# Properties

 Label 6080.2.a.e.1.1 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{15} +6.00000 q^{17} +1.00000 q^{19} -8.00000 q^{21} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} -8.00000 q^{33} -4.00000 q^{35} +8.00000 q^{37} -2.00000 q^{41} -1.00000 q^{45} +12.0000 q^{47} +9.00000 q^{49} -12.0000 q^{51} -4.00000 q^{53} -4.00000 q^{55} -2.00000 q^{57} -8.00000 q^{59} +14.0000 q^{61} +4.00000 q^{63} +2.00000 q^{67} -16.0000 q^{69} -8.00000 q^{71} -2.00000 q^{73} -2.00000 q^{75} +16.0000 q^{77} +4.00000 q^{79} -11.0000 q^{81} -12.0000 q^{83} -6.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} +16.0000 q^{93} -1.00000 q^{95} +4.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −8.00000 −1.74574
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ −8.00000 −1.39262
$$34$$ 0 0
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 0 0
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 0 0
$$69$$ −16.0000 −1.92617
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ −2.00000 −0.230940
$$76$$ 0 0
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −6.00000 −0.650791
$$86$$ 0 0
$$87$$ −12.0000 −1.28654
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 16.0000 1.65912
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ 8.00000 0.780720
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −16.0000 −1.51865
$$112$$ 0 0
$$113$$ −16.0000 −1.50515 −0.752577 0.658505i $$-0.771189\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 4.00000 0.360668
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −18.0000 −1.59724 −0.798621 0.601834i $$-0.794437\pi$$
−0.798621 + 0.601834i $$0.794437\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −24.0000 −2.02116
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ −18.0000 −1.48461
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 0 0
$$159$$ 8.00000 0.634441
$$160$$ 0 0
$$161$$ 32.0000 2.52195
$$162$$ 0 0
$$163$$ −24.0000 −1.87983 −0.939913 0.341415i $$-0.889094\pi$$
−0.939913 + 0.341415i $$0.889094\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ −6.00000 −0.464294 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 20.0000 1.52057 0.760286 0.649589i $$-0.225059\pi$$
0.760286 + 0.649589i $$0.225059\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 16.0000 1.20263
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ −28.0000 −2.06982
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 24.0000 1.75505
$$188$$ 0 0
$$189$$ 16.0000 1.16383
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 24.0000 1.68447
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 0 0
$$213$$ 16.0000 1.09630
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −32.0000 −2.17230
$$218$$ 0 0
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ −32.0000 −2.10545
$$232$$ 0 0
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ −9.00000 −0.574989
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 32.0000 2.01182
$$254$$ 0 0
$$255$$ 12.0000 0.751469
$$256$$ 0 0
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 32.0000 1.98838
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 4.00000 0.245718
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ −8.00000 −0.472225
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −4.00000 −0.233682 −0.116841 0.993151i $$-0.537277\pi$$
−0.116841 + 0.993151i $$0.537277\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ 16.0000 0.928414
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −12.0000 −0.689382
$$304$$ 0 0
$$305$$ −14.0000 −0.801638
$$306$$ 0 0
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ −4.00000 −0.225374
$$316$$ 0 0
$$317$$ −32.0000 −1.79730 −0.898650 0.438667i $$-0.855451\pi$$
−0.898650 + 0.438667i $$0.855451\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 36.0000 2.00932
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −4.00000 −0.221201
$$328$$ 0 0
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ 16.0000 0.879440 0.439720 0.898135i $$-0.355078\pi$$
0.439720 + 0.898135i $$0.355078\pi$$
$$332$$ 0 0
$$333$$ 8.00000 0.438397
$$334$$ 0 0
$$335$$ −2.00000 −0.109272
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 0 0
$$339$$ 32.0000 1.73800
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 8.00000 0.424596
$$356$$ 0 0
$$357$$ −48.0000 −2.54043
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ −16.0000 −0.830679
$$372$$ 0 0
$$373$$ 12.0000 0.621336 0.310668 0.950518i $$-0.399447\pi$$
0.310668 + 0.950518i $$0.399447\pi$$
$$374$$ 0 0
$$375$$ 2.00000 0.103280
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 36.0000 1.84434
$$382$$ 0 0
$$383$$ 22.0000 1.12415 0.562074 0.827087i $$-0.310004\pi$$
0.562074 + 0.827087i $$0.310004\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 0 0
$$393$$ −40.0000 −2.01773
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 11.0000 0.546594
$$406$$ 0 0
$$407$$ 32.0000 1.58618
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 20.0000 0.986527
$$412$$ 0 0
$$413$$ −32.0000 −1.57462
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 56.0000 2.71003
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4.00000 0.192673 0.0963366 0.995349i $$-0.469287\pi$$
0.0963366 + 0.995349i $$0.469287\pi$$
$$432$$ 0 0
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 0 0
$$435$$ 12.0000 0.575356
$$436$$ 0 0
$$437$$ 8.00000 0.382692
$$438$$ 0 0
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ −6.00000 −0.284427
$$446$$ 0 0
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 0 0
$$453$$ −24.0000 −1.12762
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ 24.0000 1.12022
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 0 0
$$465$$ −16.0000 −0.741982
$$466$$ 0 0
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −4.00000 −0.183147
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −64.0000 −2.91210
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ 0 0
$$489$$ 48.0000 2.17064
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ −32.0000 −1.43540
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ 26.0000 1.15470
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 6.00000 0.264392
$$516$$ 0 0
$$517$$ 48.0000 2.11104
$$518$$ 0 0
$$519$$ −40.0000 −1.75581
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ 0 0
$$525$$ −8.00000 −0.349149
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 18.0000 0.778208
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ −20.0000 −0.858282
$$544$$ 0 0
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ −26.0000 −1.11168 −0.555840 0.831289i $$-0.687603\pi$$
−0.555840 + 0.831289i $$0.687603\pi$$
$$548$$ 0 0
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 0 0
$$555$$ 16.0000 0.679162
$$556$$ 0 0
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −48.0000 −2.02656
$$562$$ 0 0
$$563$$ 18.0000 0.758610 0.379305 0.925272i $$-0.376163\pi$$
0.379305 + 0.925272i $$0.376163\pi$$
$$564$$ 0 0
$$565$$ 16.0000 0.673125
$$566$$ 0 0
$$567$$ −44.0000 −1.84783
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 48.0000 2.00523
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 0 0
$$579$$ −32.0000 −1.32987
$$580$$ 0 0
$$581$$ −48.0000 −1.99138
$$582$$ 0 0
$$583$$ −16.0000 −0.662652
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ −24.0000 −0.983904
$$596$$ 0 0
$$597$$ −16.0000 −0.654836
$$598$$ 0 0
$$599$$ −44.0000 −1.79779 −0.898896 0.438163i $$-0.855629\pi$$
−0.898896 + 0.438163i $$0.855629\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 0 0
$$603$$ 2.00000 0.0814463
$$604$$ 0 0
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ 0 0
$$609$$ −48.0000 −1.94506
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 0 0
$$615$$ −4.00000 −0.161296
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 32.0000 1.28412
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −8.00000 −0.319489
$$628$$ 0 0
$$629$$ 48.0000 1.91389
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ 32.0000 1.27189
$$634$$ 0 0
$$635$$ 18.0000 0.714308
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ −32.0000 −1.25611
$$650$$ 0 0
$$651$$ 64.0000 2.50836
$$652$$ 0 0
$$653$$ −26.0000 −1.01746 −0.508729 0.860927i $$-0.669885\pi$$
−0.508729 + 0.860927i $$0.669885\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 0 0
$$661$$ 46.0000 1.78919 0.894596 0.446875i $$-0.147463\pi$$
0.894596 + 0.446875i $$0.147463\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ 0 0
$$667$$ 48.0000 1.85857
$$668$$ 0 0
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 56.0000 2.16186
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ −20.0000 −0.768662 −0.384331 0.923195i $$-0.625568\pi$$
−0.384331 + 0.923195i $$0.625568\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 34.0000 1.30097 0.650487 0.759517i $$-0.274565\pi$$
0.650487 + 0.759517i $$0.274565\pi$$
$$684$$ 0 0
$$685$$ 10.0000 0.382080
$$686$$ 0 0
$$687$$ −12.0000 −0.457829
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 0 0
$$693$$ 16.0000 0.607790
$$694$$ 0 0
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ −20.0000 −0.756469
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 24.0000 0.903892
$$706$$ 0 0
$$707$$ 24.0000 0.902613
$$708$$ 0 0
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ −64.0000 −2.39682
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −48.0000 −1.79259
$$718$$ 0 0
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 0 0
$$723$$ 4.00000 0.148762
$$724$$ 0 0
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −20.0000 −0.741759 −0.370879 0.928681i $$-0.620944\pi$$
−0.370879 + 0.928681i $$0.620944\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ 18.0000 0.663940
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 30.0000 1.10059 0.550297 0.834969i $$-0.314515\pi$$
0.550297 + 0.834969i $$0.314515\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ −72.0000 −2.63082
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 0 0
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ 34.0000 1.23575 0.617876 0.786276i $$-0.287994\pi$$
0.617876 + 0.786276i $$0.287994\pi$$
$$758$$ 0 0
$$759$$ −64.0000 −2.32305
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ 8.00000 0.289619
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −24.0000 −0.864339
$$772$$ 0 0
$$773$$ −48.0000 −1.72644 −0.863220 0.504828i $$-0.831556\pi$$
−0.863220 + 0.504828i $$0.831556\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ −64.0000 −2.29599
$$778$$ 0 0
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ 0 0
$$785$$ −6.00000 −0.214149
$$786$$ 0 0
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ 0 0
$$789$$ 48.0000 1.70885
$$790$$ 0 0
$$791$$ −64.0000 −2.27558
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −8.00000 −0.283731
$$796$$ 0 0
$$797$$ −44.0000 −1.55856 −0.779280 0.626676i $$-0.784415\pi$$
−0.779280 + 0.626676i $$0.784415\pi$$
$$798$$ 0 0
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ −8.00000 −0.282314
$$804$$ 0 0
$$805$$ −32.0000 −1.12785
$$806$$ 0 0
$$807$$ 12.0000 0.422420
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 0 0
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 24.0000 0.840683
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ 0 0
$$823$$ −52.0000 −1.81261 −0.906303 0.422628i $$-0.861108\pi$$
−0.906303 + 0.422628i $$0.861108\pi$$
$$824$$ 0 0
$$825$$ −8.00000 −0.278524
$$826$$ 0 0
$$827$$ −6.00000 −0.208640 −0.104320 0.994544i $$-0.533267\pi$$
−0.104320 + 0.994544i $$0.533267\pi$$
$$828$$ 0 0
$$829$$ 42.0000 1.45872 0.729360 0.684130i $$-0.239818\pi$$
0.729360 + 0.684130i $$0.239818\pi$$
$$830$$ 0 0
$$831$$ 52.0000 1.80386
$$832$$ 0 0
$$833$$ 54.0000 1.87099
$$834$$ 0 0
$$835$$ 6.00000 0.207639
$$836$$ 0 0
$$837$$ −32.0000 −1.10608
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −60.0000 −2.06651
$$844$$ 0 0
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 64.0000 2.19389
$$852$$ 0 0
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ −28.0000 −0.956462 −0.478231 0.878234i $$-0.658722\pi$$
−0.478231 + 0.878234i $$0.658722\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 16.0000 0.545279
$$862$$ 0 0
$$863$$ −34.0000 −1.15737 −0.578687 0.815550i $$-0.696435\pi$$
−0.578687 + 0.815550i $$0.696435\pi$$
$$864$$ 0 0
$$865$$ −20.0000 −0.680020
$$866$$ 0 0
$$867$$ −38.0000 −1.29055
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −4.00000 −0.135225
$$876$$ 0 0
$$877$$ −52.0000 −1.75592 −0.877958 0.478738i $$-0.841094\pi$$
−0.877958 + 0.478738i $$0.841094\pi$$
$$878$$ 0 0
$$879$$ 8.00000 0.269833
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 8.00000 0.269221 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$884$$ 0 0
$$885$$ −16.0000 −0.537834
$$886$$ 0 0
$$887$$ −18.0000 −0.604381 −0.302190 0.953248i $$-0.597718\pi$$
−0.302190 + 0.953248i $$0.597718\pi$$
$$888$$ 0 0
$$889$$ −72.0000 −2.41480
$$890$$ 0 0
$$891$$ −44.0000 −1.47406
$$892$$ 0 0
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −10.0000 −0.332411
$$906$$ 0 0
$$907$$ 46.0000 1.52740 0.763702 0.645568i $$-0.223379\pi$$
0.763702 + 0.645568i $$0.223379\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −44.0000 −1.45779 −0.728893 0.684628i $$-0.759965\pi$$
−0.728893 + 0.684628i $$0.759965\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 28.0000 0.925651
$$916$$ 0 0
$$917$$ 80.0000 2.64183
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −44.0000 −1.44985
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ −6.00000 −0.197066
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ 0 0
$$933$$ 16.0000 0.523816
$$934$$ 0 0
$$935$$ −24.0000 −0.784884
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 28.0000 0.913745
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ −16.0000 −0.520480
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 64.0000 2.07534
$$952$$ 0 0
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ 0 0
$$957$$ −48.0000 −1.55162
$$958$$ 0 0
$$959$$ −40.0000 −1.29167
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ −16.0000 −0.515058
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −28.0000 −0.895799 −0.447900 0.894084i $$-0.647828\pi$$
−0.447900 + 0.894084i $$0.647828\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ −34.0000 −1.08443 −0.542216 0.840239i $$-0.682414\pi$$
−0.542216 + 0.840239i $$0.682414\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ −96.0000 −3.05571
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −20.0000 −0.635321 −0.317660 0.948205i $$-0.602897\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ 0 0
$$993$$ −32.0000 −1.01549
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 0 0
$$997$$ 22.0000 0.696747 0.348373 0.937356i $$-0.386734\pi$$
0.348373 + 0.937356i $$0.386734\pi$$
$$998$$ 0 0
$$999$$ 32.0000 1.01244
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 6080.2.a.e.1.1 1
4.3 odd 2 6080.2.a.s.1.1 1
8.3 odd 2 1520.2.a.c.1.1 1
8.5 even 2 760.2.a.d.1.1 1
24.5 odd 2 6840.2.a.o.1.1 1
40.13 odd 4 3800.2.d.c.3649.2 2
40.19 odd 2 7600.2.a.s.1.1 1
40.29 even 2 3800.2.a.b.1.1 1
40.37 odd 4 3800.2.d.c.3649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.d.1.1 1 8.5 even 2
1520.2.a.c.1.1 1 8.3 odd 2
3800.2.a.b.1.1 1 40.29 even 2
3800.2.d.c.3649.1 2 40.37 odd 4
3800.2.d.c.3649.2 2 40.13 odd 4
6080.2.a.e.1.1 1 1.1 even 1 trivial
6080.2.a.s.1.1 1 4.3 odd 2
6840.2.a.o.1.1 1 24.5 odd 2
7600.2.a.s.1.1 1 40.19 odd 2