 0
$$889$$ −8.87814 −0.297763
$$890$$ 0 0
$$891$$ −55.5318 −1.86039
$$892$$ 0 0
$$893$$ −1.07838 −0.0360865
$$894$$ 0 0
$$895$$ −0.894960 −0.0299152
$$896$$ 0 0
$$897$$ −5.47641 −0.182852
$$898$$ 0 0
$$899$$ −12.3135 −0.410679
$$900$$ 0 0
$$901$$ 21.6286 0.720554
$$902$$ 0 0
$$903$$ −21.9631 −0.730888
$$904$$ 0 0
$$905$$ −0.837101 −0.0278262
$$906$$ 0 0
$$907$$ 45.4778 1.51007 0.755033 0.655686i $$-0.227621\pi$$
0.755033 + 0.655686i $$0.227621\pi$$
$$908$$ 0 0
$$909$$ 0.385746 0.0127944
$$910$$ 0 0
$$911$$ 20.9483 0.694048 0.347024 0.937856i $$-0.387192\pi$$
0.347024 + 0.937856i $$0.387192\pi$$
$$912$$ 0 0
$$913$$ 90.9192 3.00899
$$914$$ 0 0
$$915$$ 9.57531 0.316550
$$916$$ 0 0
$$917$$ 1.59213 0.0525767
$$918$$ 0 0
$$919$$ 59.5174 1.96330 0.981650 0.190693i $$-0.0610735\pi$$
0.981650 + 0.190693i $$0.0610735\pi$$
$$920$$ 0 0
$$921$$ −24.9627 −0.822548
$$922$$ 0 0
$$923$$ 14.8371 0.488369
$$924$$ 0 0
$$925$$ 5.36910 0.176535
$$926$$ 0 0
$$927$$ 0.500804 0.0164486
$$928$$ 0 0
$$929$$ −16.7214 −0.548611 −0.274305 0.961643i $$-0.588448\pi$$
−0.274305 + 0.961643i $$0.588448\pi$$
$$930$$ 0 0
$$931$$ −5.83710 −0.191303
$$932$$ 0 0
$$933$$ −33.0472 −1.08192
$$934$$ 0 0
$$935$$ 20.6803 0.676320
$$936$$ 0 0
$$937$$ −32.4534 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$938$$ 0 0
$$939$$ 52.4412 1.71135
$$940$$ 0 0
$$941$$ −23.6742 −0.771757 −0.385878 0.922550i $$-0.626102\pi$$
−0.385878 + 0.922550i $$0.626102\pi$$
$$942$$ 0 0
$$943$$ 7.63317 0.248570
$$944$$ 0 0
$$945$$ 5.67420 0.184582
$$946$$ 0 0
$$947$$ −21.9733 −0.714038 −0.357019 0.934097i $$-0.616207\pi$$
−0.357019 + 0.934097i $$0.616207\pi$$
$$948$$ 0 0
$$949$$ 7.41855 0.240816
$$950$$ 0 0
$$951$$ 32.0144 1.03814
$$952$$ 0 0
$$953$$ −53.2990 −1.72652 −0.863261 0.504757i $$-0.831582\pi$$
−0.863261 + 0.504757i $$0.831582\pi$$
$$954$$ 0 0
$$955$$ −22.0410 −0.713231
$$956$$ 0 0
$$957$$ 15.3730 0.496938
$$958$$ 0 0
$$959$$ 4.25112 0.137276
$$960$$ 0 0
$$961$$ 44.3484 1.43059
$$962$$ 0 0
$$963$$ 0.179542 0.00578565
$$964$$ 0 0
$$965$$ 12.7877 0.411649
$$966$$ 0 0
$$967$$ −15.8166 −0.508627 −0.254314 0.967122i $$-0.581850\pi$$
−0.254314 + 0.967122i $$0.581850\pi$$
$$968$$ 0 0
$$969$$ 5.57531 0.179105
$$970$$ 0 0
$$971$$ 43.1506 1.38477 0.692385 0.721529i $$-0.256560\pi$$
0.692385 + 0.721529i $$0.256560\pi$$
$$972$$ 0 0
$$973$$ 9.55849 0.306431
$$974$$ 0 0
$$975$$ 2.34017 0.0749455
$$976$$ 0 0
$$977$$ −8.47414 −0.271112 −0.135556 0.990770i $$-0.543282\pi$$
−0.135556 + 0.990770i $$0.543282\pi$$
$$978$$ 0 0
$$979$$ 48.0288 1.53501
$$980$$ 0 0
$$981$$ 1.00614 0.0321237
$$982$$ 0 0
$$983$$ 5.59356 0.178407 0.0892034 0.996013i $$-0.471568\pi$$
0.0892034 + 0.996013i $$0.471568\pi$$
$$984$$ 0 0
$$985$$ −9.20394 −0.293262
$$986$$ 0 0
$$987$$ 1.98771 0.0632696
$$988$$ 0 0
$$989$$ −27.8843 −0.886669
$$990$$ 0 0
$$991$$ −32.8950 −1.04494 −0.522471 0.852657i $$-0.674990\pi$$
−0.522471 + 0.852657i $$0.674990\pi$$
$$992$$ 0 0
$$993$$ 4.68035 0.148526
$$994$$ 0 0
$$995$$ 16.1978 0.513505
$$996$$ 0 0
$$997$$ 23.2618 0.736708 0.368354 0.929686i $$-0.379921\pi$$
0.368354 + 0.929686i $$0.379921\pi$$
$$998$$ 0 0
$$999$$ −28.2511 −0.893826
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 1520.2.a.p.1.3 3
4.3 odd 2 95.2.a.a.1.3 3
5.4 even 2 7600.2.a.bx.1.1 3
8.3 odd 2 6080.2.a.bo.1.3 3
8.5 even 2 6080.2.a.by.1.1 3
12.11 even 2 855.2.a.i.1.1 3
20.3 even 4 475.2.b.d.324.1 6
20.7 even 4 475.2.b.d.324.6 6
20.19 odd 2 475.2.a.f.1.1 3
28.27 even 2 4655.2.a.u.1.3 3
60.59 even 2 4275.2.a.bk.1.3 3
76.75 even 2 1805.2.a.f.1.1 3
380.379 even 2 9025.2.a.bb.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.a.1.3 3 4.3 odd 2
475.2.a.f.1.1 3 20.19 odd 2
475.2.b.d.324.1 6 20.3 even 4
475.2.b.d.324.6 6 20.7 even 4
855.2.a.i.1.1 3 12.11 even 2
1520.2.a.p.1.3 3 1.1 even 1 trivial
1805.2.a.f.1.1 3 76.75 even 2
4275.2.a.bk.1.3 3 60.59 even 2
4655.2.a.u.1.3 3 28.27 even 2
6080.2.a.bo.1.3 3 8.3 odd 2
6080.2.a.by.1.1 3 8.5 even 2
7600.2.a.bx.1.1 3 5.4 even 2
9025.2.a.bb.1.3 3 380.379 even 2