# Properties

 Label 4275.2.a.bk.1.3 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ 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] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\chi$$ $$=$$ 4275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.17009 q^{2} +2.70928 q^{4} -1.07838 q^{7} +1.53919 q^{8} +O(q^{10})$$ $$q+2.17009 q^{2} +2.70928 q^{4} -1.07838 q^{7} +1.53919 q^{8} +6.34017 q^{11} -1.36910 q^{13} -2.34017 q^{14} -2.07838 q^{16} +3.26180 q^{17} -1.00000 q^{19} +13.7587 q^{22} +2.34017 q^{23} -2.97107 q^{26} -2.92162 q^{28} -1.41855 q^{29} +8.68035 q^{31} -7.58864 q^{32} +7.07838 q^{34} -5.36910 q^{37} -2.17009 q^{38} +3.26180 q^{41} +11.9155 q^{43} +17.1773 q^{44} +5.07838 q^{46} +1.07838 q^{47} -5.83710 q^{49} -3.70928 q^{52} +6.63090 q^{53} -1.65983 q^{56} -3.07838 q^{58} +11.4186 q^{59} +5.60197 q^{61} +18.8371 q^{62} -12.3112 q^{64} -10.3896 q^{67} +8.83710 q^{68} +10.8371 q^{71} -5.41855 q^{73} -11.6514 q^{74} -2.70928 q^{76} -6.83710 q^{77} +14.2557 q^{79} +7.07838 q^{82} -14.3402 q^{83} +25.8576 q^{86} +9.75872 q^{88} -7.57531 q^{89} +1.47641 q^{91} +6.34017 q^{92} +2.34017 q^{94} +8.88655 q^{97} -12.6670 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + q^{4} + 3 q^{8}+O(q^{10})$$ 3 * q + q^2 + q^4 + 3 * q^8 $$3 q + q^{2} + q^{4} + 3 q^{8} + 8 q^{11} - 8 q^{13} + 4 q^{14} - 3 q^{16} + 2 q^{17} - 3 q^{19} + 16 q^{22} - 4 q^{23} + 6 q^{26} - 12 q^{28} + 10 q^{29} + 4 q^{31} - 3 q^{32} + 18 q^{34} - 20 q^{37} - q^{38} + 2 q^{41} + 4 q^{43} + 12 q^{44} + 12 q^{46} + 11 q^{49} - 4 q^{52} + 16 q^{53} - 16 q^{56} - 6 q^{58} + 20 q^{59} - 2 q^{61} + 28 q^{62} - 11 q^{64} - 2 q^{67} - 2 q^{68} + 4 q^{71} - 2 q^{73} + 2 q^{74} - q^{76} + 8 q^{77} + 18 q^{82} - 32 q^{83} + 16 q^{86} + 4 q^{88} - 2 q^{89} + 20 q^{91} + 8 q^{92} - 4 q^{94} - 20 q^{97} - 15 q^{98}+O(q^{100})$$ 3 * q + q^2 + q^4 + 3 * q^8 + 8 * q^11 - 8 * q^13 + 4 * q^14 - 3 * q^16 + 2 * q^17 - 3 * q^19 + 16 * q^22 - 4 * q^23 + 6 * q^26 - 12 * q^28 + 10 * q^29 + 4 * q^31 - 3 * q^32 + 18 * q^34 - 20 * q^37 - q^38 + 2 * q^41 + 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - 4 * q^52 + 16 * q^53 - 16 * q^56 - 6 * q^58 + 20 * q^59 - 2 * q^61 + 28 * q^62 - 11 * q^64 - 2 * q^67 - 2 * q^68 + 4 * q^71 - 2 * q^73 + 2 * q^74 - q^76 + 8 * q^77 + 18 * q^82 - 32 * q^83 + 16 * q^86 + 4 * q^88 - 2 * q^89 + 20 * q^91 + 8 * q^92 - 4 * q^94 - 20 * q^97 - 15 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.17009 1.53448 0.767241 0.641358i $$-0.221629\pi$$
0.767241 + 0.641358i $$0.221629\pi$$
$$3$$ 0 0
$$4$$ 2.70928 1.35464
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.07838 −0.407588 −0.203794 0.979014i $$-0.565327\pi$$
−0.203794 + 0.979014i $$0.565327\pi$$
$$8$$ 1.53919 0.544185
$$9$$ 0 0
$$10$$ 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 $$-0.560804\pi$$
−0.189860 + 0.981811i $$0.560804\pi$$
$$14$$ −2.34017 −0.625438
$$15$$ 0 0
$$16$$ −2.07838 −0.519594
$$17$$ 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
$$21$$ 0 0
$$22$$ 13.7587 2.93337
$$23$$ 2.34017 0.487960 0.243980 0.969780i $$-0.421547\pi$$
0.243980 + 0.969780i $$0.421547\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.97107 −0.582675
$$27$$ 0 0
$$28$$ −2.92162 −0.552135
$$29$$ −1.41855 −0.263418 −0.131709 0.991288i $$-0.542046\pi$$
−0.131709 + 0.991288i $$0.542046\pi$$
$$30$$ 0 0
$$31$$ 8.68035 1.55904 0.779518 0.626380i $$-0.215464\pi$$
0.779518 + 0.626380i $$0.215464\pi$$
$$32$$ −7.58864 −1.34149
$$33$$ 0 0
$$34$$ 7.07838 1.21393
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.36910 −0.882675 −0.441337 0.897341i $$-0.645496\pi$$
−0.441337 + 0.897341i $$0.645496\pi$$
$$38$$ −2.17009 −0.352035
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.26180 0.509407 0.254703 0.967019i $$-0.418022\pi$$
0.254703 + 0.967019i $$0.418022\pi$$
$$42$$ 0 0
$$43$$ 11.9155 1.81709 0.908547 0.417783i $$-0.137193\pi$$
0.908547 + 0.417783i $$0.137193\pi$$
$$44$$ 17.1773 2.58957
$$45$$ 0 0
$$46$$ 5.07838 0.748766
$$47$$ 1.07838 0.157298 0.0786488 0.996902i $$-0.474939\pi$$
0.0786488 + 0.996902i $$0.474939\pi$$
$$48$$ 0 0
$$49$$ −5.83710 −0.833872
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −3.70928 −0.514384
$$53$$ 6.63090 0.910824 0.455412 0.890281i $$-0.349492\pi$$
0.455412 + 0.890281i $$0.349492\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.65983 −0.221804
$$57$$ 0 0
$$58$$ −3.07838 −0.404211
$$59$$ 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 $$0.383244\pi$$
$$62$$ 18.8371 2.39231
$$63$$ 0 0
$$64$$ −12.3112 −1.53891
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −10.3896 −1.26929 −0.634647 0.772802i $$-0.718855\pi$$
−0.634647 + 0.772802i $$0.718855\pi$$
$$68$$ 8.83710 1.07166
$$69$$ 0 0
$$70$$ 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 $$-0.602708\pi$$
−0.317097 + 0.948393i $$0.602708\pi$$
$$74$$ −11.6514 −1.35445
$$75$$ 0 0
$$76$$ −2.70928 −0.310775
$$77$$ −6.83710 −0.779160
$$78$$ 0 0
$$79$$ 14.2557 1.60389 0.801943 0.597400i $$-0.203800\pi$$
0.801943 + 0.597400i $$0.203800\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 7.07838 0.781676
$$83$$ −14.3402 −1.57404 −0.787019 0.616928i $$-0.788377\pi$$
−0.787019 + 0.616928i $$0.788377\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 25.8576 2.78830
$$87$$ 0 0
$$88$$ 9.75872 1.04028
$$89$$ −7.57531 −0.802981 −0.401490 0.915863i $$-0.631508\pi$$
−0.401490 + 0.915863i $$0.631508\pi$$
$$90$$ 0 0
$$91$$ 1.47641 0.154770
$$92$$ 6.34017 0.661009
$$93$$ 0 0
$$94$$ 2.34017 0.241370
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 8.88655 0.902292 0.451146 0.892450i $$-0.351015\pi$$
0.451146 + 0.892450i $$0.351015\pi$$
$$98$$ −12.6670 −1.27956
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.92162 0.489720 0.244860 0.969558i $$-0.421258\pi$$
0.244860 + 0.969558i $$0.421258\pi$$
$$102$$ 0 0
$$103$$ −6.38962 −0.629588 −0.314794 0.949160i $$-0.601935\pi$$
−0.314794 + 0.949160i $$0.601935\pi$$
$$104$$ −2.10731 −0.206638
$$105$$ 0 0
$$106$$ 14.3896 1.39764
$$107$$ 2.29072 0.221453 0.110726 0.993851i $$-0.464682\pi$$
0.110726 + 0.993851i $$0.464682\pi$$
$$108$$ 0 0
$$109$$ −12.8371 −1.22957 −0.614786 0.788694i $$-0.710757\pi$$
−0.614786 + 0.788694i $$0.710757\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.24128 0.211781
$$113$$ −12.8865 −1.21226 −0.606132 0.795364i $$-0.707280\pi$$
−0.606132 + 0.795364i $$0.707280\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.84324 −0.356836
$$117$$ 0 0
$$118$$ 24.7792 2.28111
$$119$$ −3.51745 −0.322444
$$120$$ 0 0
$$121$$ 29.1978 2.65434
$$122$$ 12.1568 1.10062
$$123$$ 0 0
$$124$$ 23.5174 2.11193
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.23287 0.730549 0.365274 0.930900i $$-0.380975\pi$$
0.365274 + 0.930900i $$0.380975\pi$$
$$128$$ −11.5392 −1.01993
$$129$$ 0 0
$$130$$ 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
$$134$$ −22.5464 −1.94771
$$135$$ 0 0
$$136$$ 5.02052 0.430506
$$137$$ −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 $$-0.377331\pi$$
0.375907 + 0.926657i $$0.377331\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 23.5174 1.97354
$$143$$ −8.68035 −0.725887
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −11.7587 −0.973159
$$147$$ 0 0
$$148$$ −14.5464 −1.19570
$$149$$ 19.7587 1.61870 0.809349 0.587328i $$-0.199820\pi$$
0.809349 + 0.587328i $$0.199820\pi$$
$$150$$ 0 0
$$151$$ 3.41855 0.278198 0.139099 0.990279i $$-0.455579\pi$$
0.139099 + 0.990279i $$0.455579\pi$$
$$152$$ −1.53919 −0.124845
$$153$$ 0 0
$$154$$ −14.8371 −1.19561
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.41855 −0.751682 −0.375841 0.926684i $$-0.622646\pi$$
−0.375841 + 0.926684i $$0.622646\pi$$
$$158$$ 30.9360 2.46114
$$159$$ 0 0
$$160$$ 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 $$0.463499\pi$$
$$164$$ 8.83710 0.690062
$$165$$ 0 0
$$166$$ −31.1194 −2.41534
$$167$$ 20.9132 1.61831 0.809156 0.587593i $$-0.199924\pi$$
0.809156 + 0.587593i $$0.199924\pi$$
$$168$$ 0 0
$$169$$ −11.1256 −0.855812
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 32.2823 2.46150
$$173$$ −1.05559 −0.0802551 −0.0401276 0.999195i $$-0.512776\pi$$
−0.0401276 + 0.999195i $$0.512776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −13.1773 −0.993274
$$177$$ 0 0
$$178$$ −16.4391 −1.23216
$$179$$ −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 $$0.509904\pi$$
$$182$$ 3.20394 0.237492
$$183$$ 0 0
$$184$$ 3.60197 0.265541
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 20.6803 1.51230
$$188$$ 2.92162 0.213081
$$189$$ 0 0
$$190$$ 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 $$-0.652236\pi$$
−0.460238 + 0.887796i $$0.652236\pi$$
$$194$$ 19.2846 1.38455
$$195$$ 0 0
$$196$$ −15.8143 −1.12959
$$197$$ −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 $$-0.694654\pi$$
−0.574116 + 0.818774i $$0.694654\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.6803 0.751467
$$203$$ 1.52973 0.107366
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −13.8660 −0.966092
$$207$$ 0 0
$$208$$ 2.84551 0.197301
$$209$$ −6.34017 −0.438559
$$210$$ 0 0
$$211$$ 7.78539 0.535968 0.267984 0.963423i $$-0.413643\pi$$
0.267984 + 0.963423i $$0.413643\pi$$
$$212$$ 17.9649 1.23384
$$213$$ 0 0
$$214$$ 4.97107 0.339815
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.36069 −0.635445
$$218$$ −27.8576 −1.88676
$$219$$ 0 0
$$220$$ 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 $$0.637999\pi$$
$$224$$ 8.18342 0.546778
$$225$$ 0 0
$$226$$ −27.9649 −1.86020
$$227$$ 2.29072 0.152041 0.0760204 0.997106i $$-0.475779\pi$$
0.0760204 + 0.997106i $$0.475779\pi$$
$$228$$ 0 0
$$229$$ −5.91548 −0.390906 −0.195453 0.980713i $$-0.562618\pi$$
−0.195453 + 0.980713i $$0.562618\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.18342 −0.143348
$$233$$ 13.5174 0.885557 0.442779 0.896631i $$-0.353993\pi$$
0.442779 + 0.896631i $$0.353993\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 30.9360 2.01376
$$237$$ 0 0
$$238$$ −7.63317 −0.494785
$$239$$ −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 $$0.424856\pi$$
$$242$$ 63.3617 4.07305
$$243$$ 0 0
$$244$$ 15.1773 0.971625
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.36910 0.0871139
$$248$$ 13.3607 0.848405
$$249$$ 0 0
$$250$$ 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
$$254$$ 17.8660 1.12101
$$255$$ 0 0
$$256$$ −0.418551 −0.0261594
$$257$$ −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 0
$$262$$ −3.20394 −0.197940
$$263$$ 5.65983 0.349000 0.174500 0.984657i $$-0.444169\pi$$
0.174500 + 0.984657i $$0.444169\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.34017 0.143485
$$267$$ 0 0
$$268$$ −28.1483 −1.71943
$$269$$ 2.31351 0.141057 0.0705286 0.997510i $$-0.477531\pi$$
0.0705286 + 0.997510i $$0.477531\pi$$
$$270$$ 0 0
$$271$$ −19.7009 −1.19674 −0.598371 0.801219i $$-0.704185\pi$$
−0.598371 + 0.801219i $$0.704185\pi$$
$$272$$ −6.77924 −0.411052
$$273$$ 0 0
$$274$$ −8.55479 −0.516814
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 25.7321 1.54609 0.773045 0.634351i $$-0.218733\pi$$
0.773045 + 0.634351i $$0.218733\pi$$
$$278$$ 19.2351 1.15365
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.58145 0.392616 0.196308 0.980542i $$-0.437105\pi$$
0.196308 + 0.980542i $$0.437105\pi$$
$$282$$ 0 0
$$283$$ 0.496928 0.0295393 0.0147697 0.999891i $$-0.495298\pi$$
0.0147697 + 0.999891i $$0.495298\pi$$
$$284$$ 29.3607 1.74224
$$285$$ 0 0
$$286$$ −18.8371 −1.11386
$$287$$ −3.51745 −0.207628
$$288$$ 0 0
$$289$$ −6.36069 −0.374158
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −14.6803 −0.859102
$$293$$ 6.63090 0.387381 0.193691 0.981063i $$-0.437954\pi$$
0.193691 + 0.981063i $$0.437954\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −8.26406 −0.480339
$$297$$ 0 0
$$298$$ 42.8781 2.48386
$$299$$ −3.20394 −0.185288
$$300$$ 0 0
$$301$$ −12.8494 −0.740626
$$302$$ 7.41855 0.426890
$$303$$ 0 0
$$304$$ 2.07838 0.119203
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −14.6042 −0.833508 −0.416754 0.909019i $$-0.636832\pi$$
−0.416754 + 0.909019i $$0.636832\pi$$
$$308$$ −18.5236 −1.05548
$$309$$ 0 0
$$310$$ 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 $$-0.834005\pi$$
−0.867078 + 0.498173i $$0.834005\pi$$
$$314$$ −20.4391 −1.15344
$$315$$ 0 0
$$316$$ 38.6225 2.17268
$$317$$ 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
$$321$$ 0 0
$$322$$ −5.47641 −0.305188
$$323$$ −3.26180 −0.181491
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 6.34017 0.351150
$$327$$ 0 0
$$328$$ 5.02052 0.277212
$$329$$ −1.16290 −0.0641127
$$330$$ 0 0
$$331$$ −2.73820 −0.150505 −0.0752527 0.997164i $$-0.523976\pi$$
−0.0752527 + 0.997164i $$0.523976\pi$$
$$332$$ −38.8515 −2.13225
$$333$$ 0 0
$$334$$ 45.3835 2.48327
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.04945 0.329534 0.164767 0.986332i $$-0.447313\pi$$
0.164767 + 0.986332i $$0.447313\pi$$
$$338$$ −24.1434 −1.31323
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 55.0349 2.98031
$$342$$ 0 0
$$343$$ 13.8432 0.747465
$$344$$ 18.3402 0.988836
$$345$$ 0 0
$$346$$ −2.29072 −0.123150
$$347$$ −5.97334 −0.320666 −0.160333 0.987063i $$-0.551257\pi$$
−0.160333 + 0.987063i $$0.551257\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −48.1133 −2.56445
$$353$$ −14.0989 −0.750409 −0.375204 0.926942i $$-0.622427\pi$$
−0.375204 + 0.926942i $$0.622427\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −20.5236 −1.08775
$$357$$ 0 0
$$358$$ −1.94214 −0.102645
$$359$$ −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
$$362$$ −1.81658 −0.0954775
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1.07838 −0.0562909 −0.0281454 0.999604i $$-0.508960\pi$$
−0.0281454 + 0.999604i $$0.508960\pi$$
$$368$$ −4.86376 −0.253541
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −7.15061 −0.371241
$$372$$ 0 0
$$373$$ −12.3051 −0.637134 −0.318567 0.947900i $$-0.603202\pi$$
−0.318567 + 0.947900i $$0.603202\pi$$
$$374$$ 44.8781 2.32059
$$375$$ 0 0
$$376$$ 1.65983 0.0855990
$$377$$ 1.94214 0.100025
$$378$$ 0 0
$$379$$ 1.04718 0.0537901 0.0268950 0.999638i $$-0.491438\pi$$
0.0268950 + 0.999638i $$0.491438\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −47.8310 −2.44724
$$383$$ 0.0806452 0.00412078 0.00206039 0.999998i $$-0.499344\pi$$
0.00206039 + 0.999998i $$0.499344\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −27.7503 −1.41245
$$387$$ 0 0
$$388$$ 24.0761 1.22228
$$389$$ 20.5236 1.04059 0.520294 0.853987i $$-0.325822\pi$$
0.520294 + 0.853987i $$0.325822\pi$$
$$390$$ 0 0
$$391$$ 7.63317 0.386026
$$392$$ −8.98440 −0.453781
$$393$$ 0 0
$$394$$ −19.9733 −1.00624
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −39.4596 −1.98042 −0.990210 0.139586i $$-0.955423\pi$$
−0.990210 + 0.139586i $$0.955423\pi$$
$$398$$ −35.1506 −1.76194
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −0.470266 −0.0234840 −0.0117420 0.999931i $$-0.503738\pi$$
−0.0117420 + 0.999931i $$0.503738\pi$$
$$402$$ 0 0
$$403$$ −11.8843 −0.591998
$$404$$ 13.3340 0.663393
$$405$$ 0 0
$$406$$ 3.31965 0.164752
$$407$$ −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
$$411$$ 0 0
$$412$$ −17.3112 −0.852864
$$413$$ −12.3135 −0.605908
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.3896 0.509393
$$417$$ 0 0
$$418$$ −13.7587 −0.672961
$$419$$ −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 $$0.819449\pi$$
$$422$$ 16.8950 0.822434
$$423$$ 0 0
$$424$$ 10.2062 0.495657
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6.04104 −0.292346
$$428$$ 6.20620 0.299988
$$429$$ 0 0
$$430$$ 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 $$-0.663720\pi$$
−0.491962 + 0.870617i $$0.663720\pi$$
$$434$$ −20.3135 −0.975080
$$435$$ 0 0
$$436$$ −34.7792 −1.66562
$$437$$ −2.34017 −0.111946
$$438$$ 0 0
$$439$$ −21.4596 −1.02421 −0.512105 0.858923i $$-0.671134\pi$$
−0.512105 + 0.858923i $$0.671134\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −9.69102 −0.460955
$$443$$ −21.5441 −1.02359 −0.511796 0.859107i $$-0.671020\pi$$
−0.511796 + 0.859107i $$0.671020\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −27.2267 −1.28922
$$447$$ 0 0
$$448$$ 13.2762 0.627240
$$449$$ 8.47027 0.399737 0.199868 0.979823i $$-0.435949\pi$$
0.199868 + 0.979823i $$0.435949\pi$$
$$450$$ 0 0
$$451$$ 20.6803 0.973799
$$452$$ −34.9132 −1.64218
$$453$$ 0 0
$$454$$ 4.97107 0.233304
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.3607 −0.531431 −0.265715 0.964052i $$-0.585608\pi$$
−0.265715 + 0.964052i $$0.585608\pi$$
$$458$$ −12.8371 −0.599838
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.04718 −0.141921 −0.0709607 0.997479i $$-0.522606\pi$$
−0.0709607 + 0.997479i $$0.522606\pi$$
$$462$$ 0 0
$$463$$ 9.97334 0.463500 0.231750 0.972775i $$-0.425555\pi$$
0.231750 + 0.972775i $$0.425555\pi$$
$$464$$ 2.94828 0.136871
$$465$$ 0 0
$$466$$ 29.3340 1.35887
$$467$$ 1.49079 0.0689853 0.0344927 0.999405i $$-0.489018\pi$$
0.0344927 + 0.999405i $$0.489018\pi$$
$$468$$ 0 0
$$469$$ 11.2039 0.517350
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 17.5753 0.808969
$$473$$ 75.5462 3.47362
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −9.52973 −0.436795
$$477$$ 0 0
$$478$$ −30.0410 −1.37405
$$479$$ 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
$$482$$ 15.7587 0.717790
$$483$$ 0 0
$$484$$ 79.1049 3.59568
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 24.2784 1.10016 0.550081 0.835112i $$-0.314597\pi$$
0.550081 + 0.835112i $$0.314597\pi$$
$$488$$ 8.62249 0.390322
$$489$$ 0 0
$$490$$ 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
$$494$$ 2.97107 0.133675
$$495$$ 0 0
$$496$$ −18.0410 −0.810067
$$497$$ −11.6865 −0.524211
$$498$$ 0 0
$$499$$ 7.33403 0.328316 0.164158 0.986434i $$-0.447509\pi$$
0.164158 + 0.986434i $$0.447509\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −22.7214 −1.01410
$$503$$ 29.0616 1.29579 0.647895 0.761729i $$-0.275649\pi$$
0.647895 + 0.761729i $$0.275649\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.1978 1.43137
$$507$$ 0 0
$$508$$ 22.3051 0.989629
$$509$$ −26.0456 −1.15445 −0.577225 0.816585i $$-0.695864\pi$$
−0.577225 + 0.816585i $$0.695864\pi$$
$$510$$ 0 0
$$511$$ 5.84324 0.258490
$$512$$ 22.1701 0.979789
$$513$$ 0 0
$$514$$ −51.2678 −2.26132
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6.83710 0.300695
$$518$$ 12.5646 0.552058
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 38.8248 1.70095 0.850473 0.526019i $$-0.176316\pi$$
0.850473 + 0.526019i $$0.176316\pi$$
$$522$$ 0 0
$$523$$ −4.59970 −0.201131 −0.100565 0.994930i $$-0.532065\pi$$
−0.100565 + 0.994930i $$0.532065\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 12.2823 0.535534
$$527$$ 28.3135 1.23336
$$528$$ 0 0
$$529$$ −17.5236 −0.761895
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.92162 0.126668
$$533$$ −4.46573 −0.193432
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −15.9916 −0.690731
$$537$$ 0 0
$$538$$ 5.02052 0.216450
$$539$$ −37.0082 −1.59406
$$540$$ 0 0
$$541$$ −12.1256 −0.521318 −0.260659 0.965431i $$-0.583940\pi$$
−0.260659 + 0.965431i $$0.583940\pi$$
$$542$$ −42.7526 −1.83638
$$543$$ 0 0
$$544$$ −24.7526 −1.06126
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9.54023 0.407911 0.203955 0.978980i $$-0.434620\pi$$
0.203955 + 0.978980i $$0.434620\pi$$
$$548$$ −10.6803 −0.456242
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.41855 0.0604323
$$552$$ 0 0
$$553$$ −15.3730 −0.653726
$$554$$ 55.8408 2.37245
$$555$$ 0 0
$$556$$ 24.0144 1.01844
$$557$$ 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
$$561$$ 0 0
$$562$$ 14.2823 0.602463
$$563$$ 23.5525 0.992620 0.496310 0.868145i $$-0.334688\pi$$
0.496310 + 0.868145i $$0.334688\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1.07838 0.0453276
$$567$$ 0 0
$$568$$ 16.6803 0.699892
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −23.5031 −0.983573 −0.491786 0.870716i $$-0.663656\pi$$
−0.491786 + 0.870716i $$0.663656\pi$$
$$572$$ −23.5174 −0.983314
$$573$$ 0 0
$$574$$ −7.63317 −0.318602
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −16.4703 −0.685666 −0.342833 0.939396i $$-0.611387\pi$$
−0.342833 + 0.939396i $$0.611387\pi$$
$$578$$ −13.8033 −0.574140
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 15.4641 0.641560
$$582$$ 0 0
$$583$$ 42.0410 1.74116
$$584$$ −8.34017 −0.345119
$$585$$ 0 0
$$586$$ 14.3896 0.594430
$$587$$ −28.8104 −1.18913 −0.594567 0.804046i $$-0.702677\pi$$
−0.594567 + 0.804046i $$0.702677\pi$$
$$588$$ 0 0
$$589$$ −8.68035 −0.357667
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 11.1590 0.458633
$$593$$ −43.2450 −1.77586 −0.887929 0.459980i $$-0.847856\pi$$
−0.887929 + 0.459980i $$0.847856\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 53.5318 2.19275
$$597$$ 0 0
$$598$$ −6.95282 −0.284322
$$599$$ 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 $$0.665122\pi$$
$$602$$ −27.8843 −1.13648
$$603$$ 0 0
$$604$$ 9.26180 0.376857
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 6.29072 0.255333 0.127666 0.991817i $$-0.459251\pi$$
0.127666 + 0.991817i $$0.459251\pi$$
$$608$$ 7.58864 0.307760
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.47641 −0.0597291
$$612$$ 0 0
$$613$$ 12.7915 0.516645 0.258322 0.966059i $$-0.416830\pi$$
0.258322 + 0.966059i $$0.416830\pi$$
$$614$$ −31.6925 −1.27900
$$615$$ 0 0
$$616$$ −10.5236 −0.424008
$$617$$ 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 $$-0.318566\pi$$
0.539626 + 0.841905i $$0.318566\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −41.9565 −1.68230
$$623$$ 8.16904 0.327286
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −66.5790 −2.66103
$$627$$ 0 0
$$628$$ −25.5174 −1.01826
$$629$$ −17.5129 −0.698286
$$630$$ 0 0
$$631$$ 35.5318 1.41450 0.707250 0.706964i $$-0.249936\pi$$
0.707250 + 0.706964i $$0.249936\pi$$
$$632$$ 21.9421 0.872812
$$633$$ 0 0
$$634$$ 40.6453 1.61423
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.99159 0.316638
$$638$$ −19.5174 −0.772703
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.9360 −0.510941 −0.255471 0.966817i $$-0.582230\pi$$
−0.255471 + 0.966817i $$0.582230\pi$$
$$642$$ 0 0
$$643$$ −8.49693 −0.335086 −0.167543 0.985865i $$-0.553583\pi$$
−0.167543 + 0.985865i $$0.553583\pi$$
$$644$$ −6.83710 −0.269420
$$645$$ 0 0
$$646$$ −7.07838 −0.278495
$$647$$ −45.4908 −1.78843 −0.894214 0.447640i $$-0.852264\pi$$
−0.894214 + 0.447640i $$0.852264\pi$$
$$648$$ 0 0
$$649$$ 72.3956 2.84178
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 7.91548 0.309994
$$653$$ 35.9421 1.40652 0.703262 0.710930i $$-0.251726\pi$$
0.703262 + 0.710930i $$0.251726\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.77924 −0.264685
$$657$$ 0 0
$$658$$ −2.52359 −0.0983798
$$659$$ −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 $$0.437075\pi$$
$$662$$ −5.94214 −0.230948
$$663$$ 0 0
$$664$$ −22.0722 −0.856569
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3.31965 −0.128538
$$668$$ 56.6596 2.19223
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 35.5174 1.37114
$$672$$ 0 0
$$673$$ 8.67194 0.334279 0.167139 0.985933i $$-0.446547\pi$$
0.167139 + 0.985933i $$0.446547\pi$$
$$674$$ 13.1278 0.505665
$$675$$ 0 0
$$676$$ −30.1422 −1.15932
$$677$$ 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
$$681$$ 0 0
$$682$$ 119.430 4.57323
$$683$$ 46.3896 1.77505 0.887525 0.460760i $$-0.152423\pi$$
0.887525 + 0.460760i $$0.152423\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 30.0410 1.14697
$$687$$ 0 0
$$688$$ −24.7649 −0.944152
$$689$$ −9.07838 −0.345859
$$690$$ 0 0
$$691$$ −34.8515 −1.32581 −0.662906 0.748702i $$-0.730677\pi$$
−0.662906 + 0.748702i $$0.730677\pi$$
$$692$$ −2.85989 −0.108717
$$693$$ 0 0
$$694$$ −12.9627 −0.492056
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 10.6393 0.402993
$$698$$ −21.7009 −0.821390
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −35.6430 −1.34622 −0.673109 0.739543i $$-0.735041\pi$$
−0.673109 + 0.739543i $$0.735041\pi$$
$$702$$ 0 0
$$703$$ 5.36910 0.202500
$$704$$ −78.0554 −2.94182
$$705$$ 0 0
$$706$$ −30.5958 −1.15149
$$707$$ −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
$$711$$ 0 0
$$712$$ −11.6598 −0.436970
$$713$$ 20.3135 0.760747
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.42469 −0.0906151
$$717$$ 0 0
$$718$$ −13.0784 −0.488081
$$719$$ 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
$$722$$ 2.17009 0.0807623
$$723$$ 0 0
$$724$$ −2.26794 −0.0842873
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −34.4391 −1.27727 −0.638637 0.769508i $$-0.720502\pi$$
−0.638637 + 0.769508i $$0.720502\pi$$
$$728$$ 2.27247 0.0842235
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 38.8659 1.43751
$$732$$ 0 0
$$733$$ 19.3607 0.715103 0.357552 0.933893i $$-0.383612\pi$$
0.357552 + 0.933893i $$0.383612\pi$$
$$734$$ −2.34017 −0.0863774
$$735$$ 0 0
$$736$$ −17.7587 −0.654595
$$737$$ −65.8720 −2.42643
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −15.5174 −0.569663
$$743$$ −12.7154 −0.466483 −0.233242 0.972419i $$-0.574933\pi$$
−0.233242 + 0.972419i $$0.574933\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −26.7031 −0.977671
$$747$$ 0 0
$$748$$ 56.0288 2.04861
$$749$$ −2.47027 −0.0902616
$$750$$ 0 0
$$751$$ 3.15836 0.115250 0.0576252 0.998338i $$-0.481647\pi$$
0.0576252 + 0.998338i $$0.481647\pi$$
$$752$$ −2.24128 −0.0817309
$$753$$ 0 0
$$754$$ 4.21461 0.153487
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −28.0410 −1.01917 −0.509584 0.860421i $$-0.670201\pi$$
−0.509584 + 0.860421i $$0.670201\pi$$
$$758$$ 2.27247 0.0825399
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 16.4924 0.597849 0.298924 0.954277i $$-0.403372\pi$$
0.298924 + 0.954277i $$0.403372\pi$$
$$762$$ 0 0
$$763$$ 13.8432 0.501159
$$764$$ −59.7152 −2.16042
$$765$$ 0 0
$$766$$ 0.175007 0.00632326
$$767$$ −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
$$771$$ 0 0
$$772$$ −34.6453 −1.24691
$$773$$ 42.1939 1.51761 0.758805 0.651318i $$-0.225784\pi$$
0.758805 + 0.651318i $$0.225784\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 13.6781 0.491014
$$777$$ 0 0
$$778$$ 44.5380 1.59676
$$779$$ −3.26180 −0.116866
$$780$$ 0 0
$$781$$ 68.7091 2.45860
$$782$$ 16.5646 0.592350
$$783$$ 0 0
$$784$$ 12.1317 0.433275
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 35.4368 1.26319 0.631593 0.775300i $$-0.282401\pi$$
0.631593 + 0.775300i $$0.282401\pi$$
$$788$$ −24.9360 −0.888308
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 13.8966 0.494105
$$792$$ 0 0
$$793$$ −7.66967 −0.272358
$$794$$ −85.6307 −3.03892
$$795$$ 0 0
$$796$$ −43.8843 −1.55544
$$797$$ 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 0
$$802$$ −1.02052 −0.0360358
$$803$$ −34.3545 −1.21235
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −25.7899 −0.908411
$$807$$ 0 0
$$808$$ 7.57531 0.266498
$$809$$ 7.16290 0.251834 0.125917 0.992041i $$-0.459813\pi$$
0.125917 + 0.992041i $$0.459813\pi$$
$$810$$ 0 0
$$811$$ −50.3545 −1.76819 −0.884094 0.467310i $$-0.845223\pi$$
−0.884094 + 0.467310i $$0.845223\pi$$
$$812$$ 4.14447 0.145442
$$813$$ 0 0
$$814$$ −73.8720 −2.58921
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −11.9155 −0.416870
$$818$$ 22.7480 0.795367
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −0.952819 −0.0332536 −0.0166268 0.999862i $$-0.505293\pi$$
−0.0166268 + 0.999862i $$0.505293\pi$$
$$822$$ 0 0
$$823$$ 44.2290 1.54173 0.770863 0.637001i $$-0.219825\pi$$
0.770863 + 0.637001i $$0.219825\pi$$
$$824$$ −9.83483 −0.342613
$$825$$ 0 0
$$826$$ −26.7214 −0.929756
$$827$$ −37.8615 −1.31657 −0.658287 0.752767i $$-0.728719\pi$$
−0.658287 + 0.752767i $$0.728719\pi$$
$$828$$ 0 0
$$829$$ −56.7214 −1.97002 −0.985008 0.172511i $$-0.944812\pi$$
−0.985008 + 0.172511i $$0.944812\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 16.8554 0.584354
$$833$$ −19.0394 −0.659677
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −17.1773 −0.594088
$$837$$ 0 0
$$838$$ −75.2327 −2.59887
$$839$$ 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
$$842$$ −75.1071 −2.58836
$$843$$ 0 0
$$844$$ 21.0928 0.726043
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −31.4863 −1.08188
$$848$$ −13.7815 −0.473259
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.5646 −0.430710
$$852$$ 0 0
$$853$$ 17.0061 0.582279 0.291140 0.956681i $$-0.405966\pi$$
0.291140 + 0.956681i $$0.405966\pi$$
$$854$$ −13.1096 −0.448600
$$855$$ 0 0
$$856$$ 3.52586 0.120511
$$857$$ 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 $$-0.722570\pi$$
−0.643623 + 0.765342i $$0.722570\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −14.6225 −0.498044
$$863$$ 32.1340 1.09385 0.546927 0.837181i $$-0.315798\pi$$
0.546927 + 0.837181i $$0.315798\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −44.4307 −1.50982
$$867$$ 0 0
$$868$$ −25.3607 −0.860798
$$869$$ 90.3833 3.06604
$$870$$ 0 0
$$871$$ 14.2245 0.481977
$$872$$ −19.7587 −0.669115
$$873$$ 0 0
$$874$$ −5.07838 −0.171779
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 19.8927 0.671729 0.335864 0.941910i $$-0.390972\pi$$
0.335864 + 0.941910i $$0.390972\pi$$
$$878$$ −46.5692 −1.57163
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −24.0722 −0.811014 −0.405507 0.914092i $$-0.632905\pi$$
−0.405507 + 0.914092i $$0.632905\pi$$
$$882$$ 0 0
$$883$$ 31.1727 1.04905 0.524523 0.851396i $$-0.324244\pi$$
0.524523 + 0.851396i $$0.324244\pi$$
$$884$$ −12.0989 −0.406930
$$885$$ 0 0
$$886$$ −46.7526 −1.57068
$$887$$ 4.86764 0.163439 0.0817197 0.996655i $$-0.473959\pi$$
0.0817197 + 0.996655i $$0.473959\pi$$
$$888$$ 0 0
$$889$$ −8.87814 −0.297763
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −33.9916 −1.13812
$$893$$ −1.07838 −0.0360865
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 12.4436 0.415712
$$897$$ 0 0
$$898$$ 18.3812 0.613389
$$899$$ −12.3135 −0.410679
$$900$$ 0 0
$$901$$ 21.6286 0.720554
$$902$$ 44.8781 1.49428
$$903$$ 0 0
$$904$$ −19.8348 −0.659697
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 45.4778 1.51007 0.755033 0.655686i $$-0.227621\pi$$
0.755033 + 0.655686i $$0.227621\pi$$
$$908$$ 6.20620 0.205960
$$909$$ 0 0
$$910$$ 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
$$914$$ −24.6537 −0.815471
$$915$$ 0 0
$$916$$ −16.0267 −0.529536
$$917$$ 1.59213 0.0525767
$$918$$ 0 0
$$919$$ −59.5174 −1.96330 −0.981650 0.190693i $$-0.938926\pi$$
−0.981650 + 0.190693i $$0.938926\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −6.61265 −0.217776
$$923$$ −14.8371 −0.488369
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 21.6430 0.711233
$$927$$ 0 0
$$928$$ 10.7649 0.353374
$$929$$ 16.7214 0.548611 0.274305 0.961643i $$-0.411552\pi$$
0.274305 + 0.961643i $$0.411552\pi$$
$$930$$ 0 0
$$931$$ 5.83710 0.191303
$$932$$ 36.6225 1.19961
$$933$$ 0 0
$$934$$ 3.23513 0.105857
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 32.4534 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$938$$ 24.3135 0.793864
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 23.6742 0.771757 0.385878 0.922550i $$-0.373898\pi$$
0.385878 + 0.922550i $$0.373898\pi$$
$$942$$ 0 0
$$943$$ 7.63317 0.248570
$$944$$ −23.7321 −0.772413
$$945$$ 0 0
$$946$$ 163.942 5.33021
$$947$$ 21.9733 0.714038 0.357019 0.934097i $$-0.383793\pi$$
0.357019 + 0.934097i $$0.383793\pi$$
$$948$$ 0 0
$$949$$ 7.41855 0.240816
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −5.41402 −0.175469
$$953$$ −53.2990 −1.72652 −0.863261 0.504757i $$-0.831582\pi$$
−0.863261 + 0.504757i $$0.831582\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −37.5052 −1.21300
$$957$$ 0 0
$$958$$ 22.0722 0.713122
$$959$$ 4.25112 0.137276
$$960$$ 0 0
$$961$$ 44.3484 1.43059
$$962$$ 15.9520 0.514313
$$963$$ 0 0
$$964$$ 19.6742 0.633663
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −15.8166 −0.508627 −0.254314 0.967122i $$-0.581850\pi$$
−0.254314 + 0.967122i $$0.581850\pi$$
$$968$$ 44.9409 1.44446
$$969$$ 0 0
$$970$$ 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
$$974$$ 52.6863 1.68818
$$975$$ 0 0
$$976$$ −11.6430 −0.372684
$$977$$ −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
$$981$$ 0 0
$$982$$ −41.6742 −1.32988
$$983$$ −5.59356 −0.178407 −0.0892034 0.996013i $$-0.528432\pi$$
−0.0892034 + 0.996013i $$0.528432\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −10.0410 −0.319772
$$987$$ 0 0
$$988$$ 3.70928 0.118008
$$989$$ 27.8843 0.886669
$$990$$ 0 0
$$991$$ 32.8950 1.04494 0.522471 0.852657i $$-0.325010\pi$$
0.522471 + 0.852657i $$0.325010\pi$$
$$992$$ −65.8720 −2.09144
$$993$$ 0 0
$$994$$ −25.3607 −0.804392
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −23.2618 −0.736708 −0.368354 0.929686i $$-0.620079\pi$$
−0.368354 + 0.929686i $$0.620079\pi$$
$$998$$ 15.9155 0.503796
$$999$$ 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 4275.2.a.bk.1.3 3
3.2 odd 2 475.2.a.f.1.1 3
5.4 even 2 855.2.a.i.1.1 3
12.11 even 2 7600.2.a.bx.1.1 3
15.2 even 4 475.2.b.d.324.1 6
15.8 even 4 475.2.b.d.324.6 6
15.14 odd 2 95.2.a.a.1.3 3
57.56 even 2 9025.2.a.bb.1.3 3
60.59 even 2 1520.2.a.p.1.3 3
105.104 even 2 4655.2.a.u.1.3 3
120.29 odd 2 6080.2.a.bo.1.3 3
120.59 even 2 6080.2.a.by.1.1 3
285.284 even 2 1805.2.a.f.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.a.1.3 3 15.14 odd 2
475.2.a.f.1.1 3 3.2 odd 2
475.2.b.d.324.1 6 15.2 even 4
475.2.b.d.324.6 6 15.8 even 4
855.2.a.i.1.1 3 5.4 even 2
1520.2.a.p.1.3 3 60.59 even 2
1805.2.a.f.1.1 3 285.284 even 2
4275.2.a.bk.1.3 3 1.1 even 1 trivial
4655.2.a.u.1.3 3 105.104 even 2
6080.2.a.bo.1.3 3 120.29 odd 2
6080.2.a.by.1.1 3 120.59 even 2
7600.2.a.bx.1.1 3 12.11 even 2
9025.2.a.bb.1.3 3 57.56 even 2