# Properties

 Label 1386.2.a.k.1.1 Level $1386$ Weight $2$ Character 1386.1 Self dual yes Analytic conductor $11.067$ 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] = [1386,2,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, 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("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.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: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1386.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} -1.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +14.0000 q^{53} -2.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -12.0000 q^{59} -14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} -12.0000 q^{71} +6.00000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{80} +6.00000 q^{82} +12.0000 q^{85} -1.00000 q^{88} +6.00000 q^{89} +2.00000 q^{91} +4.00000 q^{92} +8.00000 q^{94} -8.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 2.00000 0.316228
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −2.00000 −0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ 6.00000 0.662589
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ 0 0
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 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 $$0.5\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 2.00000 0.158114
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 12.0000 0.920358
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −10.0000 −0.760286 −0.380143 0.924928i $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 14.0000 0.961524
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ 2.00000 0.131024 0.0655122 0.997852i $$-0.479132\pi$$
0.0655122 + 0.997852i $$0.479132\pi$$
$$234$$ 0 0
$$235$$ 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 $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −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 $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 4.00000 0.248069
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 28.0000 1.72003
$$266$$ −4.00000 −0.245256
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 22.0000 1.32907
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 0 0
$$280$$ 2.00000 0.119523
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −4.00000 −0.234888
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ −28.0000 −1.60328
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 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
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ 4.00000 0.222911
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ 12.0000 0.650791
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ −24.0000 −1.27379
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 12.0000 0.628109
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −4.00000 −0.207950
$$371$$ 14.0000 0.726844
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ 36.0000 1.84920 0.924598 0.380945i $$-0.124401\pi$$
0.924598 + 0.380945i $$0.124401\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ −4.00000 −0.204658
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ −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 $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 12.0000 0.592638
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 14.0000 0.679900
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ −14.0000 −0.677507
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ −16.0000 −0.765384
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 12.0000 0.570782
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ −12.0000 −0.568216
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 8.00000 0.373002
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 2.00000 0.0926482
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 16.0000 0.738025
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 6.00000 0.275010
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −28.0000 −1.27141
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 0 0
$$490$$ 2.00000 0.0903508
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −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 $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ −20.0000 −0.889988
$$506$$ −4.00000 −0.177822
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ −2.00000 −0.0878750
$$519$$ 0 0
$$520$$ 4.00000 0.175412
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ −36.0000 −1.57417 −0.787085 0.616844i $$-0.788411\pi$$
−0.787085 + 0.616844i $$0.788411\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 28.0000 1.21624
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −30.0000 −1.29339
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 6.00000 0.257248
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ −24.0000 −1.02617 −0.513083 0.858339i $$-0.671497\pi$$
−0.513083 + 0.858339i $$0.671497\pi$$
$$548$$ 22.0000 0.939793
$$549$$ 0 0
$$550$$ 1.00000 0.0426401
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ 46.0000 1.94908 0.974541 0.224208i $$-0.0719796\pi$$
0.974541 + 0.224208i $$0.0719796\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 12.0000 0.504844
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ −4.00000 −0.166091
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −14.0000 −0.579821
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ −24.0000 −0.988064
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ −28.0000 −1.13369
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ −46.0000 −1.85792 −0.928961 0.370177i $$-0.879297\pi$$
−0.928961 + 0.370177i $$0.879297\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −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
$$636$$ 0 0
$$637$$ 2.00000 0.0792429
$$638$$ 2.00000 0.0791808
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ −10.0000 −0.394976 −0.197488 0.980305i $$-0.563278\pi$$
−0.197488 + 0.980305i $$0.563278\pi$$
$$642$$ 0 0
$$643$$ 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 $$-0.787996\pi$$
−0.786281 + 0.617869i $$0.787996\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ −20.0000 −0.783260
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ 8.00000 0.311872
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ 14.0000 0.540464
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ −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 $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ 0 0
$$685$$ 44.0000 1.68115
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 28.0000 1.06672
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 24.0000 0.910372
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 34.0000 1.28692
$$699$$ 0 0
$$700$$ −1.00000 −0.0377964
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ −10.0000 −0.376089
$$708$$ 0 0
$$709$$ −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
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ −32.0000 −1.19423
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −14.0000 −0.520306
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 0 0
$$730$$ 12.0000 0.444140
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ −48.0000 −1.76571 −0.882854 0.469647i $$-0.844381\pi$$
−0.882854 + 0.469647i $$0.844381\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 14.0000 0.513956
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ −20.0000 −0.732743
$$746$$ 26.0000 0.951928
$$747$$ 0 0
$$748$$ −6.00000 −0.219382
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 36.0000 1.30758
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ −26.0000 −0.942499 −0.471250 0.882000i $$-0.656197\pi$$
−0.471250 + 0.882000i $$0.656197\pi$$
$$762$$ 0 0
$$763$$ 10.0000 0.362024
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ 38.0000 1.37032 0.685158 0.728395i $$-0.259733\pi$$
0.685158 + 0.728395i $$0.259733\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 0 0
$$772$$ −6.00000 −0.215945
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ −18.0000 −0.645331
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ 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
$$792$$ 0 0
$$793$$ −28.0000 −0.994309
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 14.0000 0.494357
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 8.00000 0.281963
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ 50.0000 1.75791 0.878953 0.476908i $$-0.158243\pi$$
0.878953 + 0.476908i $$0.158243\pi$$
$$810$$ 0 0
$$811$$ −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
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 38.0000 1.32864
$$819$$ 0 0
$$820$$ 12.0000 0.419058
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.00000 0.0693375
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −26.0000 −0.896019
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −18.0000 −0.619219
$$846$$ 0 0
$$847$$ 1.00000 0.0343604
$$848$$ 14.0000 0.480762
$$849$$ 0 0
$$850$$ −6.00000 −0.205798
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ −14.0000 −0.479070
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.00000 0.272481
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 0 0
$$865$$ −20.0000 −0.680020
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ −4.00000 −0.135769
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 10.0000 0.338643
$$873$$ 0 0
$$874$$ −16.0000 −0.541208
$$875$$ −12.0000 −0.405674
$$876$$ 0 0
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ −34.0000 −1.14549 −0.572745 0.819734i $$-0.694121\pi$$
−0.572745 + 0.819734i $$0.694121\pi$$
$$882$$ 0 0
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ −12.0000 −0.401790
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 8.00000 0.267411
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 22.0000 0.734150
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 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
$$909$$ 0 0
$$910$$ 4.00000 0.132599
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 0 0
$$922$$ 30.0000 0.987997
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 40.0000 1.31448
$$927$$ 0 0
$$928$$ −2.00000 −0.0656532
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 2.00000 0.0655122
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 0 0
$$940$$ 16.0000 0.521862
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −44.0000 −1.42981 −0.714904 0.699223i $$-0.753530\pi$$
−0.714904 + 0.699223i $$0.753530\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 6.00000 0.194461
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ −8.00000 −0.258874
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 22.0000 0.710417
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −4.00000 −0.128965
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ −28.0000 −0.899026
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ 12.0000 0.384702
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 2.00000 0.0638877
$$981$$ 0 0
$$982$$ 28.0000 0.893516
$$983$$ 32.0000 1.02064 0.510321 0.859984i $$-0.329527\pi$$
0.510321 + 0.859984i $$0.329527\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ −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
$$999$$ 0 0
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 1386.2.a.k.1.1 1
3.2 odd 2 462.2.a.a.1.1 1
7.6 odd 2 9702.2.a.bf.1.1 1
12.11 even 2 3696.2.a.s.1.1 1
21.20 even 2 3234.2.a.n.1.1 1
33.32 even 2 5082.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.a.1.1 1 3.2 odd 2
1386.2.a.k.1.1 1 1.1 even 1 trivial
3234.2.a.n.1.1 1 21.20 even 2
3696.2.a.s.1.1 1 12.11 even 2
5082.2.a.q.1.1 1 33.32 even 2
9702.2.a.bf.1.1 1 7.6 odd 2