# Properties

 Label 592.2.a.c.1.1 Level $592$ Weight $2$ Character 592.1 Self dual yes Analytic conductor $4.727$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 296) 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} -1.00000 q^{11} -6.00000 q^{13} -2.00000 q^{15} -4.00000 q^{17} +8.00000 q^{19} -1.00000 q^{21} -6.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{33} +2.00000 q^{35} -1.00000 q^{37} -6.00000 q^{39} +7.00000 q^{41} -2.00000 q^{43} +4.00000 q^{45} -9.00000 q^{47} -6.00000 q^{49} -4.00000 q^{51} -3.00000 q^{53} +2.00000 q^{55} +8.00000 q^{57} +12.0000 q^{59} +4.00000 q^{61} +2.00000 q^{63} +12.0000 q^{65} -6.00000 q^{69} -7.00000 q^{71} +7.00000 q^{73} -1.00000 q^{75} +1.00000 q^{77} +1.00000 q^{81} -3.00000 q^{83} +8.00000 q^{85} +2.00000 q^{87} -12.0000 q^{89} +6.00000 q^{91} +4.00000 q^{93} -16.0000 q^{95} -8.00000 q^{97} +2.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 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 $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −1.00000 −0.174078
$$34$$ 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
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ 4.00000 0.596285
$$46$$ 0 0
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 8.00000 1.05963
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 0 0
$$65$$ 12.0000 1.48842
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −7.00000 −0.830747 −0.415374 0.909651i $$-0.636349\pi$$
−0.415374 + 0.909651i $$0.636349\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ −16.0000 −1.64157
$$96$$ 0 0
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −17.0000 −1.69156 −0.845782 0.533529i $$-0.820865\pi$$
−0.845782 + 0.533529i $$0.820865\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 12.0000 1.11901
$$116$$ 0 0
$$117$$ 12.0000 1.10940
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 7.00000 0.631169
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 17.0000 1.50851 0.754253 0.656584i $$-0.227999\pi$$
0.754253 + 0.656584i $$0.227999\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 0 0
$$135$$ 10.0000 0.860663
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 0 0
$$147$$ −6.00000 −0.494872
$$148$$ 0 0
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 8.00000 0.646762
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 19.0000 1.51637 0.758183 0.652042i $$-0.226088\pi$$
0.758183 + 0.652042i $$0.226088\pi$$
$$158$$ 0 0
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ 22.0000 1.72317 0.861586 0.507611i $$-0.169471\pi$$
0.861586 + 0.507611i $$0.169471\pi$$
$$164$$ 0 0
$$165$$ 2.00000 0.155700
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −16.0000 −1.22355
$$172$$ 0 0
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 22.0000 1.64436 0.822179 0.569230i $$-0.192758\pi$$
0.822179 + 0.569230i $$0.192758\pi$$
$$180$$ 0 0
$$181$$ −23.0000 −1.70958 −0.854788 0.518977i $$-0.826313\pi$$
−0.854788 + 0.518977i $$0.826313\pi$$
$$182$$ 0 0
$$183$$ 4.00000 0.295689
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 0 0
$$195$$ 12.0000 0.859338
$$196$$ 0 0
$$197$$ 7.00000 0.498729 0.249365 0.968410i $$-0.419778\pi$$
0.249365 + 0.968410i $$0.419778\pi$$
$$198$$ 0 0
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ −14.0000 −0.977802
$$206$$ 0 0
$$207$$ 12.0000 0.834058
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −9.00000 −0.619586 −0.309793 0.950804i $$-0.600260\pi$$
−0.309793 + 0.950804i $$0.600260\pi$$
$$212$$ 0 0
$$213$$ −7.00000 −0.479632
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 0 0
$$223$$ −25.0000 −1.67412 −0.837062 0.547108i $$-0.815729\pi$$
−0.837062 + 0.547108i $$0.815729\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 0 0
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ 0 0
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ 0 0
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ 0 0
$$235$$ 18.0000 1.17419
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.0000 −0.646846 −0.323423 0.946254i $$-0.604834\pi$$
−0.323423 + 0.946254i $$0.604834\pi$$
$$240$$ 0 0
$$241$$ −30.0000 −1.93247 −0.966235 0.257663i $$-0.917048\pi$$
−0.966235 + 0.257663i $$0.917048\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 12.0000 0.766652
$$246$$ 0 0
$$247$$ −48.0000 −3.05417
$$248$$ 0 0
$$249$$ −3.00000 −0.190117
$$250$$ 0 0
$$251$$ −26.0000 −1.64111 −0.820553 0.571571i $$-0.806334\pi$$
−0.820553 + 0.571571i $$0.806334\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 0 0
$$255$$ 8.00000 0.500979
$$256$$ 0 0
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 1.00000 0.0621370
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ −13.0000 −0.801614 −0.400807 0.916162i $$-0.631270\pi$$
−0.400807 + 0.916162i $$0.631270\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 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
$$271$$ 9.00000 0.546711 0.273356 0.961913i $$-0.411866\pi$$
0.273356 + 0.961913i $$0.411866\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ 0 0
$$283$$ −8.00000 −0.475551 −0.237775 0.971320i $$-0.576418\pi$$
−0.237775 + 0.971320i $$0.576418\pi$$
$$284$$ 0 0
$$285$$ −16.0000 −0.947758
$$286$$ 0 0
$$287$$ −7.00000 −0.413197
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ 0 0
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 0 0
$$299$$ 36.0000 2.08193
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ 0 0
$$303$$ −17.0000 −0.976624
$$304$$ 0 0
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ −29.0000 −1.65512 −0.827559 0.561379i $$-0.810271\pi$$
−0.827559 + 0.561379i $$0.810271\pi$$
$$308$$ 0 0
$$309$$ −2.00000 −0.113776
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ 0 0
$$315$$ −4.00000 −0.225374
$$316$$ 0 0
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ −32.0000 −1.78053
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 0 0
$$327$$ 16.0000 0.884802
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ 22.0000 1.20923 0.604615 0.796518i $$-0.293327\pi$$
0.604615 + 0.796518i $$0.293327\pi$$
$$332$$ 0 0
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −9.00000 −0.490261 −0.245131 0.969490i $$-0.578831\pi$$
−0.245131 + 0.969490i $$0.578831\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ −22.0000 −1.18102 −0.590511 0.807030i $$-0.701074\pi$$
−0.590511 + 0.807030i $$0.701074\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 30.0000 1.60128
$$352$$ 0 0
$$353$$ −36.0000 −1.91609 −0.958043 0.286623i $$-0.907467\pi$$
−0.958043 + 0.286623i $$0.907467\pi$$
$$354$$ 0 0
$$355$$ 14.0000 0.743043
$$356$$ 0 0
$$357$$ 4.00000 0.211702
$$358$$ 0 0
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ −14.0000 −0.732793
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 0 0
$$369$$ −14.0000 −0.728811
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ 0 0
$$373$$ 1.00000 0.0517780 0.0258890 0.999665i $$-0.491758\pi$$
0.0258890 + 0.999665i $$0.491758\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ −37.0000 −1.90056 −0.950281 0.311393i $$-0.899204\pi$$
−0.950281 + 0.311393i $$0.899204\pi$$
$$380$$ 0 0
$$381$$ 17.0000 0.870936
$$382$$ 0 0
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ 0 0
$$385$$ −2.00000 −0.101929
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −1.00000 −0.0501886 −0.0250943 0.999685i $$-0.507989\pi$$
−0.0250943 + 0.999685i $$0.507989\pi$$
$$398$$ 0 0
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ −10.0000 −0.499376 −0.249688 0.968326i $$-0.580328\pi$$
−0.249688 + 0.968326i $$0.580328\pi$$
$$402$$ 0 0
$$403$$ −24.0000 −1.19553
$$404$$ 0 0
$$405$$ −2.00000 −0.0993808
$$406$$ 0 0
$$407$$ 1.00000 0.0495682
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ −14.0000 −0.690569
$$412$$ 0 0
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 6.00000 0.294528
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ −5.00000 −0.244266 −0.122133 0.992514i $$-0.538973\pi$$
−0.122133 + 0.992514i $$0.538973\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 0 0
$$423$$ 18.0000 0.875190
$$424$$ 0 0
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ 0 0
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −10.0000 −0.481683 −0.240842 0.970564i $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ 0 0
$$433$$ −39.0000 −1.87422 −0.937110 0.349034i $$-0.886510\pi$$
−0.937110 + 0.349034i $$0.886510\pi$$
$$434$$ 0 0
$$435$$ −4.00000 −0.191785
$$436$$ 0 0
$$437$$ −48.0000 −2.29615
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ 13.0000 0.617649 0.308824 0.951119i $$-0.400064\pi$$
0.308824 + 0.951119i $$0.400064\pi$$
$$444$$ 0 0
$$445$$ 24.0000 1.13771
$$446$$ 0 0
$$447$$ −9.00000 −0.425685
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ −7.00000 −0.329617
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 0 0
$$459$$ 20.0000 0.933520
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 22.0000 1.02243 0.511213 0.859454i $$-0.329196\pi$$
0.511213 + 0.859454i $$0.329196\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 19.0000 0.875474
$$472$$ 0 0
$$473$$ 2.00000 0.0919601
$$474$$ 0 0
$$475$$ −8.00000 −0.367065
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ 6.00000 0.273009
$$484$$ 0 0
$$485$$ 16.0000 0.726523
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 0 0
$$489$$ 22.0000 0.994874
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ 7.00000 0.313993
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 8.00000 0.356702 0.178351 0.983967i $$-0.442924\pi$$
0.178351 + 0.983967i $$0.442924\pi$$
$$504$$ 0 0
$$505$$ 34.0000 1.51298
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 0 0
$$509$$ 45.0000 1.99459 0.997295 0.0735034i $$-0.0234180\pi$$
0.997295 + 0.0735034i $$0.0234180\pi$$
$$510$$ 0 0
$$511$$ −7.00000 −0.309662
$$512$$ 0 0
$$513$$ −40.0000 −1.76604
$$514$$ 0 0
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ 9.00000 0.395820
$$518$$ 0 0
$$519$$ −3.00000 −0.131685
$$520$$ 0 0
$$521$$ −17.0000 −0.744784 −0.372392 0.928076i $$-0.621462\pi$$
−0.372392 + 0.928076i $$0.621462\pi$$
$$522$$ 0 0
$$523$$ −6.00000 −0.262362 −0.131181 0.991358i $$-0.541877\pi$$
−0.131181 + 0.991358i $$0.541877\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −24.0000 −1.04151
$$532$$ 0 0
$$533$$ −42.0000 −1.81922
$$534$$ 0 0
$$535$$ −8.00000 −0.345870
$$536$$ 0 0
$$537$$ 22.0000 0.949370
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −36.0000 −1.54776 −0.773880 0.633332i $$-0.781687\pi$$
−0.773880 + 0.633332i $$0.781687\pi$$
$$542$$ 0 0
$$543$$ −23.0000 −0.987024
$$544$$ 0 0
$$545$$ −32.0000 −1.37073
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ 16.0000 0.681623
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 2.00000 0.0848953
$$556$$ 0 0
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ 0 0
$$563$$ 22.0000 0.927189 0.463595 0.886047i $$-0.346559\pi$$
0.463595 + 0.886047i $$0.346559\pi$$
$$564$$ 0 0
$$565$$ −4.00000 −0.168281
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 28.0000 1.17382 0.586911 0.809652i $$-0.300344\pi$$
0.586911 + 0.809652i $$0.300344\pi$$
$$570$$ 0 0
$$571$$ −21.0000 −0.878823 −0.439411 0.898286i $$-0.644813\pi$$
−0.439411 + 0.898286i $$0.644813\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ 32.0000 1.33218 0.666089 0.745873i $$-0.267967\pi$$
0.666089 + 0.745873i $$0.267967\pi$$
$$578$$ 0 0
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ 3.00000 0.124461
$$582$$ 0 0
$$583$$ 3.00000 0.124247
$$584$$ 0 0
$$585$$ −24.0000 −0.992278
$$586$$ 0 0
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 7.00000 0.287942
$$592$$ 0 0
$$593$$ 3.00000 0.123195 0.0615976 0.998101i $$-0.480380\pi$$
0.0615976 + 0.998101i $$0.480380\pi$$
$$594$$ 0 0
$$595$$ −8.00000 −0.327968
$$596$$ 0 0
$$597$$ 14.0000 0.572982
$$598$$ 0 0
$$599$$ −15.0000 −0.612883 −0.306442 0.951889i $$-0.599138\pi$$
−0.306442 + 0.951889i $$0.599138\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 20.0000 0.813116
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 54.0000 2.18461
$$612$$ 0 0
$$613$$ 35.0000 1.41364 0.706818 0.707395i $$-0.250130\pi$$
0.706818 + 0.707395i $$0.250130\pi$$
$$614$$ 0 0
$$615$$ −14.0000 −0.564534
$$616$$ 0 0
$$617$$ 9.00000 0.362326 0.181163 0.983453i $$-0.442014\pi$$
0.181163 + 0.983453i $$0.442014\pi$$
$$618$$ 0 0
$$619$$ −5.00000 −0.200967 −0.100483 0.994939i $$-0.532039\pi$$
−0.100483 + 0.994939i $$0.532039\pi$$
$$620$$ 0 0
$$621$$ 30.0000 1.20386
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ −8.00000 −0.319489
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ −9.00000 −0.357718
$$634$$ 0 0
$$635$$ −34.0000 −1.34925
$$636$$ 0 0
$$637$$ 36.0000 1.42637
$$638$$ 0 0
$$639$$ 14.0000 0.553831
$$640$$ 0 0
$$641$$ 31.0000 1.22443 0.612213 0.790693i $$-0.290279\pi$$
0.612213 + 0.790693i $$0.290279\pi$$
$$642$$ 0 0
$$643$$ −14.0000 −0.552106 −0.276053 0.961142i $$-0.589027\pi$$
−0.276053 + 0.961142i $$0.589027\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ 0 0
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ 0 0
$$653$$ −8.00000 −0.313064 −0.156532 0.987673i $$-0.550031\pi$$
−0.156532 + 0.987673i $$0.550031\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ 21.0000 0.818044 0.409022 0.912525i $$-0.365870\pi$$
0.409022 + 0.912525i $$0.365870\pi$$
$$660$$ 0 0
$$661$$ −36.0000 −1.40024 −0.700119 0.714026i $$-0.746870\pi$$
−0.700119 + 0.714026i $$0.746870\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ 16.0000 0.620453
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ −25.0000 −0.966556
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ −15.0000 −0.576497 −0.288248 0.957556i $$-0.593073\pi$$
−0.288248 + 0.957556i $$0.593073\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ −42.0000 −1.60709 −0.803543 0.595247i $$-0.797054\pi$$
−0.803543 + 0.595247i $$0.797054\pi$$
$$684$$ 0 0
$$685$$ 28.0000 1.06983
$$686$$ 0 0
$$687$$ −5.00000 −0.190762
$$688$$ 0 0
$$689$$ 18.0000 0.685745
$$690$$ 0 0
$$691$$ −52.0000 −1.97817 −0.989087 0.147335i $$-0.952930\pi$$
−0.989087 + 0.147335i $$0.952930\pi$$
$$692$$ 0 0
$$693$$ −2.00000 −0.0759737
$$694$$ 0 0
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ −28.0000 −1.06058
$$698$$ 0 0
$$699$$ −2.00000 −0.0756469
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ 17.0000 0.639351
$$708$$ 0 0
$$709$$ 12.0000 0.450669 0.225335 0.974281i $$-0.427652\pi$$
0.225335 + 0.974281i $$0.427652\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 0 0
$$717$$ −10.0000 −0.373457
$$718$$ 0 0
$$719$$ −1.00000 −0.0372937 −0.0186469 0.999826i $$-0.505936\pi$$
−0.0186469 + 0.999826i $$0.505936\pi$$
$$720$$ 0 0
$$721$$ 2.00000 0.0744839
$$722$$ 0 0
$$723$$ −30.0000 −1.11571
$$724$$ 0 0
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 19.0000 0.701781 0.350891 0.936416i $$-0.385879\pi$$
0.350891 + 0.936416i $$0.385879\pi$$
$$734$$ 0 0
$$735$$ 12.0000 0.442627
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −21.0000 −0.772497 −0.386249 0.922395i $$-0.626229\pi$$
−0.386249 + 0.922395i $$0.626229\pi$$
$$740$$ 0 0
$$741$$ −48.0000 −1.76332
$$742$$ 0 0
$$743$$ −43.0000 −1.57752 −0.788759 0.614703i $$-0.789276\pi$$
−0.788759 + 0.614703i $$0.789276\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 33.0000 1.20419 0.602094 0.798426i $$-0.294333\pi$$
0.602094 + 0.798426i $$0.294333\pi$$
$$752$$ 0 0
$$753$$ −26.0000 −0.947493
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ −16.0000 −0.578481
$$766$$ 0 0
$$767$$ −72.0000 −2.59977
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ 19.0000 0.683383 0.341691 0.939812i $$-0.389000\pi$$
0.341691 + 0.939812i $$0.389000\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 1.00000 0.0358748
$$778$$ 0 0
$$779$$ 56.0000 2.00641
$$780$$ 0 0
$$781$$ 7.00000 0.250480
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 0 0
$$785$$ −38.0000 −1.35628
$$786$$ 0 0
$$787$$ 39.0000 1.39020 0.695100 0.718913i $$-0.255360\pi$$
0.695100 + 0.718913i $$0.255360\pi$$
$$788$$ 0 0
$$789$$ −13.0000 −0.462812
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ 0 0
$$793$$ −24.0000 −0.852265
$$794$$ 0 0
$$795$$ 6.00000 0.212798
$$796$$ 0 0
$$797$$ 48.0000 1.70025 0.850124 0.526583i $$-0.176527\pi$$
0.850124 + 0.526583i $$0.176527\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ 0 0
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ −7.00000 −0.247025
$$804$$ 0 0
$$805$$ −12.0000 −0.422944
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ −45.0000 −1.58016 −0.790082 0.613001i $$-0.789962\pi$$
−0.790082 + 0.613001i $$0.789962\pi$$
$$812$$ 0 0
$$813$$ 9.00000 0.315644
$$814$$ 0 0
$$815$$ −44.0000 −1.54125
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ 29.0000 1.01211 0.506053 0.862502i $$-0.331104\pi$$
0.506053 + 0.862502i $$0.331104\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 0 0
$$827$$ 54.0000 1.87776 0.938882 0.344239i $$-0.111863\pi$$
0.938882 + 0.344239i $$0.111863\pi$$
$$828$$ 0 0
$$829$$ −40.0000 −1.38926 −0.694629 0.719368i $$-0.744431\pi$$
−0.694629 + 0.719368i $$0.744431\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.416275
$$832$$ 0 0
$$833$$ 24.0000 0.831551
$$834$$ 0 0
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ −20.0000 −0.691301
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −20.0000 −0.688837
$$844$$ 0 0
$$845$$ −46.0000 −1.58245
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 0 0
$$855$$ 32.0000 1.09438
$$856$$ 0 0
$$857$$ −52.0000 −1.77629 −0.888143 0.459567i $$-0.848005\pi$$
−0.888143 + 0.459567i $$0.848005\pi$$
$$858$$ 0 0
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ 0 0
$$861$$ −7.00000 −0.238559
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 16.0000 0.541518
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 0 0
$$887$$ −47.0000 −1.57811 −0.789053 0.614325i $$-0.789428\pi$$
−0.789053 + 0.614325i $$0.789428\pi$$
$$888$$ 0 0
$$889$$ −17.0000 −0.570162
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ −72.0000 −2.40939
$$894$$ 0 0
$$895$$ −44.0000 −1.47076
$$896$$ 0 0
$$897$$ 36.0000 1.20201
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ 2.00000 0.0665558
$$904$$ 0 0
$$905$$ 46.0000 1.52909
$$906$$ 0 0
$$907$$ 32.0000 1.06254 0.531271 0.847202i $$-0.321714\pi$$
0.531271 + 0.847202i $$0.321714\pi$$
$$908$$ 0 0
$$909$$ 34.0000 1.12771
$$910$$ 0 0
$$911$$ 54.0000 1.78910 0.894550 0.446968i $$-0.147496\pi$$
0.894550 + 0.446968i $$0.147496\pi$$
$$912$$ 0 0
$$913$$ 3.00000 0.0992855
$$914$$ 0 0
$$915$$ −8.00000 −0.264472
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 14.0000 0.461817 0.230909 0.972975i $$-0.425830\pi$$
0.230909 + 0.972975i $$0.425830\pi$$
$$920$$ 0 0
$$921$$ −29.0000 −0.955582
$$922$$ 0 0
$$923$$ 42.0000 1.38245
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −48.0000 −1.57314
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ −19.0000 −0.620703 −0.310351 0.950622i $$-0.600447\pi$$
−0.310351 + 0.950622i $$0.600447\pi$$
$$938$$ 0 0
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ −42.0000 −1.36771
$$944$$ 0 0
$$945$$ −10.0000 −0.325300
$$946$$ 0 0
$$947$$ 52.0000 1.68977 0.844886 0.534946i $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ 0 0
$$949$$ −42.0000 −1.36338
$$950$$ 0 0
$$951$$ −10.0000 −0.324272
$$952$$ 0 0
$$953$$ −19.0000 −0.615470 −0.307735 0.951472i $$-0.599571\pi$$
−0.307735 + 0.951472i $$0.599571\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −2.00000 −0.0646508
$$958$$ 0 0
$$959$$ 14.0000 0.452084
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ 0 0
$$965$$ 20.0000 0.643823
$$966$$ 0 0
$$967$$ −34.0000 −1.09337 −0.546683 0.837340i $$-0.684110\pi$$
−0.546683 + 0.837340i $$0.684110\pi$$
$$968$$ 0 0
$$969$$ −32.0000 −1.02799
$$970$$ 0 0
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ 0 0
$$975$$ 6.00000 0.192154
$$976$$ 0 0
$$977$$ 4.00000 0.127971 0.0639857 0.997951i $$-0.479619\pi$$
0.0639857 + 0.997951i $$0.479619\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −32.0000 −1.02168
$$982$$ 0 0
$$983$$ 9.00000 0.287055 0.143528 0.989646i $$-0.454155\pi$$
0.143528 + 0.989646i $$0.454155\pi$$
$$984$$ 0 0
$$985$$ −14.0000 −0.446077
$$986$$ 0 0
$$987$$ 9.00000 0.286473
$$988$$ 0 0
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ 22.0000 0.698853 0.349427 0.936964i $$-0.386376\pi$$
0.349427 + 0.936964i $$0.386376\pi$$
$$992$$ 0 0
$$993$$ 22.0000 0.698149
$$994$$ 0 0
$$995$$ −28.0000 −0.887660
$$996$$ 0 0
$$997$$ −6.00000 −0.190022 −0.0950110 0.995476i $$-0.530289\pi$$
−0.0950110 + 0.995476i $$0.530289\pi$$
$$998$$ 0 0
$$999$$ 5.00000 0.158193
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.c.1.1 1
3.2 odd 2 5328.2.a.o.1.1 1
4.3 odd 2 296.2.a.a.1.1 1
8.3 odd 2 2368.2.a.n.1.1 1
8.5 even 2 2368.2.a.g.1.1 1
12.11 even 2 2664.2.a.f.1.1 1
20.19 odd 2 7400.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.a.a.1.1 1 4.3 odd 2
592.2.a.c.1.1 1 1.1 even 1 trivial
2368.2.a.g.1.1 1 8.5 even 2
2368.2.a.n.1.1 1 8.3 odd 2
2664.2.a.f.1.1 1 12.11 even 2
5328.2.a.o.1.1 1 3.2 odd 2
7400.2.a.f.1.1 1 20.19 odd 2