# Properties

 Label 666.2.a.i.1.1 Level $666$ Weight $2$ Character 666.1 Self dual yes Analytic conductor $5.318$ 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] = [666,2,Mod(1,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$666 = 2 \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 666.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: $$5.31803677462$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 666.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +3.85410 q^{10} +1.38197 q^{11} -2.85410 q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +4.47214 q^{19} -3.85410 q^{20} -1.38197 q^{22} -2.85410 q^{23} +9.85410 q^{25} +2.85410 q^{26} -3.23607 q^{28} +9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -4.47214 q^{34} +12.4721 q^{35} -1.00000 q^{37} -4.47214 q^{38} +3.85410 q^{40} -9.61803 q^{41} -5.23607 q^{43} +1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +3.47214 q^{49} -9.85410 q^{50} -2.85410 q^{52} -0.472136 q^{53} -5.32624 q^{55} +3.23607 q^{56} -9.32624 q^{58} +4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} +1.00000 q^{64} +11.0000 q^{65} +1.09017 q^{67} +4.47214 q^{68} -12.4721 q^{70} -2.94427 q^{71} +7.09017 q^{73} +1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -8.56231 q^{79} -3.85410 q^{80} +9.61803 q^{82} +14.4721 q^{83} -17.2361 q^{85} +5.23607 q^{86} -1.38197 q^{88} +1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} -1.23607 q^{94} -17.2361 q^{95} -0.472136 q^{97} -3.47214 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - q^{20} - 5 q^{22} + q^{23} + 13 q^{25} - q^{26} - 2 q^{28} + 3 q^{29} + 17 q^{31} - 2 q^{32} + 16 q^{35} - 2 q^{37} + q^{40} - 17 q^{41} - 6 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} - 2 q^{49} - 13 q^{50} + q^{52} + 8 q^{53} + 5 q^{55} + 2 q^{56} - 3 q^{58} + 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{64} + 22 q^{65} - 9 q^{67} - 16 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} + 3 q^{79} - q^{80} + 17 q^{82} + 20 q^{83} - 30 q^{85} + 6 q^{86} - 5 q^{88} + 12 q^{89} + 14 q^{91} + q^{92} + 2 q^{94} - 30 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + 5 * q^11 + q^13 + 2 * q^14 + 2 * q^16 - q^20 - 5 * q^22 + q^23 + 13 * q^25 - q^26 - 2 * q^28 + 3 * q^29 + 17 * q^31 - 2 * q^32 + 16 * q^35 - 2 * q^37 + q^40 - 17 * q^41 - 6 * q^43 + 5 * q^44 - q^46 - 2 * q^47 - 2 * q^49 - 13 * q^50 + q^52 + 8 * q^53 + 5 * q^55 + 2 * q^56 - 3 * q^58 + 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^64 + 22 * q^65 - 9 * q^67 - 16 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 + 3 * q^79 - q^80 + 17 * q^82 + 20 * q^83 - 30 * q^85 + 6 * q^86 - 5 * q^88 + 12 * q^89 + 14 * q^91 + q^92 + 2 * q^94 - 30 * q^95 + 8 * q^97 + 2 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −3.85410 −1.72361 −0.861803 0.507242i $$-0.830665\pi$$
−0.861803 + 0.507242i $$0.830665\pi$$
$$6$$ 0 0
$$7$$ −3.23607 −1.22312 −0.611559 0.791199i $$-0.709457\pi$$
−0.611559 + 0.791199i $$0.709457\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 3.85410 1.21877
$$11$$ 1.38197 0.416678 0.208339 0.978057i $$-0.433194\pi$$
0.208339 + 0.978057i $$0.433194\pi$$
$$12$$ 0 0
$$13$$ −2.85410 −0.791585 −0.395793 0.918340i $$-0.629530\pi$$
−0.395793 + 0.918340i $$0.629530\pi$$
$$14$$ 3.23607 0.864876
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 0 0
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ −3.85410 −0.861803
$$21$$ 0 0
$$22$$ −1.38197 −0.294636
$$23$$ −2.85410 −0.595121 −0.297561 0.954703i $$-0.596173\pi$$
−0.297561 + 0.954703i $$0.596173\pi$$
$$24$$ 0 0
$$25$$ 9.85410 1.97082
$$26$$ 2.85410 0.559735
$$27$$ 0 0
$$28$$ −3.23607 −0.611559
$$29$$ 9.32624 1.73184 0.865919 0.500183i $$-0.166734\pi$$
0.865919 + 0.500183i $$0.166734\pi$$
$$30$$ 0 0
$$31$$ 7.38197 1.32584 0.662920 0.748690i $$-0.269317\pi$$
0.662920 + 0.748690i $$0.269317\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.47214 −0.766965
$$35$$ 12.4721 2.10818
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ −4.47214 −0.725476
$$39$$ 0 0
$$40$$ 3.85410 0.609387
$$41$$ −9.61803 −1.50208 −0.751042 0.660254i $$-0.770449\pi$$
−0.751042 + 0.660254i $$0.770449\pi$$
$$42$$ 0 0
$$43$$ −5.23607 −0.798493 −0.399246 0.916844i $$-0.630728\pi$$
−0.399246 + 0.916844i $$0.630728\pi$$
$$44$$ 1.38197 0.208339
$$45$$ 0 0
$$46$$ 2.85410 0.420814
$$47$$ 1.23607 0.180299 0.0901495 0.995928i $$-0.471266\pi$$
0.0901495 + 0.995928i $$0.471266\pi$$
$$48$$ 0 0
$$49$$ 3.47214 0.496019
$$50$$ −9.85410 −1.39358
$$51$$ 0 0
$$52$$ −2.85410 −0.395793
$$53$$ −0.472136 −0.0648529 −0.0324264 0.999474i $$-0.510323\pi$$
−0.0324264 + 0.999474i $$0.510323\pi$$
$$54$$ 0 0
$$55$$ −5.32624 −0.718190
$$56$$ 3.23607 0.432438
$$57$$ 0 0
$$58$$ −9.32624 −1.22460
$$59$$ 4.76393 0.620211 0.310106 0.950702i $$-0.399636\pi$$
0.310106 + 0.950702i $$0.399636\pi$$
$$60$$ 0 0
$$61$$ 10.6180 1.35950 0.679750 0.733444i $$-0.262088\pi$$
0.679750 + 0.733444i $$0.262088\pi$$
$$62$$ −7.38197 −0.937511
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 11.0000 1.36438
$$66$$ 0 0
$$67$$ 1.09017 0.133185 0.0665927 0.997780i $$-0.478787\pi$$
0.0665927 + 0.997780i $$0.478787\pi$$
$$68$$ 4.47214 0.542326
$$69$$ 0 0
$$70$$ −12.4721 −1.49071
$$71$$ −2.94427 −0.349421 −0.174710 0.984620i $$-0.555899\pi$$
−0.174710 + 0.984620i $$0.555899\pi$$
$$72$$ 0 0
$$73$$ 7.09017 0.829842 0.414921 0.909858i $$-0.363809\pi$$
0.414921 + 0.909858i $$0.363809\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 4.47214 0.512989
$$77$$ −4.47214 −0.509647
$$78$$ 0 0
$$79$$ −8.56231 −0.963335 −0.481667 0.876354i $$-0.659969\pi$$
−0.481667 + 0.876354i $$0.659969\pi$$
$$80$$ −3.85410 −0.430902
$$81$$ 0 0
$$82$$ 9.61803 1.06213
$$83$$ 14.4721 1.58852 0.794262 0.607576i $$-0.207858\pi$$
0.794262 + 0.607576i $$0.207858\pi$$
$$84$$ 0 0
$$85$$ −17.2361 −1.86951
$$86$$ 5.23607 0.564620
$$87$$ 0 0
$$88$$ −1.38197 −0.147318
$$89$$ 1.52786 0.161953 0.0809766 0.996716i $$-0.474196\pi$$
0.0809766 + 0.996716i $$0.474196\pi$$
$$90$$ 0 0
$$91$$ 9.23607 0.968203
$$92$$ −2.85410 −0.297561
$$93$$ 0 0
$$94$$ −1.23607 −0.127491
$$95$$ −17.2361 −1.76838
$$96$$ 0 0
$$97$$ −0.472136 −0.0479381 −0.0239691 0.999713i $$-0.507630\pi$$
−0.0239691 + 0.999713i $$0.507630\pi$$
$$98$$ −3.47214 −0.350739
$$99$$ 0 0
$$100$$ 9.85410 0.985410
$$101$$ −3.52786 −0.351036 −0.175518 0.984476i $$-0.556160\pi$$
−0.175518 + 0.984476i $$0.556160\pi$$
$$102$$ 0 0
$$103$$ 17.2705 1.70171 0.850857 0.525397i $$-0.176083\pi$$
0.850857 + 0.525397i $$0.176083\pi$$
$$104$$ 2.85410 0.279868
$$105$$ 0 0
$$106$$ 0.472136 0.0458579
$$107$$ 7.32624 0.708254 0.354127 0.935197i $$-0.384778\pi$$
0.354127 + 0.935197i $$0.384778\pi$$
$$108$$ 0 0
$$109$$ 2.94427 0.282010 0.141005 0.990009i $$-0.454967\pi$$
0.141005 + 0.990009i $$0.454967\pi$$
$$110$$ 5.32624 0.507837
$$111$$ 0 0
$$112$$ −3.23607 −0.305780
$$113$$ 6.94427 0.653262 0.326631 0.945152i $$-0.394087\pi$$
0.326631 + 0.945152i $$0.394087\pi$$
$$114$$ 0 0
$$115$$ 11.0000 1.02576
$$116$$ 9.32624 0.865919
$$117$$ 0 0
$$118$$ −4.76393 −0.438555
$$119$$ −14.4721 −1.32666
$$120$$ 0 0
$$121$$ −9.09017 −0.826379
$$122$$ −10.6180 −0.961312
$$123$$ 0 0
$$124$$ 7.38197 0.662920
$$125$$ −18.7082 −1.67331
$$126$$ 0 0
$$127$$ −8.47214 −0.751780 −0.375890 0.926664i $$-0.622663\pi$$
−0.375890 + 0.926664i $$0.622663\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −11.0000 −0.964764
$$131$$ 22.6525 1.97916 0.989578 0.143998i $$-0.0459958\pi$$
0.989578 + 0.143998i $$0.0459958\pi$$
$$132$$ 0 0
$$133$$ −14.4721 −1.25489
$$134$$ −1.09017 −0.0941763
$$135$$ 0 0
$$136$$ −4.47214 −0.383482
$$137$$ −3.67376 −0.313871 −0.156935 0.987609i $$-0.550161\pi$$
−0.156935 + 0.987609i $$0.550161\pi$$
$$138$$ 0 0
$$139$$ 4.85410 0.411720 0.205860 0.978581i $$-0.434001\pi$$
0.205860 + 0.978581i $$0.434001\pi$$
$$140$$ 12.4721 1.05409
$$141$$ 0 0
$$142$$ 2.94427 0.247078
$$143$$ −3.94427 −0.329837
$$144$$ 0 0
$$145$$ −35.9443 −2.98501
$$146$$ −7.09017 −0.586787
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ −16.1803 −1.32555 −0.662773 0.748821i $$-0.730620\pi$$
−0.662773 + 0.748821i $$0.730620\pi$$
$$150$$ 0 0
$$151$$ 4.29180 0.349261 0.174631 0.984634i $$-0.444127\pi$$
0.174631 + 0.984634i $$0.444127\pi$$
$$152$$ −4.47214 −0.362738
$$153$$ 0 0
$$154$$ 4.47214 0.360375
$$155$$ −28.4508 −2.28523
$$156$$ 0 0
$$157$$ −16.4721 −1.31462 −0.657310 0.753620i $$-0.728306\pi$$
−0.657310 + 0.753620i $$0.728306\pi$$
$$158$$ 8.56231 0.681180
$$159$$ 0 0
$$160$$ 3.85410 0.304694
$$161$$ 9.23607 0.727904
$$162$$ 0 0
$$163$$ −3.52786 −0.276324 −0.138162 0.990410i $$-0.544119\pi$$
−0.138162 + 0.990410i $$0.544119\pi$$
$$164$$ −9.61803 −0.751042
$$165$$ 0 0
$$166$$ −14.4721 −1.12326
$$167$$ −13.8541 −1.07206 −0.536031 0.844198i $$-0.680077\pi$$
−0.536031 + 0.844198i $$0.680077\pi$$
$$168$$ 0 0
$$169$$ −4.85410 −0.373392
$$170$$ 17.2361 1.32195
$$171$$ 0 0
$$172$$ −5.23607 −0.399246
$$173$$ 0.472136 0.0358958 0.0179479 0.999839i $$-0.494287\pi$$
0.0179479 + 0.999839i $$0.494287\pi$$
$$174$$ 0 0
$$175$$ −31.8885 −2.41055
$$176$$ 1.38197 0.104170
$$177$$ 0 0
$$178$$ −1.52786 −0.114518
$$179$$ 12.6525 0.945690 0.472845 0.881146i $$-0.343227\pi$$
0.472845 + 0.881146i $$0.343227\pi$$
$$180$$ 0 0
$$181$$ 14.4721 1.07571 0.537853 0.843039i $$-0.319236\pi$$
0.537853 + 0.843039i $$0.319236\pi$$
$$182$$ −9.23607 −0.684623
$$183$$ 0 0
$$184$$ 2.85410 0.210407
$$185$$ 3.85410 0.283359
$$186$$ 0 0
$$187$$ 6.18034 0.451951
$$188$$ 1.23607 0.0901495
$$189$$ 0 0
$$190$$ 17.2361 1.25044
$$191$$ 7.09017 0.513027 0.256513 0.966541i $$-0.417426\pi$$
0.256513 + 0.966541i $$0.417426\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0.472136 0.0338974
$$195$$ 0 0
$$196$$ 3.47214 0.248010
$$197$$ 7.52786 0.536338 0.268169 0.963372i $$-0.413581\pi$$
0.268169 + 0.963372i $$0.413581\pi$$
$$198$$ 0 0
$$199$$ 3.05573 0.216615 0.108307 0.994117i $$-0.465457\pi$$
0.108307 + 0.994117i $$0.465457\pi$$
$$200$$ −9.85410 −0.696790
$$201$$ 0 0
$$202$$ 3.52786 0.248220
$$203$$ −30.1803 −2.11824
$$204$$ 0 0
$$205$$ 37.0689 2.58900
$$206$$ −17.2705 −1.20329
$$207$$ 0 0
$$208$$ −2.85410 −0.197896
$$209$$ 6.18034 0.427503
$$210$$ 0 0
$$211$$ 11.2705 0.775894 0.387947 0.921682i $$-0.373184\pi$$
0.387947 + 0.921682i $$0.373184\pi$$
$$212$$ −0.472136 −0.0324264
$$213$$ 0 0
$$214$$ −7.32624 −0.500811
$$215$$ 20.1803 1.37629
$$216$$ 0 0
$$217$$ −23.8885 −1.62166
$$218$$ −2.94427 −0.199411
$$219$$ 0 0
$$220$$ −5.32624 −0.359095
$$221$$ −12.7639 −0.858595
$$222$$ 0 0
$$223$$ 14.1803 0.949586 0.474793 0.880098i $$-0.342523\pi$$
0.474793 + 0.880098i $$0.342523\pi$$
$$224$$ 3.23607 0.216219
$$225$$ 0 0
$$226$$ −6.94427 −0.461926
$$227$$ 4.29180 0.284857 0.142428 0.989805i $$-0.454509\pi$$
0.142428 + 0.989805i $$0.454509\pi$$
$$228$$ 0 0
$$229$$ −23.1246 −1.52812 −0.764059 0.645147i $$-0.776796\pi$$
−0.764059 + 0.645147i $$0.776796\pi$$
$$230$$ −11.0000 −0.725319
$$231$$ 0 0
$$232$$ −9.32624 −0.612298
$$233$$ 6.56231 0.429911 0.214955 0.976624i $$-0.431039\pi$$
0.214955 + 0.976624i $$0.431039\pi$$
$$234$$ 0 0
$$235$$ −4.76393 −0.310765
$$236$$ 4.76393 0.310106
$$237$$ 0 0
$$238$$ 14.4721 0.938089
$$239$$ −9.85410 −0.637409 −0.318704 0.947854i $$-0.603248\pi$$
−0.318704 + 0.947854i $$0.603248\pi$$
$$240$$ 0 0
$$241$$ −1.52786 −0.0984184 −0.0492092 0.998788i $$-0.515670\pi$$
−0.0492092 + 0.998788i $$0.515670\pi$$
$$242$$ 9.09017 0.584338
$$243$$ 0 0
$$244$$ 10.6180 0.679750
$$245$$ −13.3820 −0.854942
$$246$$ 0 0
$$247$$ −12.7639 −0.812150
$$248$$ −7.38197 −0.468755
$$249$$ 0 0
$$250$$ 18.7082 1.18321
$$251$$ 20.9443 1.32199 0.660995 0.750390i $$-0.270134\pi$$
0.660995 + 0.750390i $$0.270134\pi$$
$$252$$ 0 0
$$253$$ −3.94427 −0.247974
$$254$$ 8.47214 0.531589
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 1.05573 0.0658545 0.0329273 0.999458i $$-0.489517\pi$$
0.0329273 + 0.999458i $$0.489517\pi$$
$$258$$ 0 0
$$259$$ 3.23607 0.201079
$$260$$ 11.0000 0.682191
$$261$$ 0 0
$$262$$ −22.6525 −1.39947
$$263$$ 13.2361 0.816171 0.408085 0.912944i $$-0.366197\pi$$
0.408085 + 0.912944i $$0.366197\pi$$
$$264$$ 0 0
$$265$$ 1.81966 0.111781
$$266$$ 14.4721 0.887344
$$267$$ 0 0
$$268$$ 1.09017 0.0665927
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ 12.9443 0.786309 0.393154 0.919473i $$-0.371384\pi$$
0.393154 + 0.919473i $$0.371384\pi$$
$$272$$ 4.47214 0.271163
$$273$$ 0 0
$$274$$ 3.67376 0.221940
$$275$$ 13.6180 0.821198
$$276$$ 0 0
$$277$$ 16.7984 1.00932 0.504658 0.863319i $$-0.331619\pi$$
0.504658 + 0.863319i $$0.331619\pi$$
$$278$$ −4.85410 −0.291130
$$279$$ 0 0
$$280$$ −12.4721 −0.745353
$$281$$ −29.8885 −1.78300 −0.891501 0.453020i $$-0.850347\pi$$
−0.891501 + 0.453020i $$0.850347\pi$$
$$282$$ 0 0
$$283$$ 6.76393 0.402074 0.201037 0.979584i $$-0.435569\pi$$
0.201037 + 0.979584i $$0.435569\pi$$
$$284$$ −2.94427 −0.174710
$$285$$ 0 0
$$286$$ 3.94427 0.233230
$$287$$ 31.1246 1.83723
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 35.9443 2.11072
$$291$$ 0 0
$$292$$ 7.09017 0.414921
$$293$$ 12.6525 0.739166 0.369583 0.929198i $$-0.379501\pi$$
0.369583 + 0.929198i $$0.379501\pi$$
$$294$$ 0 0
$$295$$ −18.3607 −1.06900
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ 16.1803 0.937302
$$299$$ 8.14590 0.471089
$$300$$ 0 0
$$301$$ 16.9443 0.976652
$$302$$ −4.29180 −0.246965
$$303$$ 0 0
$$304$$ 4.47214 0.256495
$$305$$ −40.9230 −2.34324
$$306$$ 0 0
$$307$$ −12.8541 −0.733622 −0.366811 0.930295i $$-0.619550\pi$$
−0.366811 + 0.930295i $$0.619550\pi$$
$$308$$ −4.47214 −0.254824
$$309$$ 0 0
$$310$$ 28.4508 1.61590
$$311$$ 27.0344 1.53298 0.766491 0.642255i $$-0.222001\pi$$
0.766491 + 0.642255i $$0.222001\pi$$
$$312$$ 0 0
$$313$$ 28.1803 1.59285 0.796423 0.604739i $$-0.206723\pi$$
0.796423 + 0.604739i $$0.206723\pi$$
$$314$$ 16.4721 0.929576
$$315$$ 0 0
$$316$$ −8.56231 −0.481667
$$317$$ −20.9443 −1.17635 −0.588174 0.808735i $$-0.700153\pi$$
−0.588174 + 0.808735i $$0.700153\pi$$
$$318$$ 0 0
$$319$$ 12.8885 0.721620
$$320$$ −3.85410 −0.215451
$$321$$ 0 0
$$322$$ −9.23607 −0.514706
$$323$$ 20.0000 1.11283
$$324$$ 0 0
$$325$$ −28.1246 −1.56007
$$326$$ 3.52786 0.195390
$$327$$ 0 0
$$328$$ 9.61803 0.531067
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 14.4721 0.794262
$$333$$ 0 0
$$334$$ 13.8541 0.758063
$$335$$ −4.20163 −0.229559
$$336$$ 0 0
$$337$$ 12.0344 0.655558 0.327779 0.944754i $$-0.393700\pi$$
0.327779 + 0.944754i $$0.393700\pi$$
$$338$$ 4.85410 0.264028
$$339$$ 0 0
$$340$$ −17.2361 −0.934757
$$341$$ 10.2016 0.552449
$$342$$ 0 0
$$343$$ 11.4164 0.616428
$$344$$ 5.23607 0.282310
$$345$$ 0 0
$$346$$ −0.472136 −0.0253822
$$347$$ −17.2361 −0.925281 −0.462640 0.886546i $$-0.653098\pi$$
−0.462640 + 0.886546i $$0.653098\pi$$
$$348$$ 0 0
$$349$$ −10.1803 −0.544941 −0.272471 0.962164i $$-0.587841\pi$$
−0.272471 + 0.962164i $$0.587841\pi$$
$$350$$ 31.8885 1.70451
$$351$$ 0 0
$$352$$ −1.38197 −0.0736590
$$353$$ −16.2918 −0.867125 −0.433562 0.901124i $$-0.642744\pi$$
−0.433562 + 0.901124i $$0.642744\pi$$
$$354$$ 0 0
$$355$$ 11.3475 0.602264
$$356$$ 1.52786 0.0809766
$$357$$ 0 0
$$358$$ −12.6525 −0.668704
$$359$$ 4.47214 0.236030 0.118015 0.993012i $$-0.462347\pi$$
0.118015 + 0.993012i $$0.462347\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −14.4721 −0.760639
$$363$$ 0 0
$$364$$ 9.23607 0.484102
$$365$$ −27.3262 −1.43032
$$366$$ 0 0
$$367$$ 13.1246 0.685099 0.342550 0.939500i $$-0.388710\pi$$
0.342550 + 0.939500i $$0.388710\pi$$
$$368$$ −2.85410 −0.148780
$$369$$ 0 0
$$370$$ −3.85410 −0.200365
$$371$$ 1.52786 0.0793227
$$372$$ 0 0
$$373$$ 27.7082 1.43468 0.717338 0.696725i $$-0.245360\pi$$
0.717338 + 0.696725i $$0.245360\pi$$
$$374$$ −6.18034 −0.319578
$$375$$ 0 0
$$376$$ −1.23607 −0.0637453
$$377$$ −26.6180 −1.37090
$$378$$ 0 0
$$379$$ 28.0902 1.44290 0.721448 0.692469i $$-0.243477\pi$$
0.721448 + 0.692469i $$0.243477\pi$$
$$380$$ −17.2361 −0.884192
$$381$$ 0 0
$$382$$ −7.09017 −0.362765
$$383$$ −17.8885 −0.914062 −0.457031 0.889451i $$-0.651087\pi$$
−0.457031 + 0.889451i $$0.651087\pi$$
$$384$$ 0 0
$$385$$ 17.2361 0.878431
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ −0.472136 −0.0239691
$$389$$ 6.85410 0.347517 0.173758 0.984788i $$-0.444409\pi$$
0.173758 + 0.984788i $$0.444409\pi$$
$$390$$ 0 0
$$391$$ −12.7639 −0.645500
$$392$$ −3.47214 −0.175369
$$393$$ 0 0
$$394$$ −7.52786 −0.379248
$$395$$ 33.0000 1.66041
$$396$$ 0 0
$$397$$ 20.6525 1.03652 0.518259 0.855224i $$-0.326580\pi$$
0.518259 + 0.855224i $$0.326580\pi$$
$$398$$ −3.05573 −0.153170
$$399$$ 0 0
$$400$$ 9.85410 0.492705
$$401$$ 4.76393 0.237899 0.118950 0.992900i $$-0.462047\pi$$
0.118950 + 0.992900i $$0.462047\pi$$
$$402$$ 0 0
$$403$$ −21.0689 −1.04952
$$404$$ −3.52786 −0.175518
$$405$$ 0 0
$$406$$ 30.1803 1.49783
$$407$$ −1.38197 −0.0685015
$$408$$ 0 0
$$409$$ −26.1803 −1.29453 −0.647267 0.762263i $$-0.724088\pi$$
−0.647267 + 0.762263i $$0.724088\pi$$
$$410$$ −37.0689 −1.83070
$$411$$ 0 0
$$412$$ 17.2705 0.850857
$$413$$ −15.4164 −0.758592
$$414$$ 0 0
$$415$$ −55.7771 −2.73799
$$416$$ 2.85410 0.139934
$$417$$ 0 0
$$418$$ −6.18034 −0.302290
$$419$$ 10.5623 0.516002 0.258001 0.966145i $$-0.416936\pi$$
0.258001 + 0.966145i $$0.416936\pi$$
$$420$$ 0 0
$$421$$ −31.0344 −1.51253 −0.756263 0.654268i $$-0.772977\pi$$
−0.756263 + 0.654268i $$0.772977\pi$$
$$422$$ −11.2705 −0.548640
$$423$$ 0 0
$$424$$ 0.472136 0.0229289
$$425$$ 44.0689 2.13765
$$426$$ 0 0
$$427$$ −34.3607 −1.66283
$$428$$ 7.32624 0.354127
$$429$$ 0 0
$$430$$ −20.1803 −0.973182
$$431$$ −12.3607 −0.595393 −0.297696 0.954661i $$-0.596218\pi$$
−0.297696 + 0.954661i $$0.596218\pi$$
$$432$$ 0 0
$$433$$ −20.6738 −0.993518 −0.496759 0.867889i $$-0.665477\pi$$
−0.496759 + 0.867889i $$0.665477\pi$$
$$434$$ 23.8885 1.14669
$$435$$ 0 0
$$436$$ 2.94427 0.141005
$$437$$ −12.7639 −0.610582
$$438$$ 0 0
$$439$$ 7.79837 0.372196 0.186098 0.982531i $$-0.440416\pi$$
0.186098 + 0.982531i $$0.440416\pi$$
$$440$$ 5.32624 0.253918
$$441$$ 0 0
$$442$$ 12.7639 0.607118
$$443$$ −15.2705 −0.725524 −0.362762 0.931882i $$-0.618166\pi$$
−0.362762 + 0.931882i $$0.618166\pi$$
$$444$$ 0 0
$$445$$ −5.88854 −0.279144
$$446$$ −14.1803 −0.671459
$$447$$ 0 0
$$448$$ −3.23607 −0.152890
$$449$$ 28.4721 1.34368 0.671842 0.740695i $$-0.265504\pi$$
0.671842 + 0.740695i $$0.265504\pi$$
$$450$$ 0 0
$$451$$ −13.2918 −0.625886
$$452$$ 6.94427 0.326631
$$453$$ 0 0
$$454$$ −4.29180 −0.201424
$$455$$ −35.5967 −1.66880
$$456$$ 0 0
$$457$$ −24.7639 −1.15841 −0.579204 0.815183i $$-0.696637\pi$$
−0.579204 + 0.815183i $$0.696637\pi$$
$$458$$ 23.1246 1.08054
$$459$$ 0 0
$$460$$ 11.0000 0.512878
$$461$$ 38.9443 1.81382 0.906908 0.421329i $$-0.138436\pi$$
0.906908 + 0.421329i $$0.138436\pi$$
$$462$$ 0 0
$$463$$ −4.56231 −0.212028 −0.106014 0.994365i $$-0.533809\pi$$
−0.106014 + 0.994365i $$0.533809\pi$$
$$464$$ 9.32624 0.432960
$$465$$ 0 0
$$466$$ −6.56231 −0.303993
$$467$$ 24.3607 1.12728 0.563639 0.826021i $$-0.309401\pi$$
0.563639 + 0.826021i $$0.309401\pi$$
$$468$$ 0 0
$$469$$ −3.52786 −0.162902
$$470$$ 4.76393 0.219744
$$471$$ 0 0
$$472$$ −4.76393 −0.219278
$$473$$ −7.23607 −0.332715
$$474$$ 0 0
$$475$$ 44.0689 2.02202
$$476$$ −14.4721 −0.663329
$$477$$ 0 0
$$478$$ 9.85410 0.450716
$$479$$ −36.5623 −1.67057 −0.835287 0.549814i $$-0.814699\pi$$
−0.835287 + 0.549814i $$0.814699\pi$$
$$480$$ 0 0
$$481$$ 2.85410 0.130136
$$482$$ 1.52786 0.0695923
$$483$$ 0 0
$$484$$ −9.09017 −0.413190
$$485$$ 1.81966 0.0826265
$$486$$ 0 0
$$487$$ 37.3050 1.69045 0.845224 0.534412i $$-0.179467\pi$$
0.845224 + 0.534412i $$0.179467\pi$$
$$488$$ −10.6180 −0.480656
$$489$$ 0 0
$$490$$ 13.3820 0.604536
$$491$$ 28.4508 1.28397 0.641984 0.766718i $$-0.278111\pi$$
0.641984 + 0.766718i $$0.278111\pi$$
$$492$$ 0 0
$$493$$ 41.7082 1.87844
$$494$$ 12.7639 0.574276
$$495$$ 0 0
$$496$$ 7.38197 0.331460
$$497$$ 9.52786 0.427383
$$498$$ 0 0
$$499$$ 10.2918 0.460724 0.230362 0.973105i $$-0.426009\pi$$
0.230362 + 0.973105i $$0.426009\pi$$
$$500$$ −18.7082 −0.836656
$$501$$ 0 0
$$502$$ −20.9443 −0.934789
$$503$$ −19.0902 −0.851189 −0.425594 0.904914i $$-0.639935\pi$$
−0.425594 + 0.904914i $$0.639935\pi$$
$$504$$ 0 0
$$505$$ 13.5967 0.605047
$$506$$ 3.94427 0.175344
$$507$$ 0 0
$$508$$ −8.47214 −0.375890
$$509$$ 17.7082 0.784902 0.392451 0.919773i $$-0.371627\pi$$
0.392451 + 0.919773i $$0.371627\pi$$
$$510$$ 0 0
$$511$$ −22.9443 −1.01499
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −1.05573 −0.0465662
$$515$$ −66.5623 −2.93309
$$516$$ 0 0
$$517$$ 1.70820 0.0751267
$$518$$ −3.23607 −0.142185
$$519$$ 0 0
$$520$$ −11.0000 −0.482382
$$521$$ 1.41641 0.0620540 0.0310270 0.999519i $$-0.490122\pi$$
0.0310270 + 0.999519i $$0.490122\pi$$
$$522$$ 0 0
$$523$$ 11.8197 0.516838 0.258419 0.966033i $$-0.416799\pi$$
0.258419 + 0.966033i $$0.416799\pi$$
$$524$$ 22.6525 0.989578
$$525$$ 0 0
$$526$$ −13.2361 −0.577120
$$527$$ 33.0132 1.43808
$$528$$ 0 0
$$529$$ −14.8541 −0.645831
$$530$$ −1.81966 −0.0790410
$$531$$ 0 0
$$532$$ −14.4721 −0.627447
$$533$$ 27.4508 1.18903
$$534$$ 0 0
$$535$$ −28.2361 −1.22075
$$536$$ −1.09017 −0.0470882
$$537$$ 0 0
$$538$$ 4.00000 0.172452
$$539$$ 4.79837 0.206681
$$540$$ 0 0
$$541$$ −10.3262 −0.443960 −0.221980 0.975051i $$-0.571252\pi$$
−0.221980 + 0.975051i $$0.571252\pi$$
$$542$$ −12.9443 −0.556004
$$543$$ 0 0
$$544$$ −4.47214 −0.191741
$$545$$ −11.3475 −0.486075
$$546$$ 0 0
$$547$$ −16.0689 −0.687056 −0.343528 0.939142i $$-0.611622\pi$$
−0.343528 + 0.939142i $$0.611622\pi$$
$$548$$ −3.67376 −0.156935
$$549$$ 0 0
$$550$$ −13.6180 −0.580675
$$551$$ 41.7082 1.77683
$$552$$ 0 0
$$553$$ 27.7082 1.17827
$$554$$ −16.7984 −0.713695
$$555$$ 0 0
$$556$$ 4.85410 0.205860
$$557$$ −19.5623 −0.828882 −0.414441 0.910076i $$-0.636023\pi$$
−0.414441 + 0.910076i $$0.636023\pi$$
$$558$$ 0 0
$$559$$ 14.9443 0.632075
$$560$$ 12.4721 0.527044
$$561$$ 0 0
$$562$$ 29.8885 1.26077
$$563$$ −7.88854 −0.332462 −0.166231 0.986087i $$-0.553160\pi$$
−0.166231 + 0.986087i $$0.553160\pi$$
$$564$$ 0 0
$$565$$ −26.7639 −1.12597
$$566$$ −6.76393 −0.284309
$$567$$ 0 0
$$568$$ 2.94427 0.123539
$$569$$ −13.8885 −0.582238 −0.291119 0.956687i $$-0.594028\pi$$
−0.291119 + 0.956687i $$0.594028\pi$$
$$570$$ 0 0
$$571$$ −10.5623 −0.442019 −0.221009 0.975272i $$-0.570935\pi$$
−0.221009 + 0.975272i $$0.570935\pi$$
$$572$$ −3.94427 −0.164918
$$573$$ 0 0
$$574$$ −31.1246 −1.29912
$$575$$ −28.1246 −1.17288
$$576$$ 0 0
$$577$$ 10.6525 0.443468 0.221734 0.975107i $$-0.428828\pi$$
0.221734 + 0.975107i $$0.428828\pi$$
$$578$$ −3.00000 −0.124784
$$579$$ 0 0
$$580$$ −35.9443 −1.49250
$$581$$ −46.8328 −1.94295
$$582$$ 0 0
$$583$$ −0.652476 −0.0270228
$$584$$ −7.09017 −0.293393
$$585$$ 0 0
$$586$$ −12.6525 −0.522669
$$587$$ 13.0557 0.538868 0.269434 0.963019i $$-0.413163\pi$$
0.269434 + 0.963019i $$0.413163\pi$$
$$588$$ 0 0
$$589$$ 33.0132 1.36028
$$590$$ 18.3607 0.755897
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ −17.5623 −0.721197 −0.360599 0.932721i $$-0.617428\pi$$
−0.360599 + 0.932721i $$0.617428\pi$$
$$594$$ 0 0
$$595$$ 55.7771 2.28664
$$596$$ −16.1803 −0.662773
$$597$$ 0 0
$$598$$ −8.14590 −0.333111
$$599$$ 6.36068 0.259890 0.129945 0.991521i $$-0.458520\pi$$
0.129945 + 0.991521i $$0.458520\pi$$
$$600$$ 0 0
$$601$$ −24.6869 −1.00700 −0.503500 0.863995i $$-0.667955\pi$$
−0.503500 + 0.863995i $$0.667955\pi$$
$$602$$ −16.9443 −0.690597
$$603$$ 0 0
$$604$$ 4.29180 0.174631
$$605$$ 35.0344 1.42435
$$606$$ 0 0
$$607$$ 35.0344 1.42200 0.711002 0.703190i $$-0.248242\pi$$
0.711002 + 0.703190i $$0.248242\pi$$
$$608$$ −4.47214 −0.181369
$$609$$ 0 0
$$610$$ 40.9230 1.65692
$$611$$ −3.52786 −0.142722
$$612$$ 0 0
$$613$$ −13.8197 −0.558171 −0.279085 0.960266i $$-0.590031\pi$$
−0.279085 + 0.960266i $$0.590031\pi$$
$$614$$ 12.8541 0.518749
$$615$$ 0 0
$$616$$ 4.47214 0.180187
$$617$$ −0.0901699 −0.00363011 −0.00181505 0.999998i $$-0.500578\pi$$
−0.00181505 + 0.999998i $$0.500578\pi$$
$$618$$ 0 0
$$619$$ −15.2705 −0.613774 −0.306887 0.951746i $$-0.599287\pi$$
−0.306887 + 0.951746i $$0.599287\pi$$
$$620$$ −28.4508 −1.14261
$$621$$ 0 0
$$622$$ −27.0344 −1.08398
$$623$$ −4.94427 −0.198088
$$624$$ 0 0
$$625$$ 22.8328 0.913313
$$626$$ −28.1803 −1.12631
$$627$$ 0 0
$$628$$ −16.4721 −0.657310
$$629$$ −4.47214 −0.178316
$$630$$ 0 0
$$631$$ −47.3951 −1.88677 −0.943385 0.331700i $$-0.892378\pi$$
−0.943385 + 0.331700i $$0.892378\pi$$
$$632$$ 8.56231 0.340590
$$633$$ 0 0
$$634$$ 20.9443 0.831803
$$635$$ 32.6525 1.29577
$$636$$ 0 0
$$637$$ −9.90983 −0.392642
$$638$$ −12.8885 −0.510262
$$639$$ 0 0
$$640$$ 3.85410 0.152347
$$641$$ −15.5066 −0.612473 −0.306236 0.951955i $$-0.599070\pi$$
−0.306236 + 0.951955i $$0.599070\pi$$
$$642$$ 0 0
$$643$$ −28.7639 −1.13434 −0.567169 0.823601i $$-0.691962\pi$$
−0.567169 + 0.823601i $$0.691962\pi$$
$$644$$ 9.23607 0.363952
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ −30.0902 −1.18297 −0.591483 0.806317i $$-0.701457\pi$$
−0.591483 + 0.806317i $$0.701457\pi$$
$$648$$ 0 0
$$649$$ 6.58359 0.258429
$$650$$ 28.1246 1.10314
$$651$$ 0 0
$$652$$ −3.52786 −0.138162
$$653$$ 38.2705 1.49764 0.748820 0.662773i $$-0.230621\pi$$
0.748820 + 0.662773i $$0.230621\pi$$
$$654$$ 0 0
$$655$$ −87.3050 −3.41129
$$656$$ −9.61803 −0.375521
$$657$$ 0 0
$$658$$ 4.00000 0.155936
$$659$$ −40.4508 −1.57574 −0.787871 0.615841i $$-0.788816\pi$$
−0.787871 + 0.615841i $$0.788816\pi$$
$$660$$ 0 0
$$661$$ 17.3262 0.673913 0.336956 0.941520i $$-0.390603\pi$$
0.336956 + 0.941520i $$0.390603\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ −14.4721 −0.561628
$$665$$ 55.7771 2.16294
$$666$$ 0 0
$$667$$ −26.6180 −1.03065
$$668$$ −13.8541 −0.536031
$$669$$ 0 0
$$670$$ 4.20163 0.162323
$$671$$ 14.6738 0.566474
$$672$$ 0 0
$$673$$ 11.1459 0.429643 0.214821 0.976653i $$-0.431083\pi$$
0.214821 + 0.976653i $$0.431083\pi$$
$$674$$ −12.0344 −0.463549
$$675$$ 0 0
$$676$$ −4.85410 −0.186696
$$677$$ −34.6525 −1.33180 −0.665901 0.746040i $$-0.731953\pi$$
−0.665901 + 0.746040i $$0.731953\pi$$
$$678$$ 0 0
$$679$$ 1.52786 0.0586340
$$680$$ 17.2361 0.660973
$$681$$ 0 0
$$682$$ −10.2016 −0.390640
$$683$$ 8.58359 0.328442 0.164221 0.986424i $$-0.447489\pi$$
0.164221 + 0.986424i $$0.447489\pi$$
$$684$$ 0 0
$$685$$ 14.1591 0.540990
$$686$$ −11.4164 −0.435880
$$687$$ 0 0
$$688$$ −5.23607 −0.199623
$$689$$ 1.34752 0.0513366
$$690$$ 0 0
$$691$$ 35.7771 1.36102 0.680512 0.732737i $$-0.261757\pi$$
0.680512 + 0.732737i $$0.261757\pi$$
$$692$$ 0.472136 0.0179479
$$693$$ 0 0
$$694$$ 17.2361 0.654272
$$695$$ −18.7082 −0.709643
$$696$$ 0 0
$$697$$ −43.0132 −1.62924
$$698$$ 10.1803 0.385332
$$699$$ 0 0
$$700$$ −31.8885 −1.20527
$$701$$ 42.9787 1.62328 0.811642 0.584155i $$-0.198574\pi$$
0.811642 + 0.584155i $$0.198574\pi$$
$$702$$ 0 0
$$703$$ −4.47214 −0.168670
$$704$$ 1.38197 0.0520848
$$705$$ 0 0
$$706$$ 16.2918 0.613150
$$707$$ 11.4164 0.429358
$$708$$ 0 0
$$709$$ −8.21478 −0.308513 −0.154256 0.988031i $$-0.549298\pi$$
−0.154256 + 0.988031i $$0.549298\pi$$
$$710$$ −11.3475 −0.425865
$$711$$ 0 0
$$712$$ −1.52786 −0.0572591
$$713$$ −21.0689 −0.789036
$$714$$ 0 0
$$715$$ 15.2016 0.568509
$$716$$ 12.6525 0.472845
$$717$$ 0 0
$$718$$ −4.47214 −0.166899
$$719$$ −27.4164 −1.02246 −0.511230 0.859444i $$-0.670810\pi$$
−0.511230 + 0.859444i $$0.670810\pi$$
$$720$$ 0 0
$$721$$ −55.8885 −2.08140
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 14.4721 0.537853
$$725$$ 91.9017 3.41314
$$726$$ 0 0
$$727$$ −29.8541 −1.10723 −0.553614 0.832774i $$-0.686752\pi$$
−0.553614 + 0.832774i $$0.686752\pi$$
$$728$$ −9.23607 −0.342311
$$729$$ 0 0
$$730$$ 27.3262 1.01139
$$731$$ −23.4164 −0.866087
$$732$$ 0 0
$$733$$ 27.5279 1.01676 0.508382 0.861131i $$-0.330244\pi$$
0.508382 + 0.861131i $$0.330244\pi$$
$$734$$ −13.1246 −0.484438
$$735$$ 0 0
$$736$$ 2.85410 0.105204
$$737$$ 1.50658 0.0554955
$$738$$ 0 0
$$739$$ −10.9098 −0.401325 −0.200662 0.979660i $$-0.564309\pi$$
−0.200662 + 0.979660i $$0.564309\pi$$
$$740$$ 3.85410 0.141680
$$741$$ 0 0
$$742$$ −1.52786 −0.0560897
$$743$$ 48.0689 1.76348 0.881738 0.471739i $$-0.156374\pi$$
0.881738 + 0.471739i $$0.156374\pi$$
$$744$$ 0 0
$$745$$ 62.3607 2.28472
$$746$$ −27.7082 −1.01447
$$747$$ 0 0
$$748$$ 6.18034 0.225976
$$749$$ −23.7082 −0.866279
$$750$$ 0 0
$$751$$ −1.05573 −0.0385241 −0.0192620 0.999814i $$-0.506132\pi$$
−0.0192620 + 0.999814i $$0.506132\pi$$
$$752$$ 1.23607 0.0450748
$$753$$ 0 0
$$754$$ 26.6180 0.969372
$$755$$ −16.5410 −0.601989
$$756$$ 0 0
$$757$$ −4.14590 −0.150685 −0.0753426 0.997158i $$-0.524005\pi$$
−0.0753426 + 0.997158i $$0.524005\pi$$
$$758$$ −28.0902 −1.02028
$$759$$ 0 0
$$760$$ 17.2361 0.625218
$$761$$ 19.1459 0.694038 0.347019 0.937858i $$-0.387194\pi$$
0.347019 + 0.937858i $$0.387194\pi$$
$$762$$ 0 0
$$763$$ −9.52786 −0.344932
$$764$$ 7.09017 0.256513
$$765$$ 0 0
$$766$$ 17.8885 0.646339
$$767$$ −13.5967 −0.490950
$$768$$ 0 0
$$769$$ 23.8885 0.861443 0.430721 0.902485i $$-0.358259\pi$$
0.430721 + 0.902485i $$0.358259\pi$$
$$770$$ −17.2361 −0.621145
$$771$$ 0 0
$$772$$ 4.00000 0.143963
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 72.7426 2.61299
$$776$$ 0.472136 0.0169487
$$777$$ 0 0
$$778$$ −6.85410 −0.245731
$$779$$ −43.0132 −1.54111
$$780$$ 0 0
$$781$$ −4.06888 −0.145596
$$782$$ 12.7639 0.456437
$$783$$ 0 0
$$784$$ 3.47214 0.124005
$$785$$ 63.4853 2.26589
$$786$$ 0 0
$$787$$ −25.5279 −0.909970 −0.454985 0.890499i $$-0.650355\pi$$
−0.454985 + 0.890499i $$0.650355\pi$$
$$788$$ 7.52786 0.268169
$$789$$ 0 0
$$790$$ −33.0000 −1.17409
$$791$$ −22.4721 −0.799017
$$792$$ 0 0
$$793$$ −30.3050 −1.07616
$$794$$ −20.6525 −0.732929
$$795$$ 0 0
$$796$$ 3.05573 0.108307
$$797$$ −46.2705 −1.63899 −0.819493 0.573090i $$-0.805745\pi$$
−0.819493 + 0.573090i $$0.805745\pi$$
$$798$$ 0 0
$$799$$ 5.52786 0.195562
$$800$$ −9.85410 −0.348395
$$801$$ 0 0
$$802$$ −4.76393 −0.168220
$$803$$ 9.79837 0.345777
$$804$$ 0 0
$$805$$ −35.5967 −1.25462
$$806$$ 21.0689 0.742120
$$807$$ 0 0
$$808$$ 3.52786 0.124110
$$809$$ 13.1246 0.461437 0.230718 0.973021i $$-0.425892\pi$$
0.230718 + 0.973021i $$0.425892\pi$$
$$810$$ 0 0
$$811$$ −34.1033 −1.19753 −0.598765 0.800925i $$-0.704342\pi$$
−0.598765 + 0.800925i $$0.704342\pi$$
$$812$$ −30.1803 −1.05912
$$813$$ 0 0
$$814$$ 1.38197 0.0484379
$$815$$ 13.5967 0.476273
$$816$$ 0 0
$$817$$ −23.4164 −0.819236
$$818$$ 26.1803 0.915374
$$819$$ 0 0
$$820$$ 37.0689 1.29450
$$821$$ 5.41641 0.189034 0.0945170 0.995523i $$-0.469869\pi$$
0.0945170 + 0.995523i $$0.469869\pi$$
$$822$$ 0 0
$$823$$ 1.88854 0.0658305 0.0329152 0.999458i $$-0.489521\pi$$
0.0329152 + 0.999458i $$0.489521\pi$$
$$824$$ −17.2705 −0.601647
$$825$$ 0 0
$$826$$ 15.4164 0.536405
$$827$$ 2.06888 0.0719421 0.0359711 0.999353i $$-0.488548\pi$$
0.0359711 + 0.999353i $$0.488548\pi$$
$$828$$ 0 0
$$829$$ 55.7984 1.93796 0.968979 0.247144i $$-0.0794920\pi$$
0.968979 + 0.247144i $$0.0794920\pi$$
$$830$$ 55.7771 1.93605
$$831$$ 0 0
$$832$$ −2.85410 −0.0989482
$$833$$ 15.5279 0.538009
$$834$$ 0 0
$$835$$ 53.3951 1.84781
$$836$$ 6.18034 0.213752
$$837$$ 0 0
$$838$$ −10.5623 −0.364869
$$839$$ 36.6525 1.26538 0.632692 0.774404i $$-0.281950\pi$$
0.632692 + 0.774404i $$0.281950\pi$$
$$840$$ 0 0
$$841$$ 57.9787 1.99927
$$842$$ 31.0344 1.06952
$$843$$ 0 0
$$844$$ 11.2705 0.387947
$$845$$ 18.7082 0.643582
$$846$$ 0 0
$$847$$ 29.4164 1.01076
$$848$$ −0.472136 −0.0162132
$$849$$ 0 0
$$850$$ −44.0689 −1.51155
$$851$$ 2.85410 0.0978374
$$852$$ 0 0
$$853$$ 29.7426 1.01837 0.509184 0.860657i $$-0.329947\pi$$
0.509184 + 0.860657i $$0.329947\pi$$
$$854$$ 34.3607 1.17580
$$855$$ 0 0
$$856$$ −7.32624 −0.250406
$$857$$ 9.05573 0.309338 0.154669 0.987966i $$-0.450569\pi$$
0.154669 + 0.987966i $$0.450569\pi$$
$$858$$ 0 0
$$859$$ 53.4164 1.82254 0.911272 0.411805i $$-0.135101\pi$$
0.911272 + 0.411805i $$0.135101\pi$$
$$860$$ 20.1803 0.688144
$$861$$ 0 0
$$862$$ 12.3607 0.421006
$$863$$ −19.4164 −0.660942 −0.330471 0.943816i $$-0.607208\pi$$
−0.330471 + 0.943816i $$0.607208\pi$$
$$864$$ 0 0
$$865$$ −1.81966 −0.0618703
$$866$$ 20.6738 0.702523
$$867$$ 0 0
$$868$$ −23.8885 −0.810830
$$869$$ −11.8328 −0.401401
$$870$$ 0 0
$$871$$ −3.11146 −0.105428
$$872$$ −2.94427 −0.0997056
$$873$$ 0 0
$$874$$ 12.7639 0.431746
$$875$$ 60.5410 2.04666
$$876$$ 0 0
$$877$$ −16.8328 −0.568404 −0.284202 0.958764i $$-0.591729\pi$$
−0.284202 + 0.958764i $$0.591729\pi$$
$$878$$ −7.79837 −0.263182
$$879$$ 0 0
$$880$$ −5.32624 −0.179547
$$881$$ −19.7426 −0.665147 −0.332573 0.943077i $$-0.607917\pi$$
−0.332573 + 0.943077i $$0.607917\pi$$
$$882$$ 0 0
$$883$$ −33.3050 −1.12080 −0.560400 0.828222i $$-0.689353\pi$$
−0.560400 + 0.828222i $$0.689353\pi$$
$$884$$ −12.7639 −0.429297
$$885$$ 0 0
$$886$$ 15.2705 0.513023
$$887$$ −27.8885 −0.936406 −0.468203 0.883621i $$-0.655098\pi$$
−0.468203 + 0.883621i $$0.655098\pi$$
$$888$$ 0 0
$$889$$ 27.4164 0.919517
$$890$$ 5.88854 0.197384
$$891$$ 0 0
$$892$$ 14.1803 0.474793
$$893$$ 5.52786 0.184983
$$894$$ 0 0
$$895$$ −48.7639 −1.63000
$$896$$ 3.23607 0.108109
$$897$$ 0 0
$$898$$ −28.4721 −0.950127
$$899$$ 68.8460 2.29614
$$900$$ 0 0
$$901$$ −2.11146 −0.0703428
$$902$$ 13.2918 0.442568
$$903$$ 0 0
$$904$$ −6.94427 −0.230963
$$905$$ −55.7771 −1.85409
$$906$$ 0 0
$$907$$ −55.8885 −1.85575 −0.927874 0.372893i $$-0.878366\pi$$
−0.927874 + 0.372893i $$0.878366\pi$$
$$908$$ 4.29180 0.142428
$$909$$ 0 0
$$910$$ 35.5967 1.18002
$$911$$ −13.8885 −0.460148 −0.230074 0.973173i $$-0.573897\pi$$
−0.230074 + 0.973173i $$0.573897\pi$$
$$912$$ 0 0
$$913$$ 20.0000 0.661903
$$914$$ 24.7639 0.819118
$$915$$ 0 0
$$916$$ −23.1246 −0.764059
$$917$$ −73.3050 −2.42074
$$918$$ 0 0
$$919$$ −5.88854 −0.194245 −0.0971226 0.995272i $$-0.530964\pi$$
−0.0971226 + 0.995272i $$0.530964\pi$$
$$920$$ −11.0000 −0.362659
$$921$$ 0 0
$$922$$ −38.9443 −1.28256
$$923$$ 8.40325 0.276596
$$924$$ 0 0
$$925$$ −9.85410 −0.324001
$$926$$ 4.56231 0.149927
$$927$$ 0 0
$$928$$ −9.32624 −0.306149
$$929$$ 27.4508 0.900633 0.450317 0.892869i $$-0.351311\pi$$
0.450317 + 0.892869i $$0.351311\pi$$
$$930$$ 0 0
$$931$$ 15.5279 0.508905
$$932$$ 6.56231 0.214955
$$933$$ 0 0
$$934$$ −24.3607 −0.797106
$$935$$ −23.8197 −0.778986
$$936$$ 0 0
$$937$$ −48.0476 −1.56965 −0.784823 0.619720i $$-0.787246\pi$$
−0.784823 + 0.619720i $$0.787246\pi$$
$$938$$ 3.52786 0.115189
$$939$$ 0 0
$$940$$ −4.76393 −0.155382
$$941$$ 26.1803 0.853455 0.426727 0.904380i $$-0.359666\pi$$
0.426727 + 0.904380i $$0.359666\pi$$
$$942$$ 0 0
$$943$$ 27.4508 0.893923
$$944$$ 4.76393 0.155053
$$945$$ 0 0
$$946$$ 7.23607 0.235265
$$947$$ −18.8328 −0.611984 −0.305992 0.952034i $$-0.598988\pi$$
−0.305992 + 0.952034i $$0.598988\pi$$
$$948$$ 0 0
$$949$$ −20.2361 −0.656891
$$950$$ −44.0689 −1.42978
$$951$$ 0 0
$$952$$ 14.4721 0.469045
$$953$$ −11.4508 −0.370929 −0.185465 0.982651i $$-0.559379\pi$$
−0.185465 + 0.982651i $$0.559379\pi$$
$$954$$ 0 0
$$955$$ −27.3262 −0.884256
$$956$$ −9.85410 −0.318704
$$957$$ 0 0
$$958$$ 36.5623 1.18127
$$959$$ 11.8885 0.383901
$$960$$ 0 0
$$961$$ 23.4934 0.757852
$$962$$ −2.85410 −0.0920199
$$963$$ 0 0
$$964$$ −1.52786 −0.0492092
$$965$$ −15.4164 −0.496272
$$966$$ 0 0
$$967$$ 45.2705 1.45580 0.727901 0.685683i $$-0.240496\pi$$
0.727901 + 0.685683i $$0.240496\pi$$
$$968$$ 9.09017 0.292169
$$969$$ 0 0
$$970$$ −1.81966 −0.0584258
$$971$$ −42.3262 −1.35831 −0.679157 0.733993i $$-0.737654\pi$$
−0.679157 + 0.733993i $$0.737654\pi$$
$$972$$ 0 0
$$973$$ −15.7082 −0.503582
$$974$$ −37.3050 −1.19533
$$975$$ 0 0
$$976$$ 10.6180 0.339875
$$977$$ −43.5279 −1.39258 −0.696290 0.717761i $$-0.745167\pi$$
−0.696290 + 0.717761i $$0.745167\pi$$
$$978$$ 0 0
$$979$$ 2.11146 0.0674824
$$980$$ −13.3820 −0.427471
$$981$$ 0 0
$$982$$ −28.4508 −0.907903
$$983$$ 31.7771 1.01353 0.506766 0.862084i $$-0.330841\pi$$
0.506766 + 0.862084i $$0.330841\pi$$
$$984$$ 0 0
$$985$$ −29.0132 −0.924436
$$986$$ −41.7082 −1.32826
$$987$$ 0 0
$$988$$ −12.7639 −0.406075
$$989$$ 14.9443 0.475200
$$990$$ 0 0
$$991$$ 33.1033 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$992$$ −7.38197 −0.234378
$$993$$ 0 0
$$994$$ −9.52786 −0.302205
$$995$$ −11.7771 −0.373359
$$996$$ 0 0
$$997$$ −17.7771 −0.563006 −0.281503 0.959560i $$-0.590833\pi$$
−0.281503 + 0.959560i $$0.590833\pi$$
$$998$$ −10.2918 −0.325781
$$999$$ 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 666.2.a.i.1.1 2
3.2 odd 2 74.2.a.b.1.1 2
4.3 odd 2 5328.2.a.bc.1.1 2
12.11 even 2 592.2.a.g.1.2 2
15.2 even 4 1850.2.b.j.149.4 4
15.8 even 4 1850.2.b.j.149.1 4
15.14 odd 2 1850.2.a.t.1.2 2
21.20 even 2 3626.2.a.s.1.2 2
24.5 odd 2 2368.2.a.y.1.2 2
24.11 even 2 2368.2.a.u.1.1 2
33.32 even 2 8954.2.a.j.1.1 2
111.110 odd 2 2738.2.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 2 3.2 odd 2
592.2.a.g.1.2 2 12.11 even 2
666.2.a.i.1.1 2 1.1 even 1 trivial
1850.2.a.t.1.2 2 15.14 odd 2
1850.2.b.j.149.1 4 15.8 even 4
1850.2.b.j.149.4 4 15.2 even 4
2368.2.a.u.1.1 2 24.11 even 2
2368.2.a.y.1.2 2 24.5 odd 2
2738.2.a.g.1.1 2 111.110 odd 2
3626.2.a.s.1.2 2 21.20 even 2
5328.2.a.bc.1.1 2 4.3 odd 2
8954.2.a.j.1.1 2 33.32 even 2