# Properties

 Label 3040.2.a.l.1.2 Level $3040$ Weight $2$ Character 3040.1 Self dual yes Analytic conductor $24.275$ Analytic rank $1$ 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] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.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: $$24.2745222145$$ Analytic rank: $$1$$ 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: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 3040.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.539189 q^{3} +1.00000 q^{5} +0.630898 q^{7} -2.70928 q^{9} +O(q^{10})$$ $$q+0.539189 q^{3} +1.00000 q^{5} +0.630898 q^{7} -2.70928 q^{9} -1.70928 q^{11} -3.17009 q^{13} +0.539189 q^{15} +1.41855 q^{17} +1.00000 q^{19} +0.340173 q^{21} -4.04945 q^{23} +1.00000 q^{25} -3.07838 q^{27} +3.75872 q^{29} +1.41855 q^{31} -0.921622 q^{33} +0.630898 q^{35} -0.986669 q^{37} -1.70928 q^{39} -9.26180 q^{41} +5.70928 q^{43} -2.70928 q^{45} -4.04945 q^{47} -6.60197 q^{49} +0.764867 q^{51} -7.32684 q^{53} -1.70928 q^{55} +0.539189 q^{57} -13.0205 q^{59} -13.1278 q^{61} -1.70928 q^{63} -3.17009 q^{65} +6.14116 q^{67} -2.18342 q^{69} +7.94214 q^{71} -9.91548 q^{73} +0.539189 q^{75} -1.07838 q^{77} +5.02052 q^{79} +6.46800 q^{81} +3.86603 q^{83} +1.41855 q^{85} +2.02666 q^{87} +5.60197 q^{89} -2.00000 q^{91} +0.764867 q^{93} +1.00000 q^{95} -0.275126 q^{97} +4.63090 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10})$$ 3 * q + 3 * q^5 - 2 * q^7 - q^9 $$3 q + 3 q^{5} - 2 q^{7} - q^{9} + 2 q^{11} - 4 q^{13} - 10 q^{17} + 3 q^{19} - 10 q^{21} + 6 q^{23} + 3 q^{25} - 6 q^{27} - 14 q^{29} - 10 q^{31} - 6 q^{33} - 2 q^{35} - 2 q^{37} + 2 q^{39} - 20 q^{41} + 10 q^{43} - q^{45} + 6 q^{47} - q^{49} + 12 q^{51} - 10 q^{53} + 2 q^{55} - 6 q^{59} - 18 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} - 2 q^{69} - 6 q^{71} + 2 q^{73} - 18 q^{79} - 13 q^{81} - 2 q^{83} - 10 q^{85} + 8 q^{87} - 2 q^{89} - 6 q^{91} + 12 q^{93} + 3 q^{95} + 6 q^{97} + 10 q^{99}+O(q^{100})$$ 3 * q + 3 * q^5 - 2 * q^7 - q^9 + 2 * q^11 - 4 * q^13 - 10 * q^17 + 3 * q^19 - 10 * q^21 + 6 * q^23 + 3 * q^25 - 6 * q^27 - 14 * q^29 - 10 * q^31 - 6 * q^33 - 2 * q^35 - 2 * q^37 + 2 * q^39 - 20 * q^41 + 10 * q^43 - q^45 + 6 * q^47 - q^49 + 12 * q^51 - 10 * q^53 + 2 * q^55 - 6 * q^59 - 18 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 - 2 * q^69 - 6 * q^71 + 2 * q^73 - 18 * q^79 - 13 * q^81 - 2 * q^83 - 10 * q^85 + 8 * q^87 - 2 * q^89 - 6 * q^91 + 12 * q^93 + 3 * q^95 + 6 * q^97 + 10 * 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
$$3$$ 0.539189 0.311301 0.155650 0.987812i $$-0.450253\pi$$
0.155650 + 0.987812i $$0.450253\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0.630898 0.238457 0.119228 0.992867i $$-0.461958\pi$$
0.119228 + 0.992867i $$0.461958\pi$$
$$8$$ 0 0
$$9$$ −2.70928 −0.903092
$$10$$ 0 0
$$11$$ −1.70928 −0.515366 −0.257683 0.966230i $$-0.582959\pi$$
−0.257683 + 0.966230i $$0.582959\pi$$
$$12$$ 0 0
$$13$$ −3.17009 −0.879224 −0.439612 0.898188i $$-0.644884\pi$$
−0.439612 + 0.898188i $$0.644884\pi$$
$$14$$ 0 0
$$15$$ 0.539189 0.139218
$$16$$ 0 0
$$17$$ 1.41855 0.344049 0.172025 0.985093i $$-0.444969\pi$$
0.172025 + 0.985093i $$0.444969\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0.340173 0.0742318
$$22$$ 0 0
$$23$$ −4.04945 −0.844368 −0.422184 0.906510i $$-0.638736\pi$$
−0.422184 + 0.906510i $$0.638736\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −3.07838 −0.592434
$$28$$ 0 0
$$29$$ 3.75872 0.697977 0.348989 0.937127i $$-0.386525\pi$$
0.348989 + 0.937127i $$0.386525\pi$$
$$30$$ 0 0
$$31$$ 1.41855 0.254779 0.127390 0.991853i $$-0.459340\pi$$
0.127390 + 0.991853i $$0.459340\pi$$
$$32$$ 0 0
$$33$$ −0.921622 −0.160434
$$34$$ 0 0
$$35$$ 0.630898 0.106641
$$36$$ 0 0
$$37$$ −0.986669 −0.162207 −0.0811037 0.996706i $$-0.525844\pi$$
−0.0811037 + 0.996706i $$0.525844\pi$$
$$38$$ 0 0
$$39$$ −1.70928 −0.273703
$$40$$ 0 0
$$41$$ −9.26180 −1.44645 −0.723225 0.690613i $$-0.757341\pi$$
−0.723225 + 0.690613i $$0.757341\pi$$
$$42$$ 0 0
$$43$$ 5.70928 0.870656 0.435328 0.900272i $$-0.356632\pi$$
0.435328 + 0.900272i $$0.356632\pi$$
$$44$$ 0 0
$$45$$ −2.70928 −0.403875
$$46$$ 0 0
$$47$$ −4.04945 −0.590673 −0.295336 0.955393i $$-0.595432\pi$$
−0.295336 + 0.955393i $$0.595432\pi$$
$$48$$ 0 0
$$49$$ −6.60197 −0.943138
$$50$$ 0 0
$$51$$ 0.764867 0.107103
$$52$$ 0 0
$$53$$ −7.32684 −1.00642 −0.503210 0.864164i $$-0.667848\pi$$
−0.503210 + 0.864164i $$0.667848\pi$$
$$54$$ 0 0
$$55$$ −1.70928 −0.230479
$$56$$ 0 0
$$57$$ 0.539189 0.0714173
$$58$$ 0 0
$$59$$ −13.0205 −1.69513 −0.847564 0.530694i $$-0.821931\pi$$
−0.847564 + 0.530694i $$0.821931\pi$$
$$60$$ 0 0
$$61$$ −13.1278 −1.68085 −0.840423 0.541931i $$-0.817693\pi$$
−0.840423 + 0.541931i $$0.817693\pi$$
$$62$$ 0 0
$$63$$ −1.70928 −0.215348
$$64$$ 0 0
$$65$$ −3.17009 −0.393201
$$66$$ 0 0
$$67$$ 6.14116 0.750262 0.375131 0.926972i $$-0.377598\pi$$
0.375131 + 0.926972i $$0.377598\pi$$
$$68$$ 0 0
$$69$$ −2.18342 −0.262853
$$70$$ 0 0
$$71$$ 7.94214 0.942559 0.471279 0.881984i $$-0.343792\pi$$
0.471279 + 0.881984i $$0.343792\pi$$
$$72$$ 0 0
$$73$$ −9.91548 −1.16052 −0.580260 0.814432i $$-0.697049\pi$$
−0.580260 + 0.814432i $$0.697049\pi$$
$$74$$ 0 0
$$75$$ 0.539189 0.0622602
$$76$$ 0 0
$$77$$ −1.07838 −0.122893
$$78$$ 0 0
$$79$$ 5.02052 0.564853 0.282426 0.959289i $$-0.408861\pi$$
0.282426 + 0.959289i $$0.408861\pi$$
$$80$$ 0 0
$$81$$ 6.46800 0.718667
$$82$$ 0 0
$$83$$ 3.86603 0.424352 0.212176 0.977231i $$-0.431945\pi$$
0.212176 + 0.977231i $$0.431945\pi$$
$$84$$ 0 0
$$85$$ 1.41855 0.153863
$$86$$ 0 0
$$87$$ 2.02666 0.217281
$$88$$ 0 0
$$89$$ 5.60197 0.593807 0.296904 0.954907i $$-0.404046\pi$$
0.296904 + 0.954907i $$0.404046\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 0.764867 0.0793130
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −0.275126 −0.0279348 −0.0139674 0.999902i $$-0.504446\pi$$
−0.0139674 + 0.999902i $$0.504446\pi$$
$$98$$ 0 0
$$99$$ 4.63090 0.465423
$$100$$ 0 0
$$101$$ −16.7298 −1.66468 −0.832338 0.554268i $$-0.812998\pi$$
−0.832338 + 0.554268i $$0.812998\pi$$
$$102$$ 0 0
$$103$$ −17.0628 −1.68125 −0.840623 0.541621i $$-0.817811\pi$$
−0.840623 + 0.541621i $$0.817811\pi$$
$$104$$ 0 0
$$105$$ 0.340173 0.0331975
$$106$$ 0 0
$$107$$ −5.77432 −0.558225 −0.279112 0.960258i $$-0.590040\pi$$
−0.279112 + 0.960258i $$0.590040\pi$$
$$108$$ 0 0
$$109$$ 1.44521 0.138426 0.0692131 0.997602i $$-0.477951\pi$$
0.0692131 + 0.997602i $$0.477951\pi$$
$$110$$ 0 0
$$111$$ −0.532001 −0.0504953
$$112$$ 0 0
$$113$$ −3.93495 −0.370169 −0.185085 0.982723i $$-0.559256\pi$$
−0.185085 + 0.982723i $$0.559256\pi$$
$$114$$ 0 0
$$115$$ −4.04945 −0.377613
$$116$$ 0 0
$$117$$ 8.58864 0.794020
$$118$$ 0 0
$$119$$ 0.894960 0.0820409
$$120$$ 0 0
$$121$$ −8.07838 −0.734398
$$122$$ 0 0
$$123$$ −4.99386 −0.450281
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −9.71646 −0.862197 −0.431098 0.902305i $$-0.641874\pi$$
−0.431098 + 0.902305i $$0.641874\pi$$
$$128$$ 0 0
$$129$$ 3.07838 0.271036
$$130$$ 0 0
$$131$$ 14.6537 1.28030 0.640149 0.768251i $$-0.278873\pi$$
0.640149 + 0.768251i $$0.278873\pi$$
$$132$$ 0 0
$$133$$ 0.630898 0.0547058
$$134$$ 0 0
$$135$$ −3.07838 −0.264945
$$136$$ 0 0
$$137$$ 8.52359 0.728219 0.364110 0.931356i $$-0.381373\pi$$
0.364110 + 0.931356i $$0.381373\pi$$
$$138$$ 0 0
$$139$$ 22.3896 1.89906 0.949531 0.313672i $$-0.101559\pi$$
0.949531 + 0.313672i $$0.101559\pi$$
$$140$$ 0 0
$$141$$ −2.18342 −0.183877
$$142$$ 0 0
$$143$$ 5.41855 0.453122
$$144$$ 0 0
$$145$$ 3.75872 0.312145
$$146$$ 0 0
$$147$$ −3.55971 −0.293600
$$148$$ 0 0
$$149$$ 6.38962 0.523458 0.261729 0.965141i $$-0.415707\pi$$
0.261729 + 0.965141i $$0.415707\pi$$
$$150$$ 0 0
$$151$$ −4.18342 −0.340442 −0.170221 0.985406i $$-0.554448\pi$$
−0.170221 + 0.985406i $$0.554448\pi$$
$$152$$ 0 0
$$153$$ −3.84324 −0.310708
$$154$$ 0 0
$$155$$ 1.41855 0.113941
$$156$$ 0 0
$$157$$ −15.3607 −1.22592 −0.612958 0.790115i $$-0.710021\pi$$
−0.612958 + 0.790115i $$0.710021\pi$$
$$158$$ 0 0
$$159$$ −3.95055 −0.313299
$$160$$ 0 0
$$161$$ −2.55479 −0.201345
$$162$$ 0 0
$$163$$ 22.1217 1.73270 0.866352 0.499434i $$-0.166459\pi$$
0.866352 + 0.499434i $$0.166459\pi$$
$$164$$ 0 0
$$165$$ −0.921622 −0.0717482
$$166$$ 0 0
$$167$$ 1.03612 0.0801772 0.0400886 0.999196i $$-0.487236\pi$$
0.0400886 + 0.999196i $$0.487236\pi$$
$$168$$ 0 0
$$169$$ −2.95055 −0.226966
$$170$$ 0 0
$$171$$ −2.70928 −0.207183
$$172$$ 0 0
$$173$$ −12.6153 −0.959123 −0.479562 0.877508i $$-0.659204\pi$$
−0.479562 + 0.877508i $$0.659204\pi$$
$$174$$ 0 0
$$175$$ 0.630898 0.0476914
$$176$$ 0 0
$$177$$ −7.02052 −0.527695
$$178$$ 0 0
$$179$$ 23.4908 1.75578 0.877892 0.478859i $$-0.158949\pi$$
0.877892 + 0.478859i $$0.158949\pi$$
$$180$$ 0 0
$$181$$ 3.36069 0.249798 0.124899 0.992169i $$-0.460139\pi$$
0.124899 + 0.992169i $$0.460139\pi$$
$$182$$ 0 0
$$183$$ −7.07838 −0.523249
$$184$$ 0 0
$$185$$ −0.986669 −0.0725413
$$186$$ 0 0
$$187$$ −2.42469 −0.177311
$$188$$ 0 0
$$189$$ −1.94214 −0.141270
$$190$$ 0 0
$$191$$ −21.4596 −1.55276 −0.776381 0.630264i $$-0.782947\pi$$
−0.776381 + 0.630264i $$0.782947\pi$$
$$192$$ 0 0
$$193$$ 13.0856 0.941920 0.470960 0.882155i $$-0.343908\pi$$
0.470960 + 0.882155i $$0.343908\pi$$
$$194$$ 0 0
$$195$$ −1.70928 −0.122404
$$196$$ 0 0
$$197$$ −2.49693 −0.177899 −0.0889494 0.996036i $$-0.528351\pi$$
−0.0889494 + 0.996036i $$0.528351\pi$$
$$198$$ 0 0
$$199$$ −12.6803 −0.898886 −0.449443 0.893309i $$-0.648377\pi$$
−0.449443 + 0.893309i $$0.648377\pi$$
$$200$$ 0 0
$$201$$ 3.31124 0.233557
$$202$$ 0 0
$$203$$ 2.37137 0.166438
$$204$$ 0 0
$$205$$ −9.26180 −0.646872
$$206$$ 0 0
$$207$$ 10.9711 0.762542
$$208$$ 0 0
$$209$$ −1.70928 −0.118233
$$210$$ 0 0
$$211$$ 16.8638 1.16095 0.580475 0.814278i $$-0.302867\pi$$
0.580475 + 0.814278i $$0.302867\pi$$
$$212$$ 0 0
$$213$$ 4.28231 0.293419
$$214$$ 0 0
$$215$$ 5.70928 0.389369
$$216$$ 0 0
$$217$$ 0.894960 0.0607539
$$218$$ 0 0
$$219$$ −5.34632 −0.361271
$$220$$ 0 0
$$221$$ −4.49693 −0.302496
$$222$$ 0 0
$$223$$ −8.22568 −0.550832 −0.275416 0.961325i $$-0.588816\pi$$
−0.275416 + 0.961325i $$0.588816\pi$$
$$224$$ 0 0
$$225$$ −2.70928 −0.180618
$$226$$ 0 0
$$227$$ −9.19287 −0.610152 −0.305076 0.952328i $$-0.598682\pi$$
−0.305076 + 0.952328i $$0.598682\pi$$
$$228$$ 0 0
$$229$$ 3.68261 0.243354 0.121677 0.992570i $$-0.461173\pi$$
0.121677 + 0.992570i $$0.461173\pi$$
$$230$$ 0 0
$$231$$ −0.581449 −0.0382566
$$232$$ 0 0
$$233$$ 7.17727 0.470199 0.235099 0.971971i $$-0.424458\pi$$
0.235099 + 0.971971i $$0.424458\pi$$
$$234$$ 0 0
$$235$$ −4.04945 −0.264157
$$236$$ 0 0
$$237$$ 2.70701 0.175839
$$238$$ 0 0
$$239$$ −10.1256 −0.654968 −0.327484 0.944857i $$-0.606201\pi$$
−0.327484 + 0.944857i $$0.606201\pi$$
$$240$$ 0 0
$$241$$ −25.3607 −1.63363 −0.816813 0.576903i $$-0.804261\pi$$
−0.816813 + 0.576903i $$0.804261\pi$$
$$242$$ 0 0
$$243$$ 12.7226 0.816156
$$244$$ 0 0
$$245$$ −6.60197 −0.421784
$$246$$ 0 0
$$247$$ −3.17009 −0.201708
$$248$$ 0 0
$$249$$ 2.08452 0.132101
$$250$$ 0 0
$$251$$ −9.26180 −0.584599 −0.292300 0.956327i $$-0.594420\pi$$
−0.292300 + 0.956327i $$0.594420\pi$$
$$252$$ 0 0
$$253$$ 6.92162 0.435159
$$254$$ 0 0
$$255$$ 0.764867 0.0478978
$$256$$ 0 0
$$257$$ −6.34736 −0.395937 −0.197969 0.980208i $$-0.563434\pi$$
−0.197969 + 0.980208i $$0.563434\pi$$
$$258$$ 0 0
$$259$$ −0.622487 −0.0386795
$$260$$ 0 0
$$261$$ −10.1834 −0.630338
$$262$$ 0 0
$$263$$ −1.79380 −0.110610 −0.0553051 0.998470i $$-0.517613\pi$$
−0.0553051 + 0.998470i $$0.517613\pi$$
$$264$$ 0 0
$$265$$ −7.32684 −0.450084
$$266$$ 0 0
$$267$$ 3.02052 0.184853
$$268$$ 0 0
$$269$$ −6.70701 −0.408933 −0.204467 0.978874i $$-0.565546\pi$$
−0.204467 + 0.978874i $$0.565546\pi$$
$$270$$ 0 0
$$271$$ 12.8143 0.778414 0.389207 0.921150i $$-0.372749\pi$$
0.389207 + 0.921150i $$0.372749\pi$$
$$272$$ 0 0
$$273$$ −1.07838 −0.0652664
$$274$$ 0 0
$$275$$ −1.70928 −0.103073
$$276$$ 0 0
$$277$$ 4.15676 0.249755 0.124878 0.992172i $$-0.460146\pi$$
0.124878 + 0.992172i $$0.460146\pi$$
$$278$$ 0 0
$$279$$ −3.84324 −0.230089
$$280$$ 0 0
$$281$$ −29.0928 −1.73553 −0.867764 0.496976i $$-0.834444\pi$$
−0.867764 + 0.496976i $$0.834444\pi$$
$$282$$ 0 0
$$283$$ 12.1340 0.721290 0.360645 0.932703i $$-0.382557\pi$$
0.360645 + 0.932703i $$0.382557\pi$$
$$284$$ 0 0
$$285$$ 0.539189 0.0319388
$$286$$ 0 0
$$287$$ −5.84324 −0.344916
$$288$$ 0 0
$$289$$ −14.9877 −0.881630
$$290$$ 0 0
$$291$$ −0.148345 −0.00869614
$$292$$ 0 0
$$293$$ 11.9227 0.696530 0.348265 0.937396i $$-0.386771\pi$$
0.348265 + 0.937396i $$0.386771\pi$$
$$294$$ 0 0
$$295$$ −13.0205 −0.758084
$$296$$ 0 0
$$297$$ 5.26180 0.305320
$$298$$ 0 0
$$299$$ 12.8371 0.742389
$$300$$ 0 0
$$301$$ 3.60197 0.207614
$$302$$ 0 0
$$303$$ −9.02052 −0.518215
$$304$$ 0 0
$$305$$ −13.1278 −0.751697
$$306$$ 0 0
$$307$$ 3.02174 0.172460 0.0862299 0.996275i $$-0.472518\pi$$
0.0862299 + 0.996275i $$0.472518\pi$$
$$308$$ 0 0
$$309$$ −9.20006 −0.523373
$$310$$ 0 0
$$311$$ −27.3835 −1.55277 −0.776387 0.630256i $$-0.782950\pi$$
−0.776387 + 0.630256i $$0.782950\pi$$
$$312$$ 0 0
$$313$$ 15.1773 0.857870 0.428935 0.903335i $$-0.358889\pi$$
0.428935 + 0.903335i $$0.358889\pi$$
$$314$$ 0 0
$$315$$ −1.70928 −0.0963068
$$316$$ 0 0
$$317$$ 13.6092 0.764366 0.382183 0.924087i $$-0.375172\pi$$
0.382183 + 0.924087i $$0.375172\pi$$
$$318$$ 0 0
$$319$$ −6.42469 −0.359714
$$320$$ 0 0
$$321$$ −3.11345 −0.173776
$$322$$ 0 0
$$323$$ 1.41855 0.0789303
$$324$$ 0 0
$$325$$ −3.17009 −0.175845
$$326$$ 0 0
$$327$$ 0.779243 0.0430922
$$328$$ 0 0
$$329$$ −2.55479 −0.140850
$$330$$ 0 0
$$331$$ −6.92162 −0.380447 −0.190223 0.981741i $$-0.560921\pi$$
−0.190223 + 0.981741i $$0.560921\pi$$
$$332$$ 0 0
$$333$$ 2.67316 0.146488
$$334$$ 0 0
$$335$$ 6.14116 0.335527
$$336$$ 0 0
$$337$$ 22.6875 1.23587 0.617934 0.786230i $$-0.287970\pi$$
0.617934 + 0.786230i $$0.287970\pi$$
$$338$$ 0 0
$$339$$ −2.12168 −0.115234
$$340$$ 0 0
$$341$$ −2.42469 −0.131305
$$342$$ 0 0
$$343$$ −8.58145 −0.463355
$$344$$ 0 0
$$345$$ −2.18342 −0.117551
$$346$$ 0 0
$$347$$ 28.8287 1.54761 0.773803 0.633427i $$-0.218352\pi$$
0.773803 + 0.633427i $$0.218352\pi$$
$$348$$ 0 0
$$349$$ 10.2823 0.550400 0.275200 0.961387i $$-0.411256\pi$$
0.275200 + 0.961387i $$0.411256\pi$$
$$350$$ 0 0
$$351$$ 9.75872 0.520882
$$352$$ 0 0
$$353$$ −21.6163 −1.15052 −0.575261 0.817970i $$-0.695099\pi$$
−0.575261 + 0.817970i $$0.695099\pi$$
$$354$$ 0 0
$$355$$ 7.94214 0.421525
$$356$$ 0 0
$$357$$ 0.482553 0.0255394
$$358$$ 0 0
$$359$$ 15.0700 0.795362 0.397681 0.917524i $$-0.369815\pi$$
0.397681 + 0.917524i $$0.369815\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −4.35577 −0.228619
$$364$$ 0 0
$$365$$ −9.91548 −0.519000
$$366$$ 0 0
$$367$$ 29.3256 1.53078 0.765392 0.643564i $$-0.222545\pi$$
0.765392 + 0.643564i $$0.222545\pi$$
$$368$$ 0 0
$$369$$ 25.0928 1.30628
$$370$$ 0 0
$$371$$ −4.62249 −0.239988
$$372$$ 0 0
$$373$$ 0.0917087 0.00474850 0.00237425 0.999997i $$-0.499244\pi$$
0.00237425 + 0.999997i $$0.499244\pi$$
$$374$$ 0 0
$$375$$ 0.539189 0.0278436
$$376$$ 0 0
$$377$$ −11.9155 −0.613678
$$378$$ 0 0
$$379$$ −11.1506 −0.572768 −0.286384 0.958115i $$-0.592453\pi$$
−0.286384 + 0.958115i $$0.592453\pi$$
$$380$$ 0 0
$$381$$ −5.23901 −0.268403
$$382$$ 0 0
$$383$$ 14.9060 0.761662 0.380831 0.924645i $$-0.375638\pi$$
0.380831 + 0.924645i $$0.375638\pi$$
$$384$$ 0 0
$$385$$ −1.07838 −0.0549592
$$386$$ 0 0
$$387$$ −15.4680 −0.786283
$$388$$ 0 0
$$389$$ 2.71154 0.137481 0.0687403 0.997635i $$-0.478102\pi$$
0.0687403 + 0.997635i $$0.478102\pi$$
$$390$$ 0 0
$$391$$ −5.74435 −0.290504
$$392$$ 0 0
$$393$$ 7.90110 0.398558
$$394$$ 0 0
$$395$$ 5.02052 0.252610
$$396$$ 0 0
$$397$$ 33.3340 1.67299 0.836494 0.547977i $$-0.184602\pi$$
0.836494 + 0.547977i $$0.184602\pi$$
$$398$$ 0 0
$$399$$ 0.340173 0.0170299
$$400$$ 0 0
$$401$$ 7.88882 0.393949 0.196974 0.980409i $$-0.436888\pi$$
0.196974 + 0.980409i $$0.436888\pi$$
$$402$$ 0 0
$$403$$ −4.49693 −0.224008
$$404$$ 0 0
$$405$$ 6.46800 0.321397
$$406$$ 0 0
$$407$$ 1.68649 0.0835962
$$408$$ 0 0
$$409$$ −1.97334 −0.0975753 −0.0487876 0.998809i $$-0.515536\pi$$
−0.0487876 + 0.998809i $$0.515536\pi$$
$$410$$ 0 0
$$411$$ 4.59583 0.226695
$$412$$ 0 0
$$413$$ −8.21461 −0.404215
$$414$$ 0 0
$$415$$ 3.86603 0.189776
$$416$$ 0 0
$$417$$ 12.0722 0.591180
$$418$$ 0 0
$$419$$ 16.9627 0.828680 0.414340 0.910122i $$-0.364012\pi$$
0.414340 + 0.910122i $$0.364012\pi$$
$$420$$ 0 0
$$421$$ −0.366835 −0.0178784 −0.00893922 0.999960i $$-0.502845\pi$$
−0.00893922 + 0.999960i $$0.502845\pi$$
$$422$$ 0 0
$$423$$ 10.9711 0.533432
$$424$$ 0 0
$$425$$ 1.41855 0.0688098
$$426$$ 0 0
$$427$$ −8.28231 −0.400809
$$428$$ 0 0
$$429$$ 2.92162 0.141057
$$430$$ 0 0
$$431$$ 8.39803 0.404519 0.202259 0.979332i $$-0.435172\pi$$
0.202259 + 0.979332i $$0.435172\pi$$
$$432$$ 0 0
$$433$$ 31.2690 1.50269 0.751346 0.659909i $$-0.229405\pi$$
0.751346 + 0.659909i $$0.229405\pi$$
$$434$$ 0 0
$$435$$ 2.02666 0.0971710
$$436$$ 0 0
$$437$$ −4.04945 −0.193711
$$438$$ 0 0
$$439$$ −19.0784 −0.910561 −0.455281 0.890348i $$-0.650461\pi$$
−0.455281 + 0.890348i $$0.650461\pi$$
$$440$$ 0 0
$$441$$ 17.8865 0.851740
$$442$$ 0 0
$$443$$ −8.04945 −0.382441 −0.191220 0.981547i $$-0.561245\pi$$
−0.191220 + 0.981547i $$0.561245\pi$$
$$444$$ 0 0
$$445$$ 5.60197 0.265559
$$446$$ 0 0
$$447$$ 3.44521 0.162953
$$448$$ 0 0
$$449$$ −6.92162 −0.326652 −0.163326 0.986572i $$-0.552222\pi$$
−0.163326 + 0.986572i $$0.552222\pi$$
$$450$$ 0 0
$$451$$ 15.8310 0.745451
$$452$$ 0 0
$$453$$ −2.25565 −0.105980
$$454$$ 0 0
$$455$$ −2.00000 −0.0937614
$$456$$ 0 0
$$457$$ 14.6803 0.686718 0.343359 0.939204i $$-0.388435\pi$$
0.343359 + 0.939204i $$0.388435\pi$$
$$458$$ 0 0
$$459$$ −4.36683 −0.203826
$$460$$ 0 0
$$461$$ −33.1506 −1.54398 −0.771989 0.635636i $$-0.780738\pi$$
−0.771989 + 0.635636i $$0.780738\pi$$
$$462$$ 0 0
$$463$$ −18.5197 −0.860684 −0.430342 0.902666i $$-0.641607\pi$$
−0.430342 + 0.902666i $$0.641607\pi$$
$$464$$ 0 0
$$465$$ 0.764867 0.0354698
$$466$$ 0 0
$$467$$ −23.5669 −1.09055 −0.545273 0.838259i $$-0.683574\pi$$
−0.545273 + 0.838259i $$0.683574\pi$$
$$468$$ 0 0
$$469$$ 3.87444 0.178905
$$470$$ 0 0
$$471$$ −8.28231 −0.381629
$$472$$ 0 0
$$473$$ −9.75872 −0.448707
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 19.8504 0.908889
$$478$$ 0 0
$$479$$ −10.2907 −0.470195 −0.235098 0.971972i $$-0.575541\pi$$
−0.235098 + 0.971972i $$0.575541\pi$$
$$480$$ 0 0
$$481$$ 3.12783 0.142617
$$482$$ 0 0
$$483$$ −1.37751 −0.0626790
$$484$$ 0 0
$$485$$ −0.275126 −0.0124928
$$486$$ 0 0
$$487$$ −28.9926 −1.31378 −0.656891 0.753986i $$-0.728129\pi$$
−0.656891 + 0.753986i $$0.728129\pi$$
$$488$$ 0 0
$$489$$ 11.9278 0.539392
$$490$$ 0 0
$$491$$ −18.5692 −0.838015 −0.419007 0.907983i $$-0.637622\pi$$
−0.419007 + 0.907983i $$0.637622\pi$$
$$492$$ 0 0
$$493$$ 5.33194 0.240139
$$494$$ 0 0
$$495$$ 4.63090 0.208143
$$496$$ 0 0
$$497$$ 5.01068 0.224760
$$498$$ 0 0
$$499$$ 19.7671 0.884898 0.442449 0.896794i $$-0.354110\pi$$
0.442449 + 0.896794i $$0.354110\pi$$
$$500$$ 0 0
$$501$$ 0.558663 0.0249592
$$502$$ 0 0
$$503$$ 20.8287 0.928705 0.464353 0.885650i $$-0.346287\pi$$
0.464353 + 0.885650i $$0.346287\pi$$
$$504$$ 0 0
$$505$$ −16.7298 −0.744466
$$506$$ 0 0
$$507$$ −1.59090 −0.0706546
$$508$$ 0 0
$$509$$ 24.5380 1.08763 0.543813 0.839206i $$-0.316980\pi$$
0.543813 + 0.839206i $$0.316980\pi$$
$$510$$ 0 0
$$511$$ −6.25565 −0.276734
$$512$$ 0 0
$$513$$ −3.07838 −0.135914
$$514$$ 0 0
$$515$$ −17.0628 −0.751876
$$516$$ 0 0
$$517$$ 6.92162 0.304413
$$518$$ 0 0
$$519$$ −6.80203 −0.298576
$$520$$ 0 0
$$521$$ −6.58145 −0.288339 −0.144169 0.989553i $$-0.546051\pi$$
−0.144169 + 0.989553i $$0.546051\pi$$
$$522$$ 0 0
$$523$$ 27.6586 1.20943 0.604713 0.796443i $$-0.293288\pi$$
0.604713 + 0.796443i $$0.293288\pi$$
$$524$$ 0 0
$$525$$ 0.340173 0.0148464
$$526$$ 0 0
$$527$$ 2.01229 0.0876566
$$528$$ 0 0
$$529$$ −6.60197 −0.287042
$$530$$ 0 0
$$531$$ 35.2762 1.53086
$$532$$ 0 0
$$533$$ 29.3607 1.27175
$$534$$ 0 0
$$535$$ −5.77432 −0.249646
$$536$$ 0 0
$$537$$ 12.6660 0.546577
$$538$$ 0 0
$$539$$ 11.2846 0.486061
$$540$$ 0 0
$$541$$ −16.9444 −0.728497 −0.364249 0.931302i $$-0.618674\pi$$
−0.364249 + 0.931302i $$0.618674\pi$$
$$542$$ 0 0
$$543$$ 1.81205 0.0777624
$$544$$ 0 0
$$545$$ 1.44521 0.0619061
$$546$$ 0 0
$$547$$ −15.4608 −0.661057 −0.330528 0.943796i $$-0.607227\pi$$
−0.330528 + 0.943796i $$0.607227\pi$$
$$548$$ 0 0
$$549$$ 35.5669 1.51796
$$550$$ 0 0
$$551$$ 3.75872 0.160127
$$552$$ 0 0
$$553$$ 3.16743 0.134693
$$554$$ 0 0
$$555$$ −0.532001 −0.0225822
$$556$$ 0 0
$$557$$ 40.3012 1.70762 0.853809 0.520587i $$-0.174287\pi$$
0.853809 + 0.520587i $$0.174287\pi$$
$$558$$ 0 0
$$559$$ −18.0989 −0.765502
$$560$$ 0 0
$$561$$ −1.30737 −0.0551971
$$562$$ 0 0
$$563$$ −18.2134 −0.767603 −0.383801 0.923416i $$-0.625385\pi$$
−0.383801 + 0.923416i $$0.625385\pi$$
$$564$$ 0 0
$$565$$ −3.93495 −0.165545
$$566$$ 0 0
$$567$$ 4.08065 0.171371
$$568$$ 0 0
$$569$$ −4.13009 −0.173143 −0.0865713 0.996246i $$-0.527591\pi$$
−0.0865713 + 0.996246i $$0.527591\pi$$
$$570$$ 0 0
$$571$$ −25.2267 −1.05571 −0.527853 0.849336i $$-0.677003\pi$$
−0.527853 + 0.849336i $$0.677003\pi$$
$$572$$ 0 0
$$573$$ −11.5708 −0.483376
$$574$$ 0 0
$$575$$ −4.04945 −0.168874
$$576$$ 0 0
$$577$$ 34.7480 1.44658 0.723290 0.690544i $$-0.242629\pi$$
0.723290 + 0.690544i $$0.242629\pi$$
$$578$$ 0 0
$$579$$ 7.05559 0.293220
$$580$$ 0 0
$$581$$ 2.43907 0.101190
$$582$$ 0 0
$$583$$ 12.5236 0.518674
$$584$$ 0 0
$$585$$ 8.58864 0.355096
$$586$$ 0 0
$$587$$ −7.91935 −0.326867 −0.163433 0.986554i $$-0.552257\pi$$
−0.163433 + 0.986554i $$0.552257\pi$$
$$588$$ 0 0
$$589$$ 1.41855 0.0584504
$$590$$ 0 0
$$591$$ −1.34632 −0.0553800
$$592$$ 0 0
$$593$$ 0.837101 0.0343756 0.0171878 0.999852i $$-0.494529\pi$$
0.0171878 + 0.999852i $$0.494529\pi$$
$$594$$ 0 0
$$595$$ 0.894960 0.0366898
$$596$$ 0 0
$$597$$ −6.83710 −0.279824
$$598$$ 0 0
$$599$$ −24.9216 −1.01827 −0.509135 0.860687i $$-0.670035\pi$$
−0.509135 + 0.860687i $$0.670035\pi$$
$$600$$ 0 0
$$601$$ −17.7275 −0.723121 −0.361560 0.932349i $$-0.617756\pi$$
−0.361560 + 0.932349i $$0.617756\pi$$
$$602$$ 0 0
$$603$$ −16.6381 −0.677555
$$604$$ 0 0
$$605$$ −8.07838 −0.328433
$$606$$ 0 0
$$607$$ 9.40749 0.381838 0.190919 0.981606i $$-0.438853\pi$$
0.190919 + 0.981606i $$0.438853\pi$$
$$608$$ 0 0
$$609$$ 1.27862 0.0518121
$$610$$ 0 0
$$611$$ 12.8371 0.519334
$$612$$ 0 0
$$613$$ 15.3919 0.621673 0.310836 0.950463i $$-0.399391\pi$$
0.310836 + 0.950463i $$0.399391\pi$$
$$614$$ 0 0
$$615$$ −4.99386 −0.201372
$$616$$ 0 0
$$617$$ 5.28846 0.212905 0.106453 0.994318i $$-0.466051\pi$$
0.106453 + 0.994318i $$0.466051\pi$$
$$618$$ 0 0
$$619$$ 28.9132 1.16212 0.581060 0.813861i $$-0.302638\pi$$
0.581060 + 0.813861i $$0.302638\pi$$
$$620$$ 0 0
$$621$$ 12.4657 0.500233
$$622$$ 0 0
$$623$$ 3.53427 0.141597
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −0.921622 −0.0368060
$$628$$ 0 0
$$629$$ −1.39964 −0.0558073
$$630$$ 0 0
$$631$$ −6.38962 −0.254367 −0.127183 0.991879i $$-0.540594\pi$$
−0.127183 + 0.991879i $$0.540594\pi$$
$$632$$ 0 0
$$633$$ 9.09275 0.361405
$$634$$ 0 0
$$635$$ −9.71646 −0.385586
$$636$$ 0 0
$$637$$ 20.9288 0.829230
$$638$$ 0 0
$$639$$ −21.5174 −0.851217
$$640$$ 0 0
$$641$$ 1.04718 0.0413612 0.0206806 0.999786i $$-0.493417\pi$$
0.0206806 + 0.999786i $$0.493417\pi$$
$$642$$ 0 0
$$643$$ −7.46800 −0.294509 −0.147255 0.989099i $$-0.547044\pi$$
−0.147255 + 0.989099i $$0.547044\pi$$
$$644$$ 0 0
$$645$$ 3.07838 0.121211
$$646$$ 0 0
$$647$$ 2.70313 0.106271 0.0531355 0.998587i $$-0.483078\pi$$
0.0531355 + 0.998587i $$0.483078\pi$$
$$648$$ 0 0
$$649$$ 22.2557 0.873611
$$650$$ 0 0
$$651$$ 0.482553 0.0189127
$$652$$ 0 0
$$653$$ −21.4329 −0.838735 −0.419368 0.907817i $$-0.637748\pi$$
−0.419368 + 0.907817i $$0.637748\pi$$
$$654$$ 0 0
$$655$$ 14.6537 0.572567
$$656$$ 0 0
$$657$$ 26.8638 1.04806
$$658$$ 0 0
$$659$$ 19.8576 0.773543 0.386772 0.922176i $$-0.373590\pi$$
0.386772 + 0.922176i $$0.373590\pi$$
$$660$$ 0 0
$$661$$ −2.83710 −0.110350 −0.0551752 0.998477i $$-0.517572\pi$$
−0.0551752 + 0.998477i $$0.517572\pi$$
$$662$$ 0 0
$$663$$ −2.42469 −0.0941673
$$664$$ 0 0
$$665$$ 0.630898 0.0244652
$$666$$ 0 0
$$667$$ −15.2208 −0.589350
$$668$$ 0 0
$$669$$ −4.43519 −0.171475
$$670$$ 0 0
$$671$$ 22.4391 0.866251
$$672$$ 0 0
$$673$$ 24.0338 0.926437 0.463218 0.886244i $$-0.346695\pi$$
0.463218 + 0.886244i $$0.346695\pi$$
$$674$$ 0 0
$$675$$ −3.07838 −0.118487
$$676$$ 0 0
$$677$$ 19.4114 0.746039 0.373020 0.927823i $$-0.378322\pi$$
0.373020 + 0.927823i $$0.378322\pi$$
$$678$$ 0 0
$$679$$ −0.173576 −0.00666125
$$680$$ 0 0
$$681$$ −4.95669 −0.189941
$$682$$ 0 0
$$683$$ 0.821503 0.0314339 0.0157170 0.999876i $$-0.494997\pi$$
0.0157170 + 0.999876i $$0.494997\pi$$
$$684$$ 0 0
$$685$$ 8.52359 0.325670
$$686$$ 0 0
$$687$$ 1.98562 0.0757563
$$688$$ 0 0
$$689$$ 23.2267 0.884868
$$690$$ 0 0
$$691$$ −4.54638 −0.172952 −0.0864762 0.996254i $$-0.527561\pi$$
−0.0864762 + 0.996254i $$0.527561\pi$$
$$692$$ 0 0
$$693$$ 2.92162 0.110983
$$694$$ 0 0
$$695$$ 22.3896 0.849287
$$696$$ 0 0
$$697$$ −13.1383 −0.497650
$$698$$ 0 0
$$699$$ 3.86991 0.146373
$$700$$ 0 0
$$701$$ 12.6453 0.477605 0.238803 0.971068i $$-0.423245\pi$$
0.238803 + 0.971068i $$0.423245\pi$$
$$702$$ 0 0
$$703$$ −0.986669 −0.0372129
$$704$$ 0 0
$$705$$ −2.18342 −0.0822323
$$706$$ 0 0
$$707$$ −10.5548 −0.396954
$$708$$ 0 0
$$709$$ −12.8059 −0.480936 −0.240468 0.970657i $$-0.577301\pi$$
−0.240468 + 0.970657i $$0.577301\pi$$
$$710$$ 0 0
$$711$$ −13.6020 −0.510114
$$712$$ 0 0
$$713$$ −5.74435 −0.215128
$$714$$ 0 0
$$715$$ 5.41855 0.202642
$$716$$ 0 0
$$717$$ −5.45959 −0.203892
$$718$$ 0 0
$$719$$ −1.90707 −0.0711217 −0.0355608 0.999368i $$-0.511322\pi$$
−0.0355608 + 0.999368i $$0.511322\pi$$
$$720$$ 0 0
$$721$$ −10.7649 −0.400905
$$722$$ 0 0
$$723$$ −13.6742 −0.508549
$$724$$ 0 0
$$725$$ 3.75872 0.139595
$$726$$ 0 0
$$727$$ 25.9239 0.961464 0.480732 0.876868i $$-0.340371\pi$$
0.480732 + 0.876868i $$0.340371\pi$$
$$728$$ 0 0
$$729$$ −12.5441 −0.464597
$$730$$ 0 0
$$731$$ 8.09890 0.299549
$$732$$ 0 0
$$733$$ 39.1605 1.44642 0.723212 0.690626i $$-0.242665\pi$$
0.723212 + 0.690626i $$0.242665\pi$$
$$734$$ 0 0
$$735$$ −3.55971 −0.131302
$$736$$ 0 0
$$737$$ −10.4969 −0.386659
$$738$$ 0 0
$$739$$ 8.31351 0.305817 0.152909 0.988240i $$-0.451136\pi$$
0.152909 + 0.988240i $$0.451136\pi$$
$$740$$ 0 0
$$741$$ −1.70928 −0.0627918
$$742$$ 0 0
$$743$$ −47.5597 −1.74480 −0.872398 0.488796i $$-0.837436\pi$$
−0.872398 + 0.488796i $$0.837436\pi$$
$$744$$ 0 0
$$745$$ 6.38962 0.234098
$$746$$ 0 0
$$747$$ −10.4741 −0.383229
$$748$$ 0 0
$$749$$ −3.64301 −0.133113
$$750$$ 0 0
$$751$$ −23.8622 −0.870742 −0.435371 0.900251i $$-0.643383\pi$$
−0.435371 + 0.900251i $$0.643383\pi$$
$$752$$ 0 0
$$753$$ −4.99386 −0.181986
$$754$$ 0 0
$$755$$ −4.18342 −0.152250
$$756$$ 0 0
$$757$$ 44.3402 1.61157 0.805785 0.592208i $$-0.201743\pi$$
0.805785 + 0.592208i $$0.201743\pi$$
$$758$$ 0 0
$$759$$ 3.73206 0.135465
$$760$$ 0 0
$$761$$ 0.577574 0.0209370 0.0104685 0.999945i $$-0.496668\pi$$
0.0104685 + 0.999945i $$0.496668\pi$$
$$762$$ 0 0
$$763$$ 0.911781 0.0330087
$$764$$ 0 0
$$765$$ −3.84324 −0.138953
$$766$$ 0 0
$$767$$ 41.2762 1.49040
$$768$$ 0 0
$$769$$ 21.7938 0.785904 0.392952 0.919559i $$-0.371454\pi$$
0.392952 + 0.919559i $$0.371454\pi$$
$$770$$ 0 0
$$771$$ −3.42243 −0.123256
$$772$$ 0 0
$$773$$ −43.4135 −1.56147 −0.780737 0.624860i $$-0.785156\pi$$
−0.780737 + 0.624860i $$0.785156\pi$$
$$774$$ 0 0
$$775$$ 1.41855 0.0509558
$$776$$ 0 0
$$777$$ −0.335638 −0.0120410
$$778$$ 0 0
$$779$$ −9.26180 −0.331838
$$780$$ 0 0
$$781$$ −13.5753 −0.485763
$$782$$ 0 0
$$783$$ −11.5708 −0.413506
$$784$$ 0 0
$$785$$ −15.3607 −0.548247
$$786$$ 0 0
$$787$$ 14.2134 0.506653 0.253326 0.967381i $$-0.418475\pi$$
0.253326 + 0.967381i $$0.418475\pi$$
$$788$$ 0 0
$$789$$ −0.967195 −0.0344331
$$790$$ 0 0
$$791$$ −2.48255 −0.0882694
$$792$$ 0 0
$$793$$ 41.6163 1.47784
$$794$$ 0 0
$$795$$ −3.95055 −0.140112
$$796$$ 0 0
$$797$$ −22.0482 −0.780988 −0.390494 0.920605i $$-0.627696\pi$$
−0.390494 + 0.920605i $$0.627696\pi$$
$$798$$ 0 0
$$799$$ −5.74435 −0.203220
$$800$$ 0 0
$$801$$ −15.1773 −0.536263
$$802$$ 0 0
$$803$$ 16.9483 0.598092
$$804$$ 0 0
$$805$$ −2.55479 −0.0900444
$$806$$ 0 0
$$807$$ −3.61634 −0.127301
$$808$$ 0 0
$$809$$ 7.49854 0.263635 0.131817 0.991274i $$-0.457919\pi$$
0.131817 + 0.991274i $$0.457919\pi$$
$$810$$ 0 0
$$811$$ 43.4063 1.52420 0.762100 0.647459i $$-0.224168\pi$$
0.762100 + 0.647459i $$0.224168\pi$$
$$812$$ 0 0
$$813$$ 6.90934 0.242321
$$814$$ 0 0
$$815$$ 22.1217 0.774889
$$816$$ 0 0
$$817$$ 5.70928 0.199742
$$818$$ 0 0
$$819$$ 5.41855 0.189339
$$820$$ 0 0
$$821$$ −26.7115 −0.932239 −0.466120 0.884722i $$-0.654348\pi$$
−0.466120 + 0.884722i $$0.654348\pi$$
$$822$$ 0 0
$$823$$ 22.2206 0.774561 0.387280 0.921962i $$-0.373415\pi$$
0.387280 + 0.921962i $$0.373415\pi$$
$$824$$ 0 0
$$825$$ −0.921622 −0.0320868
$$826$$ 0 0
$$827$$ −31.7431 −1.10382 −0.551908 0.833905i $$-0.686100\pi$$
−0.551908 + 0.833905i $$0.686100\pi$$
$$828$$ 0 0
$$829$$ 40.9093 1.42084 0.710420 0.703778i $$-0.248505\pi$$
0.710420 + 0.703778i $$0.248505\pi$$
$$830$$ 0 0
$$831$$ 2.24128 0.0777490
$$832$$ 0 0
$$833$$ −9.36523 −0.324486
$$834$$ 0 0
$$835$$ 1.03612 0.0358563
$$836$$ 0 0
$$837$$ −4.36683 −0.150940
$$838$$ 0 0
$$839$$ −14.3980 −0.497075 −0.248538 0.968622i $$-0.579950\pi$$
−0.248538 + 0.968622i $$0.579950\pi$$
$$840$$ 0 0
$$841$$ −14.8720 −0.512827
$$842$$ 0 0
$$843$$ −15.6865 −0.540271
$$844$$ 0 0
$$845$$ −2.95055 −0.101502
$$846$$ 0 0
$$847$$ −5.09663 −0.175122
$$848$$ 0 0
$$849$$ 6.54250 0.224538
$$850$$ 0 0
$$851$$ 3.99547 0.136963
$$852$$ 0 0
$$853$$ −24.1399 −0.826536 −0.413268 0.910610i $$-0.635613\pi$$
−0.413268 + 0.910610i $$0.635613\pi$$
$$854$$ 0 0
$$855$$ −2.70928 −0.0926553
$$856$$ 0 0
$$857$$ 47.4089 1.61946 0.809729 0.586804i $$-0.199614\pi$$
0.809729 + 0.586804i $$0.199614\pi$$
$$858$$ 0 0
$$859$$ −38.9504 −1.32897 −0.664485 0.747302i $$-0.731349\pi$$
−0.664485 + 0.747302i $$0.731349\pi$$
$$860$$ 0 0
$$861$$ −3.15061 −0.107373
$$862$$ 0 0
$$863$$ 20.5392 0.699162 0.349581 0.936906i $$-0.386324\pi$$
0.349581 + 0.936906i $$0.386324\pi$$
$$864$$ 0 0
$$865$$ −12.6153 −0.428933
$$866$$ 0 0
$$867$$ −8.08121 −0.274452
$$868$$ 0 0
$$869$$ −8.58145 −0.291106
$$870$$ 0 0
$$871$$ −19.4680 −0.659648
$$872$$ 0 0
$$873$$ 0.745393 0.0252277
$$874$$ 0 0
$$875$$ 0.630898 0.0213282
$$876$$ 0 0
$$877$$ 2.56198 0.0865118 0.0432559 0.999064i $$-0.486227\pi$$
0.0432559 + 0.999064i $$0.486227\pi$$
$$878$$ 0 0
$$879$$ 6.42857 0.216830
$$880$$ 0 0
$$881$$ −25.0289 −0.843246 −0.421623 0.906771i $$-0.638539\pi$$
−0.421623 + 0.906771i $$0.638539\pi$$
$$882$$ 0 0
$$883$$ 57.2411 1.92632 0.963158 0.268936i $$-0.0866719\pi$$
0.963158 + 0.268936i $$0.0866719\pi$$
$$884$$ 0 0
$$885$$ −7.02052 −0.235992
$$886$$ 0 0
$$887$$ 31.0049 1.04104 0.520522 0.853848i $$-0.325737\pi$$
0.520522 + 0.853848i $$0.325737\pi$$
$$888$$ 0 0
$$889$$ −6.13009 −0.205597
$$890$$ 0 0
$$891$$ −11.0556 −0.370376
$$892$$ 0 0
$$893$$ −4.04945 −0.135510
$$894$$ 0 0
$$895$$ 23.4908 0.785210
$$896$$ 0 0
$$897$$ 6.92162 0.231106
$$898$$ 0 0
$$899$$ 5.33194 0.177830
$$900$$ 0 0
$$901$$ −10.3935 −0.346258
$$902$$ 0 0
$$903$$ 1.94214 0.0646304
$$904$$ 0 0
$$905$$ 3.36069 0.111713
$$906$$ 0 0
$$907$$ −24.9783 −0.829389 −0.414695 0.909961i $$-0.636112\pi$$
−0.414695 + 0.909961i $$0.636112\pi$$
$$908$$ 0 0
$$909$$ 45.3256 1.50336
$$910$$ 0 0
$$911$$ 54.4678 1.80460 0.902300 0.431109i $$-0.141878\pi$$
0.902300 + 0.431109i $$0.141878\pi$$
$$912$$ 0 0
$$913$$ −6.60811 −0.218697
$$914$$ 0 0
$$915$$ −7.07838 −0.234004
$$916$$ 0 0
$$917$$ 9.24497 0.305296
$$918$$ 0 0
$$919$$ 19.7998 0.653134 0.326567 0.945174i $$-0.394108\pi$$
0.326567 + 0.945174i $$0.394108\pi$$
$$920$$ 0 0
$$921$$ 1.62929 0.0536869
$$922$$ 0 0
$$923$$ −25.1773 −0.828720
$$924$$ 0 0
$$925$$ −0.986669 −0.0324415
$$926$$ 0 0
$$927$$ 46.2278 1.51832
$$928$$ 0 0
$$929$$ −7.78992 −0.255579 −0.127790 0.991801i $$-0.540788\pi$$
−0.127790 + 0.991801i $$0.540788\pi$$
$$930$$ 0 0
$$931$$ −6.60197 −0.216371
$$932$$ 0 0
$$933$$ −14.7649 −0.483380
$$934$$ 0 0
$$935$$ −2.42469 −0.0792960
$$936$$ 0 0
$$937$$ 2.88058 0.0941046 0.0470523 0.998892i $$-0.485017\pi$$
0.0470523 + 0.998892i $$0.485017\pi$$
$$938$$ 0 0
$$939$$ 8.18342 0.267056
$$940$$ 0 0
$$941$$ −53.6163 −1.74784 −0.873921 0.486067i $$-0.838431\pi$$
−0.873921 + 0.486067i $$0.838431\pi$$
$$942$$ 0 0
$$943$$ 37.5052 1.22134
$$944$$ 0 0
$$945$$ −1.94214 −0.0631779
$$946$$ 0 0
$$947$$ 26.7708 0.869935 0.434968 0.900446i $$-0.356760\pi$$
0.434968 + 0.900446i $$0.356760\pi$$
$$948$$ 0 0
$$949$$ 31.4329 1.02036
$$950$$ 0 0
$$951$$ 7.33791 0.237948
$$952$$ 0 0
$$953$$ 16.1061 0.521727 0.260864 0.965376i $$-0.415993\pi$$
0.260864 + 0.965376i $$0.415993\pi$$
$$954$$ 0 0
$$955$$ −21.4596 −0.694416
$$956$$ 0 0
$$957$$ −3.46412 −0.111979
$$958$$ 0 0
$$959$$ 5.37751 0.173649
$$960$$ 0 0
$$961$$ −28.9877 −0.935088
$$962$$ 0 0
$$963$$ 15.6442 0.504128
$$964$$ 0 0
$$965$$ 13.0856 0.421239
$$966$$ 0 0
$$967$$ −17.8537 −0.574138 −0.287069 0.957910i $$-0.592681\pi$$
−0.287069 + 0.957910i $$0.592681\pi$$
$$968$$ 0 0
$$969$$ 0.764867 0.0245711
$$970$$ 0 0
$$971$$ −58.1978 −1.86766 −0.933828 0.357722i $$-0.883553\pi$$
−0.933828 + 0.357722i $$0.883553\pi$$
$$972$$ 0 0
$$973$$ 14.1256 0.452845
$$974$$ 0 0
$$975$$ −1.70928 −0.0547406
$$976$$ 0 0
$$977$$ 12.7766 0.408759 0.204380 0.978892i $$-0.434482\pi$$
0.204380 + 0.978892i $$0.434482\pi$$
$$978$$ 0 0
$$979$$ −9.57531 −0.306028
$$980$$ 0 0
$$981$$ −3.91548 −0.125012
$$982$$ 0 0
$$983$$ −40.0833 −1.27846 −0.639229 0.769016i $$-0.720747\pi$$
−0.639229 + 0.769016i $$0.720747\pi$$
$$984$$ 0 0
$$985$$ −2.49693 −0.0795588
$$986$$ 0 0
$$987$$ −1.37751 −0.0438467
$$988$$ 0 0
$$989$$ −23.1194 −0.735155
$$990$$ 0 0
$$991$$ −27.5318 −0.874577 −0.437289 0.899321i $$-0.644061\pi$$
−0.437289 + 0.899321i $$0.644061\pi$$
$$992$$ 0 0
$$993$$ −3.73206 −0.118433
$$994$$ 0 0
$$995$$ −12.6803 −0.401994
$$996$$ 0 0
$$997$$ 20.1399 0.637838 0.318919 0.947782i $$-0.396680\pi$$
0.318919 + 0.947782i $$0.396680\pi$$
$$998$$ 0 0
$$999$$ 3.03734 0.0960972
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 3040.2.a.l.1.2 3
4.3 odd 2 3040.2.a.m.1.2 yes 3
8.3 odd 2 6080.2.a.bu.1.2 3
8.5 even 2 6080.2.a.bt.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.l.1.2 3 1.1 even 1 trivial
3040.2.a.m.1.2 yes 3 4.3 odd 2
6080.2.a.bt.1.2 3 8.5 even 2
6080.2.a.bu.1.2 3 8.3 odd 2