# Properties

 Label 462.2.a.a.1.1 Level $462$ Weight $2$ Character 462.1 Self dual yes Analytic conductor $3.689$ 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] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, 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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.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: $$3.68908857338$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 462.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +1.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} +2.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -1.00000 q^{84} +12.0000 q^{85} -2.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +2.00000 q^{91} -4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 1.00000 0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ −1.00000 −0.218218
$$22$$ −1.00000 −0.213201
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 6.00000 1.02899
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 4.00000 0.648886
$$39$$ −2.00000 −0.320256
$$40$$ 2.00000 0.316228
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −2.00000 −0.298142
$$46$$ 4.00000 0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 6.00000 0.840168
$$52$$ 2.00000 0.277350
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −2.00000 −0.269680
$$56$$ −1.00000 −0.133631
$$57$$ 4.00000 0.529813
$$58$$ −2.00000 −0.262613
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 1.00000 0.123091
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 4.00000 0.481543
$$70$$ 2.00000 0.239046
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 1.00000 0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 1.00000 0.113961
$$78$$ 2.00000 0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 12.0000 1.30158
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ −1.00000 −0.106600
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 2.00000 0.209657
$$92$$ −4.00000 −0.417029
$$93$$ 4.00000 0.414781
$$94$$ 8.00000 0.825137
$$95$$ 8.00000 0.820783
$$96$$ 1.00000 0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 1.00000 0.100504
$$100$$ −1.00000 −0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 2.00000 0.195180
$$106$$ 14.0000 1.35980
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 2.00000 0.189832
$$112$$ 1.00000 0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 8.00000 0.746004
$$116$$ 2.00000 0.185695
$$117$$ 2.00000 0.184900
$$118$$ −12.0000 −1.10469
$$119$$ −6.00000 −0.550019
$$120$$ −2.00000 −0.182574
$$121$$ 1.00000 0.0909091
$$122$$ 14.0000 1.26750
$$123$$ 6.00000 0.541002
$$124$$ −4.00000 −0.359211
$$125$$ 12.0000 1.07331
$$126$$ −1.00000 −0.0890871
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ −4.00000 −0.346844
$$134$$ −4.00000 −0.345547
$$135$$ 2.00000 0.172133
$$136$$ 6.00000 0.514496
$$137$$ −22.0000 −1.87959 −0.939793 0.341743i $$-0.888983\pi$$
−0.939793 + 0.341743i $$0.888983\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 8.00000 0.673722
$$142$$ −12.0000 −1.00702
$$143$$ 2.00000 0.167248
$$144$$ 1.00000 0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ −6.00000 −0.496564
$$147$$ −1.00000 −0.0824786
$$148$$ −2.00000 −0.164399
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −6.00000 −0.485071
$$154$$ −1.00000 −0.0805823
$$155$$ 8.00000 0.642575
$$156$$ −2.00000 −0.160128
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 14.0000 1.11027
$$160$$ 2.00000 0.158114
$$161$$ −4.00000 −0.315244
$$162$$ −1.00000 −0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 2.00000 0.155700
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ −12.0000 −0.920358
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ 2.00000 0.151620
$$175$$ −1.00000 −0.0755929
$$176$$ 1.00000 0.0753778
$$177$$ −12.0000 −0.901975
$$178$$ 6.00000 0.449719
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 14.0000 1.03491
$$184$$ 4.00000 0.294884
$$185$$ 4.00000 0.294086
$$186$$ −4.00000 −0.293294
$$187$$ −6.00000 −0.438763
$$188$$ −8.00000 −0.583460
$$189$$ −1.00000 −0.0727393
$$190$$ −8.00000 −0.580381
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 4.00000 0.286446
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ −10.0000 −0.703598
$$203$$ 2.00000 0.140372
$$204$$ 6.00000 0.420084
$$205$$ 12.0000 0.838116
$$206$$ 4.00000 0.278693
$$207$$ −4.00000 −0.278019
$$208$$ 2.00000 0.138675
$$209$$ −4.00000 −0.276686
$$210$$ −2.00000 −0.138013
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ −14.0000 −0.961524
$$213$$ −12.0000 −0.822226
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −4.00000 −0.271538
$$218$$ −10.0000 −0.677285
$$219$$ −6.00000 −0.405442
$$220$$ −2.00000 −0.134840
$$221$$ −12.0000 −0.807207
$$222$$ −2.00000 −0.134231
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ 6.00000 0.399114
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ −1.00000 −0.0657952
$$232$$ −2.00000 −0.131306
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 16.0000 1.04372
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 6.00000 0.388922
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ −14.0000 −0.896258
$$245$$ −2.00000 −0.127775
$$246$$ −6.00000 −0.382546
$$247$$ −8.00000 −0.509028
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −4.00000 −0.251478
$$254$$ −16.0000 −1.00393
$$255$$ −12.0000 −0.751469
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ −4.00000 −0.248069
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 28.0000 1.72003
$$266$$ 4.00000 0.245256
$$267$$ 6.00000 0.367194
$$268$$ 4.00000 0.244339
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ −2.00000 −0.121046
$$274$$ 22.0000 1.32907
$$275$$ −1.00000 −0.0603023
$$276$$ 4.00000 0.240772
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ −4.00000 −0.239474
$$280$$ 2.00000 0.119523
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 12.0000 0.712069
$$285$$ −8.00000 −0.473879
$$286$$ −2.00000 −0.118262
$$287$$ −6.00000 −0.354169
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 4.00000 0.234888
$$291$$ 14.0000 0.820695
$$292$$ 6.00000 0.351123
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −24.0000 −1.39733
$$296$$ 2.00000 0.116248
$$297$$ −1.00000 −0.0580259
$$298$$ −10.0000 −0.579284
$$299$$ −8.00000 −0.462652
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −10.0000 −0.574485
$$304$$ −4.00000 −0.229416
$$305$$ 28.0000 1.60328
$$306$$ 6.00000 0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ 4.00000 0.227552
$$310$$ −8.00000 −0.454369
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 2.00000 0.113228
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ −2.00000 −0.112687
$$316$$ 0 0
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ −14.0000 −0.785081
$$319$$ 2.00000 0.111979
$$320$$ −2.00000 −0.111803
$$321$$ −12.0000 −0.669775
$$322$$ 4.00000 0.222911
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ 20.0000 1.10770
$$327$$ −10.0000 −0.553001
$$328$$ 6.00000 0.331295
$$329$$ −8.00000 −0.441054
$$330$$ −2.00000 −0.110096
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ −1.00000 −0.0545545
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 6.00000 0.325875
$$340$$ 12.0000 0.650791
$$341$$ −4.00000 −0.216612
$$342$$ 4.00000 0.216295
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −8.00000 −0.430706
$$346$$ −10.0000 −0.537603
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −2.00000 −0.106752
$$352$$ −1.00000 −0.0533002
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 12.0000 0.637793
$$355$$ −24.0000 −1.27379
$$356$$ −6.00000 −0.317999
$$357$$ 6.00000 0.317554
$$358$$ 4.00000 0.211407
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −3.00000 −0.157895
$$362$$ 14.0000 0.735824
$$363$$ −1.00000 −0.0524864
$$364$$ 2.00000 0.104828
$$365$$ −12.0000 −0.628109
$$366$$ −14.0000 −0.731792
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −6.00000 −0.312348
$$370$$ −4.00000 −0.207950
$$371$$ −14.0000 −0.726844
$$372$$ 4.00000 0.207390
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 6.00000 0.310253
$$375$$ −12.0000 −0.619677
$$376$$ 8.00000 0.412568
$$377$$ 4.00000 0.206010
$$378$$ 1.00000 0.0514344
$$379$$ 36.0000 1.84920 0.924598 0.380945i $$-0.124401\pi$$
0.924598 + 0.380945i $$0.124401\pi$$
$$380$$ 8.00000 0.410391
$$381$$ −16.0000 −0.819705
$$382$$ −4.00000 −0.204658
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −2.00000 −0.101929
$$386$$ 6.00000 0.305392
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ 24.0000 1.21373
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 12.0000 0.601506
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −8.00000 −0.398508
$$404$$ 10.0000 0.497519
$$405$$ −2.00000 −0.0993808
$$406$$ −2.00000 −0.0992583
$$407$$ −2.00000 −0.0991363
$$408$$ −6.00000 −0.297044
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ 22.0000 1.08518
$$412$$ −4.00000 −0.197066
$$413$$ 12.0000 0.590481
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ −12.0000 −0.587643
$$418$$ 4.00000 0.195646
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 0 0
$$423$$ −8.00000 −0.388973
$$424$$ 14.0000 0.679900
$$425$$ 6.00000 0.291043
$$426$$ 12.0000 0.581402
$$427$$ −14.0000 −0.677507
$$428$$ 12.0000 0.580042
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 4.00000 0.191785
$$436$$ 10.0000 0.478913
$$437$$ 16.0000 0.765384
$$438$$ 6.00000 0.286691
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000 0.570782
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 12.0000 0.568855
$$446$$ 12.0000 0.568216
$$447$$ −10.0000 −0.472984
$$448$$ 1.00000 0.0472456
$$449$$ −22.0000 −1.03824 −0.519122 0.854700i $$-0.673741\pi$$
−0.519122 + 0.854700i $$0.673741\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −6.00000 −0.282529
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.00000 −0.187523
$$456$$ −4.00000 −0.187317
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 6.00000 0.280056
$$460$$ 8.00000 0.373002
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 1.00000 0.0465242
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ −8.00000 −0.370991
$$466$$ 2.00000 0.0926482
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 4.00000 0.184703
$$470$$ −16.0000 −0.738025
$$471$$ −2.00000 −0.0921551
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ −6.00000 −0.275010
$$477$$ −14.0000 −0.641016
$$478$$ −24.0000 −1.09773
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ −4.00000 −0.182384
$$482$$ 26.0000 1.18427
$$483$$ 4.00000 0.182006
$$484$$ 1.00000 0.0454545
$$485$$ 28.0000 1.27141
$$486$$ 1.00000 0.0453609
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 14.0000 0.633750
$$489$$ 20.0000 0.904431
$$490$$ 2.00000 0.0903508
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −12.0000 −0.540453
$$494$$ 8.00000 0.359937
$$495$$ −2.00000 −0.0898933
$$496$$ −4.00000 −0.179605
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 0 0
$$502$$ −20.0000 −0.892644
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ −20.0000 −0.889988
$$506$$ 4.00000 0.177822
$$507$$ 9.00000 0.399704
$$508$$ 16.0000 0.709885
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 12.0000 0.531369
$$511$$ 6.00000 0.265424
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ −18.0000 −0.793946
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ 2.00000 0.0878750
$$519$$ −10.0000 −0.438951
$$520$$ 4.00000 0.175412
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −36.0000 −1.57417 −0.787085 0.616844i $$-0.788411\pi$$
−0.787085 + 0.616844i $$0.788411\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ −1.00000 −0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ −28.0000 −1.21624
$$531$$ 12.0000 0.520756
$$532$$ −4.00000 −0.173422
$$533$$ −12.0000 −0.519778
$$534$$ −6.00000 −0.259645
$$535$$ −24.0000 −1.03761
$$536$$ −4.00000 −0.172774
$$537$$ 4.00000 0.172613
$$538$$ −30.0000 −1.29339
$$539$$ 1.00000 0.0430730
$$540$$ 2.00000 0.0860663
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ 0 0
$$543$$ 14.0000 0.600798
$$544$$ 6.00000 0.257248
$$545$$ −20.0000 −0.856706
$$546$$ 2.00000 0.0855921
$$547$$ −24.0000 −1.02617 −0.513083 0.858339i $$-0.671497\pi$$
−0.513083 + 0.858339i $$0.671497\pi$$
$$548$$ −22.0000 −0.939793
$$549$$ −14.0000 −0.597505
$$550$$ 1.00000 0.0426401
$$551$$ −8.00000 −0.340811
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ −4.00000 −0.169791
$$556$$ 12.0000 0.508913
$$557$$ −46.0000 −1.94908 −0.974541 0.224208i $$-0.928020\pi$$
−0.974541 + 0.224208i $$0.928020\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 6.00000 0.253320
$$562$$ 18.0000 0.759284
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 12.0000 0.504844
$$566$$ 4.00000 0.168133
$$567$$ 1.00000 0.0419961
$$568$$ −12.0000 −0.503509
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 8.00000 0.335083
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ −4.00000 −0.167102
$$574$$ 6.00000 0.250435
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 6.00000 0.249351
$$580$$ −4.00000 −0.166091
$$581$$ 0 0
$$582$$ −14.0000 −0.580319
$$583$$ −14.0000 −0.579821
$$584$$ −6.00000 −0.248282
$$585$$ −4.00000 −0.165380
$$586$$ 14.0000 0.578335
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 16.0000 0.659269
$$590$$ 24.0000 0.988064
$$591$$ 6.00000 0.246807
$$592$$ −2.00000 −0.0821995
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 12.0000 0.491952
$$596$$ 10.0000 0.409616
$$597$$ 12.0000 0.491127
$$598$$ 8.00000 0.327144
$$599$$ −44.0000 −1.79779 −0.898896 0.438163i $$-0.855629\pi$$
−0.898896 + 0.438163i $$0.855629\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ −2.00000 −0.0813116
$$606$$ 10.0000 0.406222
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 4.00000 0.162221
$$609$$ −2.00000 −0.0810441
$$610$$ −28.0000 −1.13369
$$611$$ −16.0000 −0.647291
$$612$$ −6.00000 −0.242536
$$613$$ −46.0000 −1.85792 −0.928961 0.370177i $$-0.879297\pi$$
−0.928961 + 0.370177i $$0.879297\pi$$
$$614$$ 12.0000 0.484281
$$615$$ −12.0000 −0.483887
$$616$$ −1.00000 −0.0402911
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 4.00000 0.160514
$$622$$ −8.00000 −0.320771
$$623$$ −6.00000 −0.240385
$$624$$ −2.00000 −0.0800641
$$625$$ −19.0000 −0.760000
$$626$$ 14.0000 0.559553
$$627$$ 4.00000 0.159745
$$628$$ 2.00000 0.0798087
$$629$$ 12.0000 0.478471
$$630$$ 2.00000 0.0796819
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −10.0000 −0.397151
$$635$$ −32.0000 −1.26988
$$636$$ 14.0000 0.555136
$$637$$ 2.00000 0.0792429
$$638$$ −2.00000 −0.0791808
$$639$$ 12.0000 0.474713
$$640$$ 2.00000 0.0790569
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 40.0000 1.57256 0.786281 0.617869i $$-0.212004\pi$$
0.786281 + 0.617869i $$0.212004\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 12.0000 0.471041
$$650$$ 2.00000 0.0784465
$$651$$ 4.00000 0.156772
$$652$$ −20.0000 −0.783260
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000 0.234082
$$658$$ 8.00000 0.311872
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 8.00000 0.310227
$$666$$ 2.00000 0.0774984
$$667$$ −8.00000 −0.309761
$$668$$ 0 0
$$669$$ 12.0000 0.463947
$$670$$ 8.00000 0.309067
$$671$$ −14.0000 −0.540464
$$672$$ 1.00000 0.0385758
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 1.00000 0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −14.0000 −0.537271
$$680$$ −12.0000 −0.460179
$$681$$ 0 0
$$682$$ 4.00000 0.153168
$$683$$ −28.0000 −1.07139 −0.535695 0.844411i $$-0.679950\pi$$
−0.535695 + 0.844411i $$0.679950\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 44.0000 1.68115
$$686$$ −1.00000 −0.0381802
$$687$$ 14.0000 0.534133
$$688$$ 0 0
$$689$$ −28.0000 −1.06672
$$690$$ 8.00000 0.304555
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 10.0000 0.380143
$$693$$ 1.00000 0.0379869
$$694$$ 28.0000 1.06287
$$695$$ −24.0000 −0.910372
$$696$$ 2.00000 0.0758098
$$697$$ 36.0000 1.36360
$$698$$ −34.0000 −1.28692
$$699$$ 2.00000 0.0756469
$$700$$ −1.00000 −0.0377964
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ 8.00000 0.301726
$$704$$ 1.00000 0.0376889
$$705$$ −16.0000 −0.602595
$$706$$ 14.0000 0.526897
$$707$$ 10.0000 0.376089
$$708$$ −12.0000 −0.450988
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 24.0000 0.900704
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 16.0000 0.599205
$$714$$ −6.00000 −0.224544
$$715$$ −4.00000 −0.149592
$$716$$ −4.00000 −0.149487
$$717$$ −24.0000 −0.896296
$$718$$ −32.0000 −1.19423
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ −4.00000 −0.148968
$$722$$ 3.00000 0.111648
$$723$$ 26.0000 0.966950
$$724$$ −14.0000 −0.520306
$$725$$ −2.00000 −0.0742781
$$726$$ 1.00000 0.0371135
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ 0 0
$$732$$ 14.0000 0.517455
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 2.00000 0.0737711
$$736$$ 4.00000 0.147442
$$737$$ 4.00000 0.147342
$$738$$ 6.00000 0.220863
$$739$$ −48.0000 −1.76571 −0.882854 0.469647i $$-0.844381\pi$$
−0.882854 + 0.469647i $$0.844381\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 8.00000 0.293887
$$742$$ 14.0000 0.513956
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ −20.0000 −0.732743
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ −6.00000 −0.219382
$$749$$ 12.0000 0.438470
$$750$$ 12.0000 0.438178
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −20.0000 −0.728841
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −36.0000 −1.30758
$$759$$ 4.00000 0.145191
$$760$$ −8.00000 −0.290191
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 10.0000 0.362024
$$764$$ 4.00000 0.144715
$$765$$ 12.0000 0.433861
$$766$$ −16.0000 −0.578103
$$767$$ 24.0000 0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ 38.0000 1.37032 0.685158 0.728395i $$-0.259733\pi$$
0.685158 + 0.728395i $$0.259733\pi$$
$$770$$ 2.00000 0.0720750
$$771$$ −18.0000 −0.648254
$$772$$ −6.00000 −0.215945
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 14.0000 0.502571
$$777$$ 2.00000 0.0717496
$$778$$ −18.0000 −0.645331
$$779$$ 24.0000 0.859889
$$780$$ 4.00000 0.143223
$$781$$ 12.0000 0.429394
$$782$$ −24.0000 −0.858238
$$783$$ −2.00000 −0.0714742
$$784$$ 1.00000 0.0357143
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ −1.00000 −0.0355335
$$793$$ −28.0000 −0.994309
$$794$$ −18.0000 −0.638796
$$795$$ −28.0000 −0.993058
$$796$$ −12.0000 −0.425329
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 48.0000 1.69812
$$800$$ 1.00000 0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ 14.0000 0.494357
$$803$$ 6.00000 0.211735
$$804$$ −4.00000 −0.141069
$$805$$ 8.00000 0.281963
$$806$$ 8.00000 0.281788
$$807$$ −30.0000 −1.05605
$$808$$ −10.0000 −0.351799
$$809$$ −50.0000 −1.75791 −0.878953 0.476908i $$-0.841757\pi$$
−0.878953 + 0.476908i $$0.841757\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 0 0
$$814$$ 2.00000 0.0701000
$$815$$ 40.0000 1.40114
$$816$$ 6.00000 0.210042
$$817$$ 0 0
$$818$$ −38.0000 −1.32864
$$819$$ 2.00000 0.0698857
$$820$$ 12.0000 0.419058
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ −22.0000 −0.767338
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 1.00000 0.0348155
$$826$$ −12.0000 −0.417533
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 2.00000 0.0693375
$$833$$ −6.00000 −0.207888
$$834$$ 12.0000 0.415526
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 4.00000 0.138260
$$838$$ 12.0000 0.414533
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000 0.896019
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ 18.0000 0.619219
$$846$$ 8.00000 0.275046
$$847$$ 1.00000 0.0343604
$$848$$ −14.0000 −0.480762
$$849$$ 4.00000 0.137280
$$850$$ −6.00000 −0.205798
$$851$$ 8.00000 0.274236
$$852$$ −12.0000 −0.411113
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 14.0000 0.479070
$$855$$ 8.00000 0.273594
$$856$$ −12.0000 −0.410152
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ 8.00000 0.272481
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −20.0000 −0.680020
$$866$$ −34.0000 −1.15537
$$867$$ −19.0000 −0.645274
$$868$$ −4.00000 −0.135769
$$869$$ 0 0
$$870$$ −4.00000 −0.135613
$$871$$ 8.00000 0.271070
$$872$$ −10.0000 −0.338643
$$873$$ −14.0000 −0.473828
$$874$$ −16.0000 −0.541208
$$875$$ 12.0000 0.405674
$$876$$ −6.00000 −0.202721
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 14.0000 0.472208
$$880$$ −2.00000 −0.0674200
$$881$$ 34.0000 1.14549 0.572745 0.819734i $$-0.305879\pi$$
0.572745 + 0.819734i $$0.305879\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 24.0000 0.806751
$$886$$ −12.0000 −0.403148
$$887$$ 16.0000 0.537227 0.268614 0.963248i $$-0.413434\pi$$
0.268614 + 0.963248i $$0.413434\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 16.0000 0.536623
$$890$$ −12.0000 −0.402241
$$891$$ 1.00000 0.0335013
$$892$$ −12.0000 −0.401790
$$893$$ 32.0000 1.07084
$$894$$ 10.0000 0.334450
$$895$$ 8.00000 0.267411
$$896$$ −1.00000 −0.0334077
$$897$$ 8.00000 0.267112
$$898$$ 22.0000 0.734150
$$899$$ −8.00000 −0.266815
$$900$$ −1.00000 −0.0333333
$$901$$ 84.0000 2.79845
$$902$$ 6.00000 0.199778
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 28.0000 0.930751
$$906$$ 0 0
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ 0 0
$$909$$ 10.0000 0.331679
$$910$$ 4.00000 0.132599
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ −26.0000 −0.860004
$$915$$ −28.0000 −0.925651
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 12.0000 0.395413
$$922$$ 30.0000 0.987997
$$923$$ 24.0000 0.789970
$$924$$ −1.00000 −0.0328976
$$925$$ 2.00000 0.0657596
$$926$$ −40.0000 −1.31448
$$927$$ −4.00000 −0.131377
$$928$$ −2.00000 −0.0656532
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 8.00000 0.262330
$$931$$ −4.00000 −0.131095
$$932$$ −2.00000 −0.0655122
$$933$$ −8.00000 −0.261908
$$934$$ −20.0000 −0.654420
$$935$$ 12.0000 0.392442
$$936$$ −2.00000 −0.0653720
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 14.0000 0.456873
$$940$$ 16.0000 0.521862
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ 24.0000 0.781548
$$944$$ 12.0000 0.390567
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 44.0000 1.42981 0.714904 0.699223i $$-0.246470\pi$$
0.714904 + 0.699223i $$0.246470\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ −4.00000 −0.129777
$$951$$ −10.0000 −0.324272
$$952$$ 6.00000 0.194461
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 14.0000 0.453267
$$955$$ −8.00000 −0.258874
$$956$$ 24.0000 0.776215
$$957$$ −2.00000 −0.0646508
$$958$$ 24.0000 0.775405
$$959$$ −22.0000 −0.710417
$$960$$ 2.00000 0.0645497
$$961$$ −15.0000 −0.483871
$$962$$ 4.00000 0.128965
$$963$$ 12.0000 0.386695
$$964$$ −26.0000 −0.837404
$$965$$ 12.0000 0.386294
$$966$$ −4.00000 −0.128698
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ −24.0000 −0.770991
$$970$$ −28.0000 −0.899026
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 12.0000 0.384702
$$974$$ 0 0
$$975$$ 2.00000 0.0640513
$$976$$ −14.0000 −0.448129
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ −6.00000 −0.191761
$$980$$ −2.00000 −0.0638877
$$981$$ 10.0000 0.319275
$$982$$ 28.0000 0.893516
$$983$$ −32.0000 −1.02064 −0.510321 0.859984i $$-0.670473\pi$$
−0.510321 + 0.859984i $$0.670473\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 12.0000 0.382352
$$986$$ 12.0000 0.382158
$$987$$ 8.00000 0.254643
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 2.00000 0.0635642
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 20.0000 0.634681
$$994$$ −12.0000 −0.380617
$$995$$ 24.0000 0.760851
$$996$$ 0 0
$$997$$ 26.0000 0.823428 0.411714 0.911313i $$-0.364930\pi$$
0.411714 + 0.911313i $$0.364930\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ 2.00000 0.0632772
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 462.2.a.a.1.1 1
3.2 odd 2 1386.2.a.k.1.1 1
4.3 odd 2 3696.2.a.s.1.1 1
7.6 odd 2 3234.2.a.n.1.1 1
11.10 odd 2 5082.2.a.q.1.1 1
21.20 even 2 9702.2.a.bf.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.a.1.1 1 1.1 even 1 trivial
1386.2.a.k.1.1 1 3.2 odd 2
3234.2.a.n.1.1 1 7.6 odd 2
3696.2.a.s.1.1 1 4.3 odd 2
5082.2.a.q.1.1 1 11.10 odd 2
9702.2.a.bf.1.1 1 21.20 even 2