# Properties

 Label 4410.2.a.bd.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +6.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -2.00000 q^{19} -1.00000 q^{20} +6.00000 q^{22} +3.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} +3.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} +9.00000 q^{41} -7.00000 q^{43} +6.00000 q^{44} +3.00000 q^{46} +1.00000 q^{50} +4.00000 q^{52} +6.00000 q^{53} -6.00000 q^{55} +3.00000 q^{58} -6.00000 q^{59} -5.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +5.00000 q^{67} +6.00000 q^{71} +16.0000 q^{73} -4.00000 q^{74} -2.00000 q^{76} +2.00000 q^{79} -1.00000 q^{80} +9.00000 q^{82} +3.00000 q^{83} -7.00000 q^{86} +6.00000 q^{88} -15.0000 q^{89} +3.00000 q^{92} +2.00000 q^{95} -14.0000 q^{97} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 6.00000 1.27920
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ −7.00000 −1.06749 −0.533745 0.845645i $$-0.679216\pi$$
−0.533745 + 0.845645i $$0.679216\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −6.00000 −0.809040
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 3.00000 0.393919
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −5.00000 −0.640184 −0.320092 0.947386i $$-0.603714\pi$$
−0.320092 + 0.947386i $$0.603714\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 16.0000 1.87266 0.936329 0.351123i $$-0.114200\pi$$
0.936329 + 0.351123i $$0.114200\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 9.00000 0.993884
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −7.00000 −0.754829
$$87$$ 0 0
$$88$$ 6.00000 0.639602
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 3.00000 0.312772
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 15.0000 1.45010 0.725052 0.688694i $$-0.241816\pi$$
0.725052 + 0.688694i $$0.241816\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ −6.00000 −0.572078
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ −6.00000 −0.552345
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ −5.00000 −0.452679
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 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
$$133$$ 0 0
$$134$$ 5.00000 0.431934
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 24.0000 2.00698
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 16.0000 1.32417
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 2.00000 0.159111
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 9.00000 0.702782
$$165$$ 0 0
$$166$$ 3.00000 0.232845
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −7.00000 −0.533745
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 0 0
$$178$$ −15.0000 −1.12430
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ −11.0000 −0.817624 −0.408812 0.912619i $$-0.634057\pi$$
−0.408812 + 0.912619i $$0.634057\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 3.00000 0.221163
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 2.00000 0.145095
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 15.0000 1.05540
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ 1.00000 0.0696733
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 15.0000 1.02538
$$215$$ 7.00000 0.477396
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 11.0000 0.745014
$$219$$ 0 0
$$220$$ −6.00000 −0.404520
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 28.0000 1.87502 0.937509 0.347960i $$-0.113126\pi$$
0.937509 + 0.347960i $$0.113126\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −3.00000 −0.197814
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 25.0000 1.60706
$$243$$ 0 0
$$244$$ −5.00000 −0.320092
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ −8.00000 −0.508001
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 5.00000 0.305424
$$269$$ 15.0000 0.914566 0.457283 0.889321i $$-0.348823\pi$$
0.457283 + 0.889321i $$0.348823\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 10.0000 0.599760
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ −3.00000 −0.176166
$$291$$ 0 0
$$292$$ 16.0000 0.936329
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ −4.00000 −0.232495
$$297$$ 0 0
$$298$$ −15.0000 −0.868927
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4.00000 −0.230174
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 5.00000 0.286299
$$306$$ 0 0
$$307$$ −5.00000 −0.285365 −0.142683 0.989769i $$-0.545573\pi$$
−0.142683 + 0.989769i $$0.545573\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8.00000 0.454369
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ −8.00000 −0.452187 −0.226093 0.974106i $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ −12.0000 −0.673987 −0.336994 0.941507i $$-0.609410\pi$$
−0.336994 + 0.941507i $$0.609410\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 9.00000 0.496942
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 3.00000 0.164646
$$333$$ 0 0
$$334$$ −3.00000 −0.164153
$$335$$ −5.00000 −0.273179
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 3.00000 0.163178
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −48.0000 −2.59935
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −7.00000 −0.377415
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9.00000 −0.483145 −0.241573 0.970383i $$-0.577663\pi$$
−0.241573 + 0.970383i $$0.577663\pi$$
$$348$$ 0 0
$$349$$ −17.0000 −0.909989 −0.454995 0.890494i $$-0.650359\pi$$
−0.454995 + 0.890494i $$0.650359\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6.00000 0.319801
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −6.00000 −0.318447
$$356$$ −15.0000 −0.794998
$$357$$ 0 0
$$358$$ 24.0000 1.26844
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −11.0000 −0.578147
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −16.0000 −0.837478
$$366$$ 0 0
$$367$$ −35.0000 −1.82699 −0.913493 0.406855i $$-0.866625\pi$$
−0.913493 + 0.406855i $$0.866625\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 0 0
$$382$$ 6.00000 0.306987
$$383$$ −15.0000 −0.766464 −0.383232 0.923652i $$-0.625189\pi$$
−0.383232 + 0.923652i $$0.625189\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ −2.00000 −0.100631
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −15.0000 −0.749064 −0.374532 0.927214i $$-0.622197\pi$$
−0.374532 + 0.927214i $$0.622197\pi$$
$$402$$ 0 0
$$403$$ −32.0000 −1.59403
$$404$$ 15.0000 0.746278
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 13.0000 0.642809 0.321404 0.946942i $$-0.395845\pi$$
0.321404 + 0.946942i $$0.395845\pi$$
$$410$$ −9.00000 −0.444478
$$411$$ 0 0
$$412$$ 1.00000 0.0492665
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3.00000 −0.147264
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ −10.0000 −0.486792
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 15.0000 0.725052
$$429$$ 0 0
$$430$$ 7.00000 0.337570
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 0 0
$$433$$ 22.0000 1.05725 0.528626 0.848855i $$-0.322707\pi$$
0.528626 + 0.848855i $$0.322707\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ −6.00000 −0.286039
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −21.0000 −0.997740 −0.498870 0.866677i $$-0.666252\pi$$
−0.498870 + 0.866677i $$0.666252\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 28.0000 1.32584
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9.00000 0.424736 0.212368 0.977190i $$-0.431882\pi$$
0.212368 + 0.977190i $$0.431882\pi$$
$$450$$ 0 0
$$451$$ 54.0000 2.54276
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.0000 1.49690 0.748448 0.663193i $$-0.230799\pi$$
0.748448 + 0.663193i $$0.230799\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ −3.00000 −0.139876
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ −13.0000 −0.604161 −0.302081 0.953282i $$-0.597681\pi$$
−0.302081 + 0.953282i $$0.597681\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ −15.0000 −0.694117 −0.347059 0.937843i $$-0.612820\pi$$
−0.347059 + 0.937843i $$0.612820\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −6.00000 −0.276172
$$473$$ −42.0000 −1.93116
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −5.00000 −0.226339
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ 0 0
$$505$$ −15.0000 −0.667491
$$506$$ 18.0000 0.800198
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1.00000 −0.0440653
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −4.00000 −0.175412
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 21.0000 0.915644
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 36.0000 1.55933
$$534$$ 0 0
$$535$$ −15.0000 −0.648507
$$536$$ 5.00000 0.215967
$$537$$ 0 0
$$538$$ 15.0000 0.646696
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −11.0000 −0.471188
$$546$$ 0 0
$$547$$ −19.0000 −0.812381 −0.406191 0.913788i $$-0.633143\pi$$
−0.406191 + 0.913788i $$0.633143\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 0 0
$$550$$ 6.00000 0.255841
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ 10.0000 0.424094
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −28.0000 −1.18427
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 27.0000 1.13791 0.568957 0.822367i $$-0.307347\pi$$
0.568957 + 0.822367i $$0.307347\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 6.00000 0.251754
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −22.0000 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$572$$ 24.0000 1.00349
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.00000 0.125109
$$576$$ 0 0
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 0 0
$$580$$ −3.00000 −0.124568
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 16.0000 0.662085
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 6.00000 0.247016
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −15.0000 −0.614424
$$597$$ 0 0
$$598$$ 12.0000 0.490716
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 46.0000 1.87638 0.938190 0.346122i $$-0.112502\pi$$
0.938190 + 0.346122i $$0.112502\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −4.00000 −0.162758
$$605$$ −25.0000 −1.01639
$$606$$ 0 0
$$607$$ −23.0000 −0.933541 −0.466771 0.884378i $$-0.654583\pi$$
−0.466771 + 0.884378i $$0.654583\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 5.00000 0.202444
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −5.00000 −0.201784
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 14.0000 0.557331 0.278666 0.960388i $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ 2.00000 0.0795557
$$633$$ 0 0
$$634$$ −12.0000 −0.476581
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 18.0000 0.712627
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 48.0000 1.87839 0.939193 0.343391i $$-0.111576\pi$$
0.939193 + 0.343391i $$0.111576\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 9.00000 0.351391
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −41.0000 −1.59472 −0.797358 0.603507i $$-0.793769\pi$$
−0.797358 + 0.603507i $$0.793769\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ 3.00000 0.116423
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ −3.00000 −0.116073
$$669$$ 0 0
$$670$$ −5.00000 −0.193167
$$671$$ −30.0000 −1.15814
$$672$$ 0 0
$$673$$ 8.00000 0.308377 0.154189 0.988041i $$-0.450724\pi$$
0.154189 + 0.988041i $$0.450724\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −48.0000 −1.83801
$$683$$ −9.00000 −0.344375 −0.172188 0.985064i $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −7.00000 −0.266872
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 22.0000 0.836919 0.418460 0.908235i $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −9.00000 −0.341635
$$695$$ −10.0000 −0.379322
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −17.0000 −0.643459
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 3.00000 0.113308 0.0566542 0.998394i $$-0.481957\pi$$
0.0566542 + 0.998394i $$0.481957\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −31.0000 −1.16423 −0.582115 0.813107i $$-0.697775\pi$$
−0.582115 + 0.813107i $$0.697775\pi$$
$$710$$ −6.00000 −0.225176
$$711$$ 0 0
$$712$$ −15.0000 −0.562149
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ −24.0000 −0.897549
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −15.0000 −0.558242
$$723$$ 0 0
$$724$$ −11.0000 −0.408812
$$725$$ 3.00000 0.111417
$$726$$ 0 0
$$727$$ 19.0000 0.704671 0.352335 0.935874i $$-0.385388\pi$$
0.352335 + 0.935874i $$0.385388\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −16.0000 −0.592187
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −35.0000 −1.29187
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ 30.0000 1.10506
$$738$$ 0 0
$$739$$ 26.0000 0.956425 0.478213 0.878244i $$-0.341285\pi$$
0.478213 + 0.878244i $$0.341285\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −39.0000 −1.43077 −0.715386 0.698730i $$-0.753749\pi$$
−0.715386 + 0.698730i $$0.753749\pi$$
$$744$$ 0 0
$$745$$ 15.0000 0.549557
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 4.00000 0.145575
$$756$$ 0 0
$$757$$ −28.0000 −1.01768 −0.508839 0.860862i $$-0.669925\pi$$
−0.508839 + 0.860862i $$0.669925\pi$$
$$758$$ −34.0000 −1.23494
$$759$$ 0 0
$$760$$ 2.00000 0.0725476
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ −15.0000 −0.541972
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −50.0000 −1.80305 −0.901523 0.432731i $$-0.857550\pi$$
−0.901523 + 0.432731i $$0.857550\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −22.0000 −0.785214
$$786$$ 0 0
$$787$$ 43.0000 1.53278 0.766392 0.642373i $$-0.222050\pi$$
0.766392 + 0.642373i $$0.222050\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ −2.00000 −0.0711568
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ −48.0000 −1.70025 −0.850124 0.526583i $$-0.823473\pi$$
−0.850124 + 0.526583i $$0.823473\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −15.0000 −0.529668
$$803$$ 96.0000 3.38777
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −32.0000 −1.12715
$$807$$ 0 0
$$808$$ 15.0000 0.527698
$$809$$ −21.0000 −0.738321 −0.369160 0.929366i $$-0.620355\pi$$
−0.369160 + 0.929366i $$0.620355\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 14.0000 0.489798
$$818$$ 13.0000 0.454534
$$819$$ 0 0
$$820$$ −9.00000 −0.314294
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −19.0000 −0.662298 −0.331149 0.943578i $$-0.607436\pi$$
−0.331149 + 0.943578i $$0.607436\pi$$
$$824$$ 1.00000 0.0348367
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −15.0000 −0.521601 −0.260801 0.965393i $$-0.583986\pi$$
−0.260801 + 0.965393i $$0.583986\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ −3.00000 −0.104132
$$831$$ 0 0
$$832$$ 4.00000 0.138675
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ −12.0000 −0.415029
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 17.0000 0.585859
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 15.0000 0.512689
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 7.00000 0.238698
$$861$$ 0 0
$$862$$ −30.0000 −1.02180
$$863$$ −27.0000 −0.919091 −0.459545 0.888154i $$-0.651988\pi$$
−0.459545 + 0.888154i $$0.651988\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 22.0000 0.747590
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 11.0000 0.372507
$$873$$ 0 0
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 0 0
$$880$$ −6.00000 −0.202260
$$881$$ 57.0000 1.92038 0.960189 0.279350i $$-0.0901189\pi$$
0.960189 + 0.279350i $$0.0901189\pi$$
$$882$$ 0 0
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −21.0000 −0.705509
$$887$$ −21.0000 −0.705111 −0.352555 0.935791i $$-0.614687\pi$$
−0.352555 + 0.935791i $$0.614687\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 15.0000 0.502801
$$891$$ 0 0
$$892$$ 28.0000 0.937509
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 9.00000 0.300334
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 54.0000 1.79800
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 11.0000 0.365652
$$906$$ 0 0
$$907$$ −25.0000 −0.830111 −0.415056 0.909796i $$-0.636238\pi$$
−0.415056 + 0.909796i $$0.636238\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 0 0
$$913$$ 18.0000 0.595713
$$914$$ 32.0000 1.05847
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 14.0000 0.461817 0.230909 0.972975i $$-0.425830\pi$$
0.230909 + 0.972975i $$0.425830\pi$$
$$920$$ −3.00000 −0.0989071
$$921$$ 0 0
$$922$$ 18.0000 0.592798
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ −13.0000 −0.427207
$$927$$ 0 0
$$928$$ 3.00000 0.0984798
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ −15.0000 −0.490815
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 28.0000 0.914720 0.457360 0.889282i $$-0.348795\pi$$
0.457360 + 0.889282i $$0.348795\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 0 0
$$943$$ 27.0000 0.879241
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −42.0000 −1.36554
$$947$$ 3.00000 0.0974869 0.0487435 0.998811i $$-0.484478\pi$$
0.0487435 + 0.998811i $$0.484478\pi$$
$$948$$ 0 0
$$949$$ 64.0000 2.07753
$$950$$ −2.00000 −0.0648886
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −60.0000 −1.94359 −0.971795 0.235826i $$-0.924220\pi$$
−0.971795 + 0.235826i $$0.924220\pi$$
$$954$$ 0 0
$$955$$ −6.00000 −0.194155
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 12.0000 0.387702
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −16.0000 −0.515861
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 35.0000 1.12552 0.562762 0.826619i $$-0.309739\pi$$
0.562762 + 0.826619i $$0.309739\pi$$
$$968$$ 25.0000 0.803530
$$969$$ 0 0
$$970$$ 14.0000 0.449513
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −5.00000 −0.160046
$$977$$ 6.00000 0.191957 0.0959785 0.995383i $$-0.469402\pi$$
0.0959785 + 0.995383i $$0.469402\pi$$
$$978$$ 0 0
$$979$$ −90.0000 −2.87641
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 39.0000 1.24391 0.621953 0.783054i $$-0.286339\pi$$
0.621953 + 0.783054i $$0.286339\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ −21.0000 −0.667761
$$990$$ 0 0
$$991$$ −28.0000 −0.889449 −0.444725 0.895667i $$-0.646698\pi$$
−0.444725 + 0.895667i $$0.646698\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ −22.0000 −0.696398
$$999$$ 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 4410.2.a.bd.1.1 1
3.2 odd 2 490.2.a.b.1.1 1
7.3 odd 6 630.2.k.b.541.1 2
7.5 odd 6 630.2.k.b.361.1 2
7.6 odd 2 4410.2.a.bm.1.1 1
12.11 even 2 3920.2.a.bc.1.1 1
15.2 even 4 2450.2.c.l.99.1 2
15.8 even 4 2450.2.c.l.99.2 2
15.14 odd 2 2450.2.a.bc.1.1 1
21.2 odd 6 490.2.e.h.361.1 2
21.5 even 6 70.2.e.c.11.1 2
21.11 odd 6 490.2.e.h.471.1 2
21.17 even 6 70.2.e.c.51.1 yes 2
21.20 even 2 490.2.a.c.1.1 1
84.47 odd 6 560.2.q.g.81.1 2
84.59 odd 6 560.2.q.g.401.1 2
84.83 odd 2 3920.2.a.p.1.1 1
105.17 odd 12 350.2.j.b.149.2 4
105.38 odd 12 350.2.j.b.149.1 4
105.47 odd 12 350.2.j.b.249.1 4
105.59 even 6 350.2.e.e.51.1 2
105.62 odd 4 2450.2.c.g.99.1 2
105.68 odd 12 350.2.j.b.249.2 4
105.83 odd 4 2450.2.c.g.99.2 2
105.89 even 6 350.2.e.e.151.1 2
105.104 even 2 2450.2.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.c.11.1 2 21.5 even 6
70.2.e.c.51.1 yes 2 21.17 even 6
350.2.e.e.51.1 2 105.59 even 6
350.2.e.e.151.1 2 105.89 even 6
350.2.j.b.149.1 4 105.38 odd 12
350.2.j.b.149.2 4 105.17 odd 12
350.2.j.b.249.1 4 105.47 odd 12
350.2.j.b.249.2 4 105.68 odd 12
490.2.a.b.1.1 1 3.2 odd 2
490.2.a.c.1.1 1 21.20 even 2
490.2.e.h.361.1 2 21.2 odd 6
490.2.e.h.471.1 2 21.11 odd 6
560.2.q.g.81.1 2 84.47 odd 6
560.2.q.g.401.1 2 84.59 odd 6
630.2.k.b.361.1 2 7.5 odd 6
630.2.k.b.541.1 2 7.3 odd 6
2450.2.a.w.1.1 1 105.104 even 2
2450.2.a.bc.1.1 1 15.14 odd 2
2450.2.c.g.99.1 2 105.62 odd 4
2450.2.c.g.99.2 2 105.83 odd 4
2450.2.c.l.99.1 2 15.2 even 4
2450.2.c.l.99.2 2 15.8 even 4
3920.2.a.p.1.1 1 84.83 odd 2
3920.2.a.bc.1.1 1 12.11 even 2
4410.2.a.bd.1.1 1 1.1 even 1 trivial
4410.2.a.bm.1.1 1 7.6 odd 2