# Properties

 Label 1638.2.a.q.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +5.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +2.00000 q^{19} +5.00000 q^{22} -5.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +7.00000 q^{37} +2.00000 q^{38} +9.00000 q^{41} -12.0000 q^{43} +5.00000 q^{44} -5.00000 q^{46} +7.00000 q^{47} +1.00000 q^{49} -5.00000 q^{50} -1.00000 q^{52} +4.00000 q^{53} +1.00000 q^{56} -4.00000 q^{58} +6.00000 q^{59} +13.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +11.0000 q^{67} +4.00000 q^{68} +7.00000 q^{73} +7.00000 q^{74} +2.00000 q^{76} +5.00000 q^{77} -17.0000 q^{79} +9.00000 q^{82} -4.00000 q^{83} -12.0000 q^{86} +5.00000 q^{88} -14.0000 q^{89} -1.00000 q^{91} -5.00000 q^{92} +7.00000 q^{94} +5.00000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 5.00000 1.06600
$$23$$ −5.00000 −1.04257 −0.521286 0.853382i $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 5.00000 0.753778
$$45$$ 0 0
$$46$$ −5.00000 −0.737210
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −5.00000 −0.707107
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 1.00000 0.127000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 5.00000 0.569803
$$78$$ 0 0
$$79$$ −17.0000 −1.91265 −0.956325 0.292306i $$-0.905577\pi$$
−0.956325 + 0.292306i $$0.905577\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 9.00000 0.993884
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ 0 0
$$88$$ 5.00000 0.533002
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ −5.00000 −0.521286
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.00000 0.507673 0.253837 0.967247i $$-0.418307\pi$$
0.253837 + 0.967247i $$0.418307\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ −15.0000 −1.49256 −0.746278 0.665635i $$-0.768161\pi$$
−0.746278 + 0.665635i $$0.768161\pi$$
$$102$$ 0 0
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 13.0000 1.17696
$$123$$ 0 0
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 9.00000 0.798621 0.399310 0.916816i $$-0.369250\pi$$
0.399310 + 0.916816i $$0.369250\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 11.0000 0.950255
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −5.00000 −0.418121
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ 0 0
$$148$$ 7.00000 0.575396
$$149$$ −7.00000 −0.573462 −0.286731 0.958011i $$-0.592569\pi$$
−0.286731 + 0.958011i $$0.592569\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ 5.00000 0.402911
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1.00000 0.0798087 0.0399043 0.999204i $$-0.487295\pi$$
0.0399043 + 0.999204i $$0.487295\pi$$
$$158$$ −17.0000 −1.35245
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.00000 −0.394055
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 9.00000 0.702782
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −12.0000 −0.914991
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ −5.00000 −0.377964
$$176$$ 5.00000 0.376889
$$177$$ 0 0
$$178$$ −14.0000 −1.04934
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ −5.00000 −0.368605
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 20.0000 1.46254
$$188$$ 7.00000 0.510527
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 5.00000 0.358979
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −27.0000 −1.92367 −0.961835 0.273629i $$-0.911776\pi$$
−0.961835 + 0.273629i $$0.911776\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ −5.00000 −0.353553
$$201$$ 0 0
$$202$$ −15.0000 −1.05540
$$203$$ −4.00000 −0.280745
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 10.0000 0.691714
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ 4.00000 0.274721
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1.00000 0.0678844
$$218$$ −18.0000 −1.21911
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 3.00000 0.200895 0.100447 0.994942i $$-0.467973\pi$$
0.100447 + 0.994942i $$0.467973\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −1.00000 −0.0665190
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 16.0000 1.05731 0.528655 0.848837i $$-0.322697\pi$$
0.528655 + 0.848837i $$0.322697\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −30.0000 −1.93247 −0.966235 0.257663i $$-0.917048\pi$$
−0.966235 + 0.257663i $$0.917048\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 0 0
$$244$$ 13.0000 0.832240
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 1.00000 0.0635001
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 13.0000 0.820553 0.410276 0.911961i $$-0.365432\pi$$
0.410276 + 0.911961i $$0.365432\pi$$
$$252$$ 0 0
$$253$$ −25.0000 −1.57174
$$254$$ 9.00000 0.564710
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −28.0000 −1.74659 −0.873296 0.487190i $$-0.838022\pi$$
−0.873296 + 0.487190i $$0.838022\pi$$
$$258$$ 0 0
$$259$$ 7.00000 0.434959
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ 11.0000 0.671932
$$269$$ −13.0000 −0.792624 −0.396312 0.918116i $$-0.629710\pi$$
−0.396312 + 0.918116i $$0.629710\pi$$
$$270$$ 0 0
$$271$$ −1.00000 −0.0607457 −0.0303728 0.999539i $$-0.509669\pi$$
−0.0303728 + 0.999539i $$0.509669\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ −25.0000 −1.50756
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ 3.00000 0.178331 0.0891657 0.996017i $$-0.471580\pi$$
0.0891657 + 0.996017i $$0.471580\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −5.00000 −0.295656
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7.00000 0.409644
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7.00000 0.406867
$$297$$ 0 0
$$298$$ −7.00000 −0.405499
$$299$$ 5.00000 0.289157
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ −12.0000 −0.690522
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.0000 1.82634 0.913168 0.407583i $$-0.133628\pi$$
0.913168 + 0.407583i $$0.133628\pi$$
$$308$$ 5.00000 0.284901
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −26.0000 −1.47432 −0.737162 0.675716i $$-0.763835\pi$$
−0.737162 + 0.675716i $$0.763835\pi$$
$$312$$ 0 0
$$313$$ −30.0000 −1.69570 −0.847850 0.530236i $$-0.822103\pi$$
−0.847850 + 0.530236i $$0.822103\pi$$
$$314$$ 1.00000 0.0564333
$$315$$ 0 0
$$316$$ −17.0000 −0.956325
$$317$$ −11.0000 −0.617822 −0.308911 0.951091i $$-0.599964\pi$$
−0.308911 + 0.951091i $$0.599964\pi$$
$$318$$ 0 0
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −5.00000 −0.278639
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 5.00000 0.277350
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 9.00000 0.496942
$$329$$ 7.00000 0.385922
$$330$$ 0 0
$$331$$ 3.00000 0.164895 0.0824475 0.996595i $$-0.473726\pi$$
0.0824475 + 0.996595i $$0.473726\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 0 0
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 9.00000 0.490261 0.245131 0.969490i $$-0.421169\pi$$
0.245131 + 0.969490i $$0.421169\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 5.00000 0.270765
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 2.00000 0.105703
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −5.00000 −0.262794
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −5.00000 −0.260643
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 20.0000 1.03418
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ 9.00000 0.459879 0.229939 0.973205i $$-0.426147\pi$$
0.229939 + 0.973205i $$0.426147\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 0 0
$$388$$ 5.00000 0.253837
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ −20.0000 −1.01144
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ −27.0000 −1.36024
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 28.0000 1.39825 0.699127 0.714998i $$-0.253572\pi$$
0.699127 + 0.714998i $$0.253572\pi$$
$$402$$ 0 0
$$403$$ −1.00000 −0.0498135
$$404$$ −15.0000 −0.746278
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 35.0000 1.73489
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 6.00000 0.295599
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 10.0000 0.489116
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ 0 0
$$421$$ −35.0000 −1.70580 −0.852898 0.522078i $$-0.825157\pi$$
−0.852898 + 0.522078i $$0.825157\pi$$
$$422$$ −14.0000 −0.681509
$$423$$ 0 0
$$424$$ 4.00000 0.194257
$$425$$ −20.0000 −0.970143
$$426$$ 0 0
$$427$$ 13.0000 0.629114
$$428$$ 8.00000 0.386695
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.00000 0.289010 0.144505 0.989504i $$-0.453841\pi$$
0.144505 + 0.989504i $$0.453841\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 1.00000 0.0480015
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ −10.0000 −0.478365
$$438$$ 0 0
$$439$$ −2.00000 −0.0954548 −0.0477274 0.998860i $$-0.515198\pi$$
−0.0477274 + 0.998860i $$0.515198\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 3.00000 0.142054
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 45.0000 2.11897
$$452$$ −1.00000 −0.0470360
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 16.0000 0.747631
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 12.0000 0.557687 0.278844 0.960337i $$-0.410049\pi$$
0.278844 + 0.960337i $$0.410049\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ 21.0000 0.972806
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 11.0000 0.507933
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ −60.0000 −2.75880
$$474$$ 0 0
$$475$$ −10.0000 −0.458831
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ −30.0000 −1.36646
$$483$$ 0 0
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 13.0000 0.588482
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ −16.0000 −0.720604
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.00000 0.0447661 0.0223831 0.999749i $$-0.492875\pi$$
0.0223831 + 0.999749i $$0.492875\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 13.0000 0.580218
$$503$$ 40.0000 1.78351 0.891756 0.452517i $$-0.149474\pi$$
0.891756 + 0.452517i $$0.149474\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −25.0000 −1.11139
$$507$$ 0 0
$$508$$ 9.00000 0.399310
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −28.0000 −1.23503
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 35.0000 1.53930
$$518$$ 7.00000 0.307562
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 0 0
$$523$$ −13.0000 −0.568450 −0.284225 0.958758i $$-0.591736\pi$$
−0.284225 + 0.958758i $$0.591736\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ 2.00000 0.0869565
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000 0.0867110
$$533$$ −9.00000 −0.389833
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 11.0000 0.475128
$$537$$ 0 0
$$538$$ −13.0000 −0.560470
$$539$$ 5.00000 0.215365
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ −1.00000 −0.0429537
$$543$$ 0 0
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 0 0
$$550$$ −25.0000 −1.06600
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ −17.0000 −0.722914
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 9.00000 0.381342 0.190671 0.981654i $$-0.438934\pi$$
0.190671 + 0.981654i $$0.438934\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 16.0000 0.674919
$$563$$ −35.0000 −1.47507 −0.737537 0.675307i $$-0.764011\pi$$
−0.737537 + 0.675307i $$0.764011\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 3.00000 0.126099
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −21.0000 −0.880366 −0.440183 0.897908i $$-0.645086\pi$$
−0.440183 + 0.897908i $$0.645086\pi$$
$$570$$ 0 0
$$571$$ 22.0000 0.920671 0.460336 0.887745i $$-0.347729\pi$$
0.460336 + 0.887745i $$0.347729\pi$$
$$572$$ −5.00000 −0.209061
$$573$$ 0 0
$$574$$ 9.00000 0.375653
$$575$$ 25.0000 1.04257
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 20.0000 0.828315
$$584$$ 7.00000 0.289662
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 0 0
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 7.00000 0.287698
$$593$$ 46.0000 1.88899 0.944497 0.328521i $$-0.106550\pi$$
0.944497 + 0.328521i $$0.106550\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −7.00000 −0.286731
$$597$$ 0 0
$$598$$ 5.00000 0.204465
$$599$$ 9.00000 0.367730 0.183865 0.982952i $$-0.441139\pi$$
0.183865 + 0.982952i $$0.441139\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ 0 0
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.00000 −0.283190
$$612$$ 0 0
$$613$$ −23.0000 −0.928961 −0.464481 0.885583i $$-0.653759\pi$$
−0.464481 + 0.885583i $$0.653759\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 6.00000 0.241160 0.120580 0.992704i $$-0.461525\pi$$
0.120580 + 0.992704i $$0.461525\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −26.0000 −1.04251
$$623$$ −14.0000 −0.560898
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −30.0000 −1.19904
$$627$$ 0 0
$$628$$ 1.00000 0.0399043
$$629$$ 28.0000 1.11643
$$630$$ 0 0
$$631$$ −26.0000 −1.03504 −0.517522 0.855670i $$-0.673145\pi$$
−0.517522 + 0.855670i $$0.673145\pi$$
$$632$$ −17.0000 −0.676224
$$633$$ 0 0
$$634$$ −11.0000 −0.436866
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ −20.0000 −0.791808
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 33.0000 1.30342 0.651711 0.758468i $$-0.274052\pi$$
0.651711 + 0.758468i $$0.274052\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ −5.00000 −0.197028
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 30.0000 1.17760
$$650$$ 5.00000 0.196116
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 12.0000 0.469596 0.234798 0.972044i $$-0.424557\pi$$
0.234798 + 0.972044i $$0.424557\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 9.00000 0.351391
$$657$$ 0 0
$$658$$ 7.00000 0.272888
$$659$$ −26.0000 −1.01282 −0.506408 0.862294i $$-0.669027\pi$$
−0.506408 + 0.862294i $$0.669027\pi$$
$$660$$ 0 0
$$661$$ −20.0000 −0.777910 −0.388955 0.921257i $$-0.627164\pi$$
−0.388955 + 0.921257i $$0.627164\pi$$
$$662$$ 3.00000 0.116598
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.0000 0.774403
$$668$$ −8.00000 −0.309529
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 65.0000 2.50930
$$672$$ 0 0
$$673$$ −9.00000 −0.346925 −0.173462 0.984841i $$-0.555495\pi$$
−0.173462 + 0.984841i $$0.555495\pi$$
$$674$$ 9.00000 0.346667
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −3.00000 −0.115299 −0.0576497 0.998337i $$-0.518361\pi$$
−0.0576497 + 0.998337i $$0.518361\pi$$
$$678$$ 0 0
$$679$$ 5.00000 0.191882
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 5.00000 0.191460
$$683$$ −3.00000 −0.114792 −0.0573959 0.998351i $$-0.518280\pi$$
−0.0573959 + 0.998351i $$0.518280\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ −12.0000 −0.457496
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ −16.0000 −0.607352
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 2.00000 0.0757011
$$699$$ 0 0
$$700$$ −5.00000 −0.188982
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 0 0
$$703$$ 14.0000 0.528020
$$704$$ 5.00000 0.188445
$$705$$ 0 0
$$706$$ −3.00000 −0.112906
$$707$$ −15.0000 −0.564133
$$708$$ 0 0
$$709$$ 29.0000 1.08912 0.544559 0.838723i $$-0.316697\pi$$
0.544559 + 0.838723i $$0.316697\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −14.0000 −0.524672
$$713$$ −5.00000 −0.187251
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 2.00000 0.0747435
$$717$$ 0 0
$$718$$ 4.00000 0.149279
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 6.00000 0.223452
$$722$$ −15.0000 −0.558242
$$723$$ 0 0
$$724$$ −5.00000 −0.185824
$$725$$ 20.0000 0.742781
$$726$$ 0 0
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ 0 0
$$733$$ 20.0000 0.738717 0.369358 0.929287i $$-0.379577\pi$$
0.369358 + 0.929287i $$0.379577\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −5.00000 −0.184302
$$737$$ 55.0000 2.02595
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ 20.0000 0.731272
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ 25.0000 0.912263 0.456131 0.889912i $$-0.349235\pi$$
0.456131 + 0.889912i $$0.349235\pi$$
$$752$$ 7.00000 0.255264
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.0000 1.19625 0.598125 0.801403i $$-0.295913\pi$$
0.598125 + 0.801403i $$0.295913\pi$$
$$762$$ 0 0
$$763$$ −18.0000 −0.651644
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 9.00000 0.325183
$$767$$ −6.00000 −0.216647
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.00000 0.143963
$$773$$ −34.0000 −1.22290 −0.611448 0.791285i $$-0.709412\pi$$
−0.611448 + 0.791285i $$0.709412\pi$$
$$774$$ 0 0
$$775$$ −5.00000 −0.179605
$$776$$ 5.00000 0.179490
$$777$$ 0 0
$$778$$ −8.00000 −0.286814
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −20.0000 −0.715199
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 8.00000 0.285169 0.142585 0.989783i $$-0.454459\pi$$
0.142585 + 0.989783i $$0.454459\pi$$
$$788$$ −27.0000 −0.961835
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −1.00000 −0.0355559
$$792$$ 0 0
$$793$$ −13.0000 −0.461644
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ −3.00000 −0.106265 −0.0531327 0.998587i $$-0.516921\pi$$
−0.0531327 + 0.998587i $$0.516921\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ −5.00000 −0.176777
$$801$$ 0 0
$$802$$ 28.0000 0.988714
$$803$$ 35.0000 1.23512
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.00000 −0.0352235
$$807$$ 0 0
$$808$$ −15.0000 −0.527698
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ 35.0000 1.22675
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 0 0
$$823$$ −15.0000 −0.522867 −0.261434 0.965221i $$-0.584195\pi$$
−0.261434 + 0.965221i $$0.584195\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ −54.0000 −1.87550 −0.937749 0.347314i $$-0.887094\pi$$
−0.937749 + 0.347314i $$0.887094\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 10.0000 0.345857
$$837$$ 0 0
$$838$$ 9.00000 0.310900
$$839$$ −47.0000 −1.62262 −0.811310 0.584616i $$-0.801245\pi$$
−0.811310 + 0.584616i $$0.801245\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −35.0000 −1.20618
$$843$$ 0 0
$$844$$ −14.0000 −0.481900
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ 4.00000 0.137361
$$849$$ 0 0
$$850$$ −20.0000 −0.685994
$$851$$ −35.0000 −1.19978
$$852$$ 0 0
$$853$$ −16.0000 −0.547830 −0.273915 0.961754i $$-0.588319\pi$$
−0.273915 + 0.961754i $$0.588319\pi$$
$$854$$ 13.0000 0.444851
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ −2.00000 −0.0683187 −0.0341593 0.999416i $$-0.510875\pi$$
−0.0341593 + 0.999416i $$0.510875\pi$$
$$858$$ 0 0
$$859$$ 29.0000 0.989467 0.494734 0.869045i $$-0.335266\pi$$
0.494734 + 0.869045i $$0.335266\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 6.00000 0.204361
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ 0 0
$$868$$ 1.00000 0.0339422
$$869$$ −85.0000 −2.88343
$$870$$ 0 0
$$871$$ −11.0000 −0.372721
$$872$$ −18.0000 −0.609557
$$873$$ 0 0
$$874$$ −10.0000 −0.338255
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ −2.00000 −0.0674967
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 30.0000 1.00958 0.504790 0.863242i $$-0.331570\pi$$
0.504790 + 0.863242i $$0.331570\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ −54.0000 −1.81314 −0.906571 0.422053i $$-0.861310\pi$$
−0.906571 + 0.422053i $$0.861310\pi$$
$$888$$ 0 0
$$889$$ 9.00000 0.301850
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 3.00000 0.100447
$$893$$ 14.0000 0.468492
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 16.0000 0.533037
$$902$$ 45.0000 1.49834
$$903$$ 0 0
$$904$$ −1.00000 −0.0332595
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 14.0000 0.464862 0.232431 0.972613i $$-0.425332\pi$$
0.232431 + 0.972613i $$0.425332\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −28.0000 −0.927681 −0.463841 0.885919i $$-0.653529\pi$$
−0.463841 + 0.885919i $$0.653529\pi$$
$$912$$ 0 0
$$913$$ −20.0000 −0.661903
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 16.0000 0.528655
$$917$$ −8.00000 −0.264183
$$918$$ 0 0
$$919$$ 31.0000 1.02260 0.511298 0.859404i $$-0.329165\pi$$
0.511298 + 0.859404i $$0.329165\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −12.0000 −0.395199
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −35.0000 −1.15079
$$926$$ 12.0000 0.394344
$$927$$ 0 0
$$928$$ −4.00000 −0.131306
$$929$$ 29.0000 0.951459 0.475730 0.879592i $$-0.342184\pi$$
0.475730 + 0.879592i $$0.342184\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 21.0000 0.687878
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ 11.0000 0.359163
$$939$$ 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
$$943$$ −45.0000 −1.46540
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ −60.0000 −1.95077
$$947$$ 16.0000 0.519930 0.259965 0.965618i $$-0.416289\pi$$
0.259965 + 0.965618i $$0.416289\pi$$
$$948$$ 0 0
$$949$$ −7.00000 −0.227230
$$950$$ −10.0000 −0.324443
$$951$$ 0 0
$$952$$ 4.00000 0.129641
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 16.0000 0.516937
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ −7.00000 −0.225689
$$963$$ 0 0
$$964$$ −30.0000 −0.966235
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.00000 0.0643157 0.0321578 0.999483i $$-0.489762\pi$$
0.0321578 + 0.999483i $$0.489762\pi$$
$$968$$ 14.0000 0.449977
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −9.00000 −0.288824 −0.144412 0.989518i $$-0.546129\pi$$
−0.144412 + 0.989518i $$0.546129\pi$$
$$972$$ 0 0
$$973$$ 4.00000 0.128234
$$974$$ −26.0000 −0.833094
$$975$$ 0 0
$$976$$ 13.0000 0.416120
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 0 0
$$979$$ −70.0000 −2.23721
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −28.0000 −0.893516
$$983$$ 56.0000 1.78612 0.893061 0.449935i $$-0.148553\pi$$
0.893061 + 0.449935i $$0.148553\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −16.0000 −0.509544
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 60.0000 1.90789
$$990$$ 0 0
$$991$$ 59.0000 1.87420 0.937098 0.349065i $$-0.113501\pi$$
0.937098 + 0.349065i $$0.113501\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 33.0000 1.04512 0.522560 0.852602i $$-0.324977\pi$$
0.522560 + 0.852602i $$0.324977\pi$$
$$998$$ 1.00000 0.0316544
$$999$$ 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 1638.2.a.q.1.1 1
3.2 odd 2 182.2.a.b.1.1 1
12.11 even 2 1456.2.a.b.1.1 1
15.14 odd 2 4550.2.a.o.1.1 1
21.2 odd 6 1274.2.f.m.1145.1 2
21.5 even 6 1274.2.f.u.1145.1 2
21.11 odd 6 1274.2.f.m.79.1 2
21.17 even 6 1274.2.f.u.79.1 2
21.20 even 2 1274.2.a.a.1.1 1
24.5 odd 2 5824.2.a.a.1.1 1
24.11 even 2 5824.2.a.be.1.1 1
39.5 even 4 2366.2.d.i.337.2 2
39.8 even 4 2366.2.d.i.337.1 2
39.38 odd 2 2366.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.b.1.1 1 3.2 odd 2
1274.2.a.a.1.1 1 21.20 even 2
1274.2.f.m.79.1 2 21.11 odd 6
1274.2.f.m.1145.1 2 21.2 odd 6
1274.2.f.u.79.1 2 21.17 even 6
1274.2.f.u.1145.1 2 21.5 even 6
1456.2.a.b.1.1 1 12.11 even 2
1638.2.a.q.1.1 1 1.1 even 1 trivial
2366.2.a.o.1.1 1 39.38 odd 2
2366.2.d.i.337.1 2 39.8 even 4
2366.2.d.i.337.2 2 39.5 even 4
4550.2.a.o.1.1 1 15.14 odd 2
5824.2.a.a.1.1 1 24.5 odd 2
5824.2.a.be.1.1 1 24.11 even 2