# Properties

 Label 592.2.a.e.1.1 Level $592$ Weight $2$ Character 592.1 Self dual yes Analytic conductor $4.727$ 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] = [592,2,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 592.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: $$4.72714379966$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 592.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +5.00000 q^{11} -2.00000 q^{13} -6.00000 q^{15} +3.00000 q^{21} -2.00000 q^{23} -1.00000 q^{25} +9.00000 q^{27} +6.00000 q^{29} +4.00000 q^{31} +15.0000 q^{33} -2.00000 q^{35} -1.00000 q^{37} -6.00000 q^{39} -9.00000 q^{41} -2.00000 q^{43} -12.0000 q^{45} +9.00000 q^{47} -6.00000 q^{49} +1.00000 q^{53} -10.0000 q^{55} -8.00000 q^{59} -8.00000 q^{61} +6.00000 q^{63} +4.00000 q^{65} -8.00000 q^{67} -6.00000 q^{69} -9.00000 q^{71} -1.00000 q^{73} -3.00000 q^{75} +5.00000 q^{77} -4.00000 q^{79} +9.00000 q^{81} +15.0000 q^{83} +18.0000 q^{87} +4.00000 q^{89} -2.00000 q^{91} +12.0000 q^{93} +4.00000 q^{97} +30.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 0 0
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −6.00000 −1.54919
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 9.00000 1.73205
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 15.0000 2.61116
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ −12.0000 −1.78885
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 0 0
$$55$$ −10.0000 −1.34840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ 6.00000 0.755929
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −9.00000 −1.06810 −0.534052 0.845452i $$-0.679331\pi$$
−0.534052 + 0.845452i $$0.679331\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ 0 0
$$75$$ −3.00000 −0.346410
$$76$$ 0 0
$$77$$ 5.00000 0.569803
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 15.0000 1.64646 0.823232 0.567705i $$-0.192169\pi$$
0.823232 + 0.567705i $$0.192169\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 18.0000 1.92980
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 12.0000 1.24434
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.00000 0.406138 0.203069 0.979164i $$-0.434908\pi$$
0.203069 + 0.979164i $$0.434908\pi$$
$$98$$ 0 0
$$99$$ 30.0000 3.01511
$$100$$ 0 0
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ −18.0000 −1.77359 −0.886796 0.462160i $$-0.847074\pi$$
−0.886796 + 0.462160i $$0.847074\pi$$
$$104$$ 0 0
$$105$$ −6.00000 −0.585540
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ −12.0000 −1.10940
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −27.0000 −2.43451
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −1.00000 −0.0887357 −0.0443678 0.999015i $$-0.514127\pi$$
−0.0443678 + 0.999015i $$0.514127\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −18.0000 −1.54919
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 27.0000 2.27381
$$142$$ 0 0
$$143$$ −10.0000 −0.836242
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ −18.0000 −1.48461
$$148$$ 0 0
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 23.0000 1.83560 0.917800 0.397043i $$-0.129964\pi$$
0.917800 + 0.397043i $$0.129964\pi$$
$$158$$ 0 0
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 18.0000 1.40987 0.704934 0.709273i $$-0.250976\pi$$
0.704934 + 0.709273i $$0.250976\pi$$
$$164$$ 0 0
$$165$$ −30.0000 −2.33550
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −24.0000 −1.80395
$$178$$ 0 0
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ −24.0000 −1.77413
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 9.00000 0.654654
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ 0 0
$$195$$ 12.0000 0.859338
$$196$$ 0 0
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ −24.0000 −1.69283
$$202$$ 0 0
$$203$$ 6.00000 0.421117
$$204$$ 0 0
$$205$$ 18.0000 1.25717
$$206$$ 0 0
$$207$$ −12.0000 −0.834058
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 0 0
$$213$$ −27.0000 −1.85001
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ −3.00000 −0.202721
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 17.0000 1.13840 0.569202 0.822198i $$-0.307252\pi$$
0.569202 + 0.822198i $$0.307252\pi$$
$$224$$ 0 0
$$225$$ −6.00000 −0.400000
$$226$$ 0 0
$$227$$ 16.0000 1.06196 0.530979 0.847385i $$-0.321824\pi$$
0.530979 + 0.847385i $$0.321824\pi$$
$$228$$ 0 0
$$229$$ 7.00000 0.462573 0.231287 0.972886i $$-0.425707\pi$$
0.231287 + 0.972886i $$0.425707\pi$$
$$230$$ 0 0
$$231$$ 15.0000 0.986928
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −18.0000 −1.17419
$$236$$ 0 0
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 12.0000 0.766652
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 45.0000 2.85176
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ −10.0000 −0.628695
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ 36.0000 2.22834
$$262$$ 0 0
$$263$$ −19.0000 −1.17159 −0.585795 0.810459i $$-0.699218\pi$$
−0.585795 + 0.810459i $$0.699218\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 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
$$271$$ 31.0000 1.88312 0.941558 0.336851i $$-0.109362\pi$$
0.941558 + 0.336851i $$0.109362\pi$$
$$272$$ 0 0
$$273$$ −6.00000 −0.363137
$$274$$ 0 0
$$275$$ −5.00000 −0.301511
$$276$$ 0 0
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 0 0
$$279$$ 24.0000 1.43684
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −9.00000 −0.531253
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ 0 0
$$293$$ −2.00000 −0.116841 −0.0584206 0.998292i $$-0.518606\pi$$
−0.0584206 + 0.998292i $$0.518606\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ 0 0
$$297$$ 45.0000 2.61116
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −2.00000 −0.115278
$$302$$ 0 0
$$303$$ 9.00000 0.517036
$$304$$ 0 0
$$305$$ 16.0000 0.916157
$$306$$ 0 0
$$307$$ 17.0000 0.970241 0.485121 0.874447i $$-0.338776\pi$$
0.485121 + 0.874447i $$0.338776\pi$$
$$308$$ 0 0
$$309$$ −54.0000 −3.07195
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ −12.0000 −0.676123
$$316$$ 0 0
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 0 0
$$319$$ 30.0000 1.67968
$$320$$ 0 0
$$321$$ 36.0000 2.00932
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ −48.0000 −2.65441
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 0 0
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ 16.0000 0.874173
$$336$$ 0 0
$$337$$ −25.0000 −1.36184 −0.680918 0.732359i $$-0.738419\pi$$
−0.680918 + 0.732359i $$0.738419\pi$$
$$338$$ 0 0
$$339$$ −54.0000 −2.93288
$$340$$ 0 0
$$341$$ 20.0000 1.08306
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ 10.0000 0.536828 0.268414 0.963304i $$-0.413500\pi$$
0.268414 + 0.963304i $$0.413500\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ −18.0000 −0.960769
$$352$$ 0 0
$$353$$ 8.00000 0.425797 0.212899 0.977074i $$-0.431710\pi$$
0.212899 + 0.977074i $$0.431710\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 42.0000 2.20443
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −54.0000 −2.81113
$$370$$ 0 0
$$371$$ 1.00000 0.0519174
$$372$$ 0 0
$$373$$ −19.0000 −0.983783 −0.491891 0.870657i $$-0.663694\pi$$
−0.491891 + 0.870657i $$0.663694\pi$$
$$374$$ 0 0
$$375$$ 36.0000 1.85903
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ −15.0000 −0.770498 −0.385249 0.922813i $$-0.625884\pi$$
−0.385249 + 0.922813i $$0.625884\pi$$
$$380$$ 0 0
$$381$$ −3.00000 −0.153695
$$382$$ 0 0
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 0 0
$$385$$ −10.0000 −0.509647
$$386$$ 0 0
$$387$$ −12.0000 −0.609994
$$388$$ 0 0
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −5.00000 −0.250943 −0.125471 0.992097i $$-0.540044\pi$$
−0.125471 + 0.992097i $$0.540044\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ −18.0000 −0.894427
$$406$$ 0 0
$$407$$ −5.00000 −0.247841
$$408$$ 0 0
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ −30.0000 −1.47264
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ −7.00000 −0.341972 −0.170986 0.985273i $$-0.554695\pi$$
−0.170986 + 0.985273i $$0.554695\pi$$
$$420$$ 0 0
$$421$$ −24.0000 −1.16969 −0.584844 0.811146i $$-0.698844\pi$$
−0.584844 + 0.811146i $$0.698844\pi$$
$$422$$ 0 0
$$423$$ 54.0000 2.62557
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 0 0
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ 9.00000 0.432512 0.216256 0.976337i $$-0.430615\pi$$
0.216256 + 0.976337i $$0.430615\pi$$
$$434$$ 0 0
$$435$$ −36.0000 −1.72607
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −36.0000 −1.71429
$$442$$ 0 0
$$443$$ −1.00000 −0.0475114 −0.0237557 0.999718i $$-0.507562\pi$$
−0.0237557 + 0.999718i $$0.507562\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ 0 0
$$447$$ −15.0000 −0.709476
$$448$$ 0 0
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 0 0
$$451$$ −45.0000 −2.11897
$$452$$ 0 0
$$453$$ −48.0000 −2.25524
$$454$$ 0 0
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 22.0000 1.02243 0.511213 0.859454i $$-0.329196\pi$$
0.511213 + 0.859454i $$0.329196\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ 2.00000 0.0925490 0.0462745 0.998929i $$-0.485265\pi$$
0.0462745 + 0.998929i $$0.485265\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 69.0000 3.17935
$$472$$ 0 0
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ −14.0000 −0.639676 −0.319838 0.947472i $$-0.603629\pi$$
−0.319838 + 0.947472i $$0.603629\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 0 0
$$485$$ −8.00000 −0.363261
$$486$$ 0 0
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ 0 0
$$489$$ 54.0000 2.44196
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −60.0000 −2.69680
$$496$$ 0 0
$$497$$ −9.00000 −0.403705
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 36.0000 1.60836
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ −27.0000 −1.19911
$$508$$ 0 0
$$509$$ −31.0000 −1.37405 −0.687025 0.726633i $$-0.741084\pi$$
−0.687025 + 0.726633i $$0.741084\pi$$
$$510$$ 0 0
$$511$$ −1.00000 −0.0442374
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 36.0000 1.58635
$$516$$ 0 0
$$517$$ 45.0000 1.97910
$$518$$ 0 0
$$519$$ 27.0000 1.18517
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ 0 0
$$523$$ 22.0000 0.961993 0.480996 0.876723i $$-0.340275\pi$$
0.480996 + 0.876723i $$0.340275\pi$$
$$524$$ 0 0
$$525$$ −3.00000 −0.130931
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ −48.0000 −2.08302
$$532$$ 0 0
$$533$$ 18.0000 0.779667
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ 0 0
$$537$$ −54.0000 −2.33027
$$538$$ 0 0
$$539$$ −30.0000 −1.29219
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 0 0
$$543$$ 15.0000 0.643712
$$544$$ 0 0
$$545$$ 32.0000 1.37073
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ −48.0000 −2.04859
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −4.00000 −0.170097
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 30.0000 1.26435 0.632175 0.774826i $$-0.282163\pi$$
0.632175 + 0.774826i $$0.282163\pi$$
$$564$$ 0 0
$$565$$ 36.0000 1.51453
$$566$$ 0 0
$$567$$ 9.00000 0.377964
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −7.00000 −0.292941 −0.146470 0.989215i $$-0.546791\pi$$
−0.146470 + 0.989215i $$0.546791\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 2.00000 0.0834058
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 0 0
$$579$$ −78.0000 −3.24157
$$580$$ 0 0
$$581$$ 15.0000 0.622305
$$582$$ 0 0
$$583$$ 5.00000 0.207079
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ 0 0
$$587$$ 32.0000 1.32078 0.660391 0.750922i $$-0.270391\pi$$
0.660391 + 0.750922i $$0.270391\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 9.00000 0.370211
$$592$$ 0 0
$$593$$ −5.00000 −0.205325 −0.102663 0.994716i $$-0.532736\pi$$
−0.102663 + 0.994716i $$0.532736\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6.00000 −0.245564
$$598$$ 0 0
$$599$$ −1.00000 −0.0408589 −0.0204294 0.999791i $$-0.506503\pi$$
−0.0204294 + 0.999791i $$0.506503\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ −48.0000 −1.95471
$$604$$ 0 0
$$605$$ −28.0000 −1.13836
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 18.0000 0.729397
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ 0 0
$$613$$ 15.0000 0.605844 0.302922 0.953015i $$-0.402038\pi$$
0.302922 + 0.953015i $$0.402038\pi$$
$$614$$ 0 0
$$615$$ 54.0000 2.17749
$$616$$ 0 0
$$617$$ 17.0000 0.684394 0.342197 0.939628i $$-0.388829\pi$$
0.342197 + 0.939628i $$0.388829\pi$$
$$618$$ 0 0
$$619$$ 1.00000 0.0401934 0.0200967 0.999798i $$-0.493603\pi$$
0.0200967 + 0.999798i $$0.493603\pi$$
$$620$$ 0 0
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ 4.00000 0.160257
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 28.0000 1.11466 0.557331 0.830290i $$-0.311825\pi$$
0.557331 + 0.830290i $$0.311825\pi$$
$$632$$ 0 0
$$633$$ 39.0000 1.55011
$$634$$ 0 0
$$635$$ 2.00000 0.0793676
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ −54.0000 −2.13621
$$640$$ 0 0
$$641$$ −1.00000 −0.0394976 −0.0197488 0.999805i $$-0.506287\pi$$
−0.0197488 + 0.999805i $$0.506287\pi$$
$$642$$ 0 0
$$643$$ −14.0000 −0.552106 −0.276053 0.961142i $$-0.589027\pi$$
−0.276053 + 0.961142i $$0.589027\pi$$
$$644$$ 0 0
$$645$$ 12.0000 0.472500
$$646$$ 0 0
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ 0 0
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 0 0
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ 51.0000 1.97177
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 27.0000 1.04077 0.520387 0.853931i $$-0.325788\pi$$
0.520387 + 0.853931i $$0.325788\pi$$
$$674$$ 0 0
$$675$$ −9.00000 −0.346410
$$676$$ 0 0
$$677$$ −11.0000 −0.422764 −0.211382 0.977403i $$-0.567796\pi$$
−0.211382 + 0.977403i $$0.567796\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ 48.0000 1.83936
$$682$$ 0 0
$$683$$ −18.0000 −0.688751 −0.344375 0.938832i $$-0.611909\pi$$
−0.344375 + 0.938832i $$0.611909\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 21.0000 0.801200
$$688$$ 0 0
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ 30.0000 1.13961
$$694$$ 0 0
$$695$$ 8.00000 0.303457
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −12.0000 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −54.0000 −2.03376
$$706$$ 0 0
$$707$$ 3.00000 0.112827
$$708$$ 0 0
$$709$$ 40.0000 1.50223 0.751116 0.660171i $$-0.229516\pi$$
0.751116 + 0.660171i $$0.229516\pi$$
$$710$$ 0 0
$$711$$ −24.0000 −0.900070
$$712$$ 0 0
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ 20.0000 0.747958
$$716$$ 0 0
$$717$$ 18.0000 0.672222
$$718$$ 0 0
$$719$$ −39.0000 −1.45445 −0.727227 0.686397i $$-0.759191\pi$$
−0.727227 + 0.686397i $$0.759191\pi$$
$$720$$ 0 0
$$721$$ −18.0000 −0.670355
$$722$$ 0 0
$$723$$ 42.0000 1.56200
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 7.00000 0.258551 0.129275 0.991609i $$-0.458735\pi$$
0.129275 + 0.991609i $$0.458735\pi$$
$$734$$ 0 0
$$735$$ 36.0000 1.32788
$$736$$ 0 0
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ 9.00000 0.331070 0.165535 0.986204i $$-0.447065\pi$$
0.165535 + 0.986204i $$0.447065\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ 0 0
$$745$$ 10.0000 0.366372
$$746$$ 0 0
$$747$$ 90.0000 3.29293
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −25.0000 −0.912263 −0.456131 0.889912i $$-0.650765\pi$$
−0.456131 + 0.889912i $$0.650765\pi$$
$$752$$ 0 0
$$753$$ 6.00000 0.218652
$$754$$ 0 0
$$755$$ 32.0000 1.16460
$$756$$ 0 0
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ 0 0
$$759$$ −30.0000 −1.08893
$$760$$ 0 0
$$761$$ −35.0000 −1.26875 −0.634375 0.773026i $$-0.718742\pi$$
−0.634375 + 0.773026i $$0.718742\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −9.00000 −0.323708 −0.161854 0.986815i $$-0.551747\pi$$
−0.161854 + 0.986815i $$0.551747\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ −3.00000 −0.107624
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −45.0000 −1.61023
$$782$$ 0 0
$$783$$ 54.0000 1.92980
$$784$$ 0 0
$$785$$ −46.0000 −1.64181
$$786$$ 0 0
$$787$$ 5.00000 0.178231 0.0891154 0.996021i $$-0.471596\pi$$
0.0891154 + 0.996021i $$0.471596\pi$$
$$788$$ 0 0
$$789$$ −57.0000 −2.02925
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ 52.0000 1.84193 0.920967 0.389640i $$-0.127401\pi$$
0.920967 + 0.389640i $$0.127401\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ −5.00000 −0.176446
$$804$$ 0 0
$$805$$ 4.00000 0.140981
$$806$$ 0 0
$$807$$ −18.0000 −0.633630
$$808$$ 0 0
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ −47.0000 −1.65039 −0.825197 0.564846i $$-0.808936\pi$$
−0.825197 + 0.564846i $$0.808936\pi$$
$$812$$ 0 0
$$813$$ 93.0000 3.26165
$$814$$ 0 0
$$815$$ −36.0000 −1.26102
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −47.0000 −1.64031 −0.820156 0.572140i $$-0.806113\pi$$
−0.820156 + 0.572140i $$0.806113\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ −15.0000 −0.522233
$$826$$ 0 0
$$827$$ −22.0000 −0.765015 −0.382507 0.923952i $$-0.624939\pi$$
−0.382507 + 0.923952i $$0.624939\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ 36.0000 1.24883
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ 0 0
$$837$$ 36.0000 1.24434
$$838$$ 0 0
$$839$$ −44.0000 −1.51905 −0.759524 0.650479i $$-0.774568\pi$$
−0.759524 + 0.650479i $$0.774568\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 36.0000 1.23991
$$844$$ 0 0
$$845$$ 18.0000 0.619219
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ 0 0
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −27.0000 −0.920158
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −51.0000 −1.73205
$$868$$ 0 0
$$869$$ −20.0000 −0.678454
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 24.0000 0.812277
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 0 0
$$877$$ 50.0000 1.68838 0.844190 0.536044i $$-0.180082\pi$$
0.844190 + 0.536044i $$0.180082\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 0 0
$$883$$ −48.0000 −1.61533 −0.807664 0.589643i $$-0.799269\pi$$
−0.807664 + 0.589643i $$0.799269\pi$$
$$884$$ 0 0
$$885$$ 48.0000 1.61350
$$886$$ 0 0
$$887$$ −25.0000 −0.839418 −0.419709 0.907659i $$-0.637868\pi$$
−0.419709 + 0.907659i $$0.637868\pi$$
$$888$$ 0 0
$$889$$ −1.00000 −0.0335389
$$890$$ 0 0
$$891$$ 45.0000 1.50756
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 36.0000 1.20335
$$896$$ 0 0
$$897$$ 12.0000 0.400668
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −6.00000 −0.199667
$$904$$ 0 0
$$905$$ −10.0000 −0.332411
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 0 0
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −26.0000 −0.861418 −0.430709 0.902491i $$-0.641737\pi$$
−0.430709 + 0.902491i $$0.641737\pi$$
$$912$$ 0 0
$$913$$ 75.0000 2.48214
$$914$$ 0 0
$$915$$ 48.0000 1.58683
$$916$$ 0 0
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 58.0000 1.91324 0.956622 0.291333i $$-0.0940987\pi$$
0.956622 + 0.291333i $$0.0940987\pi$$
$$920$$ 0 0
$$921$$ 51.0000 1.68051
$$922$$ 0 0
$$923$$ 18.0000 0.592477
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 0 0
$$927$$ −108.000 −3.54719
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 37.0000 1.20874 0.604369 0.796705i $$-0.293425\pi$$
0.604369 + 0.796705i $$0.293425\pi$$
$$938$$ 0 0
$$939$$ 66.0000 2.15383
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ 18.0000 0.586161
$$944$$ 0 0
$$945$$ −18.0000 −0.585540
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ 66.0000 2.14020
$$952$$ 0 0
$$953$$ 61.0000 1.97598 0.987992 0.154506i $$-0.0493785\pi$$
0.987992 + 0.154506i $$0.0493785\pi$$
$$954$$ 0 0
$$955$$ −8.00000 −0.258874
$$956$$ 0 0
$$957$$ 90.0000 2.90929
$$958$$ 0 0
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 72.0000 2.32017
$$964$$ 0 0
$$965$$ 52.0000 1.67394
$$966$$ 0 0
$$967$$ 14.0000 0.450210 0.225105 0.974335i $$-0.427728\pi$$
0.225105 + 0.974335i $$0.427728\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 8.00000 0.256732 0.128366 0.991727i $$-0.459027\pi$$
0.128366 + 0.991727i $$0.459027\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ 0 0
$$975$$ 6.00000 0.192154
$$976$$ 0 0
$$977$$ 28.0000 0.895799 0.447900 0.894084i $$-0.352172\pi$$
0.447900 + 0.894084i $$0.352172\pi$$
$$978$$ 0 0
$$979$$ 20.0000 0.639203
$$980$$ 0 0
$$981$$ −96.0000 −3.06504
$$982$$ 0 0
$$983$$ −9.00000 −0.287055 −0.143528 0.989646i $$-0.545845\pi$$
−0.143528 + 0.989646i $$0.545845\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 27.0000 0.859419
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 18.0000 0.571789 0.285894 0.958261i $$-0.407709\pi$$
0.285894 + 0.958261i $$0.407709\pi$$
$$992$$ 0 0
$$993$$ 6.00000 0.190404
$$994$$ 0 0
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ 0 0
$$999$$ −9.00000 −0.284747
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 592.2.a.e.1.1 1
3.2 odd 2 5328.2.a.r.1.1 1
4.3 odd 2 37.2.a.a.1.1 1
8.3 odd 2 2368.2.a.q.1.1 1
8.5 even 2 2368.2.a.b.1.1 1
12.11 even 2 333.2.a.d.1.1 1
20.3 even 4 925.2.b.b.149.2 2
20.7 even 4 925.2.b.b.149.1 2
20.19 odd 2 925.2.a.e.1.1 1
28.27 even 2 1813.2.a.a.1.1 1
44.43 even 2 4477.2.a.b.1.1 1
52.51 odd 2 6253.2.a.c.1.1 1
60.59 even 2 8325.2.a.e.1.1 1
148.31 even 4 1369.2.b.c.1368.1 2
148.43 even 4 1369.2.b.c.1368.2 2
148.147 odd 2 1369.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 1 4.3 odd 2
333.2.a.d.1.1 1 12.11 even 2
592.2.a.e.1.1 1 1.1 even 1 trivial
925.2.a.e.1.1 1 20.19 odd 2
925.2.b.b.149.1 2 20.7 even 4
925.2.b.b.149.2 2 20.3 even 4
1369.2.a.e.1.1 1 148.147 odd 2
1369.2.b.c.1368.1 2 148.31 even 4
1369.2.b.c.1368.2 2 148.43 even 4
1813.2.a.a.1.1 1 28.27 even 2
2368.2.a.b.1.1 1 8.5 even 2
2368.2.a.q.1.1 1 8.3 odd 2
4477.2.a.b.1.1 1 44.43 even 2
5328.2.a.r.1.1 1 3.2 odd 2
6253.2.a.c.1.1 1 52.51 odd 2
8325.2.a.e.1.1 1 60.59 even 2