# Properties

 Label 1728.2.a.h.1.1 Level $1728$ Weight $2$ Character 1728.1 Self dual yes Analytic conductor $13.798$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{5} +3.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{5} +3.00000 q^{7} -6.00000 q^{11} +3.00000 q^{13} -2.00000 q^{17} +3.00000 q^{19} +6.00000 q^{23} -1.00000 q^{25} +8.00000 q^{29} -6.00000 q^{35} -7.00000 q^{37} +8.00000 q^{41} +12.0000 q^{43} +6.00000 q^{47} +2.00000 q^{49} -4.00000 q^{53} +12.0000 q^{55} -6.00000 q^{59} +1.00000 q^{61} -6.00000 q^{65} +3.00000 q^{67} +12.0000 q^{71} -15.0000 q^{73} -18.0000 q^{77} +9.00000 q^{79} +12.0000 q^{83} +4.00000 q^{85} -10.0000 q^{89} +9.00000 q^{91} -6.00000 q^{95} +9.00000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −6.00000 −1.01419
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ 12.0000 1.61808
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −15.0000 −1.75562 −0.877809 0.479012i $$-0.840995\pi$$
−0.877809 + 0.479012i $$0.840995\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −18.0000 −2.05129
$$78$$ 0 0
$$79$$ 9.00000 1.01258 0.506290 0.862364i $$-0.331017\pi$$
0.506290 + 0.862364i $$0.331017\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 9.00000 0.943456
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ 9.00000 0.913812 0.456906 0.889515i $$-0.348958\pi$$
0.456906 + 0.889515i $$0.348958\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.0000 1.59206 0.796030 0.605257i $$-0.206930\pi$$
0.796030 + 0.605257i $$0.206930\pi$$
$$102$$ 0 0
$$103$$ 15.0000 1.47799 0.738997 0.673709i $$-0.235300\pi$$
0.738997 + 0.673709i $$0.235300\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ −12.0000 −1.11901
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 9.00000 0.780399
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 21.0000 1.78120 0.890598 0.454791i $$-0.150286\pi$$
0.890598 + 0.454791i $$0.150286\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −18.0000 −1.50524
$$144$$ 0 0
$$145$$ −16.0000 −1.32873
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −3.00000 −0.244137 −0.122068 0.992522i $$-0.538953\pi$$
−0.122068 + 0.992522i $$0.538953\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ 0 0
$$163$$ −3.00000 −0.234978 −0.117489 0.993074i $$-0.537485\pi$$
−0.117489 + 0.993074i $$0.537485\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −8.00000 −0.608229 −0.304114 0.952636i $$-0.598361\pi$$
−0.304114 + 0.952636i $$0.598361\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −9.00000 −0.668965 −0.334482 0.942402i $$-0.608561\pi$$
−0.334482 + 0.942402i $$0.608561\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 14.0000 1.02930
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ 0 0
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 15.0000 1.06332 0.531661 0.846957i $$-0.321568\pi$$
0.531661 + 0.846957i $$0.321568\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 24.0000 1.68447
$$204$$ 0 0
$$205$$ −16.0000 −1.11749
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −18.0000 −1.24509
$$210$$ 0 0
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −24.0000 −1.63679
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −3.00000 −0.193247 −0.0966235 0.995321i $$-0.530804\pi$$
−0.0966235 + 0.995321i $$0.530804\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.00000 −0.255551
$$246$$ 0 0
$$247$$ 9.00000 0.572656
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −36.0000 −2.26330
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −20.0000 −1.24757 −0.623783 0.781598i $$-0.714405\pi$$
−0.623783 + 0.781598i $$0.714405\pi$$
$$258$$ 0 0
$$259$$ −21.0000 −1.30488
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 8.00000 0.491436
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 18.0000 1.04097
$$300$$ 0 0
$$301$$ 36.0000 2.07501
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4.00000 0.224662 0.112331 0.993671i $$-0.464168\pi$$
0.112331 + 0.993671i $$0.464168\pi$$
$$318$$ 0 0
$$319$$ −48.0000 −2.68748
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ −3.00000 −0.166410
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 18.0000 0.992372
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −6.00000 −0.327815
$$336$$ 0 0
$$337$$ 9.00000 0.490261 0.245131 0.969490i $$-0.421169\pi$$
0.245131 + 0.969490i $$0.421169\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 1.00000 0.0535288 0.0267644 0.999642i $$-0.491480\pi$$
0.0267644 + 0.999642i $$0.491480\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 20.0000 1.06449 0.532246 0.846590i $$-0.321348\pi$$
0.532246 + 0.846590i $$0.321348\pi$$
$$354$$ 0 0
$$355$$ −24.0000 −1.27379
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 18.0000 0.950004 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 30.0000 1.57027
$$366$$ 0 0
$$367$$ 3.00000 0.156599 0.0782994 0.996930i $$-0.475051\pi$$
0.0782994 + 0.996930i $$0.475051\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −23.0000 −1.19089 −0.595447 0.803394i $$-0.703025\pi$$
−0.595447 + 0.803394i $$0.703025\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 27.0000 1.38690 0.693448 0.720506i $$-0.256091\pi$$
0.693448 + 0.720506i $$0.256091\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 36.0000 1.83473
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −34.0000 −1.72387 −0.861934 0.507020i $$-0.830747\pi$$
−0.861934 + 0.507020i $$0.830747\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −18.0000 −0.905678
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −20.0000 −0.998752 −0.499376 0.866385i $$-0.666437\pi$$
−0.499376 + 0.866385i $$0.666437\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 42.0000 2.08186
$$408$$ 0 0
$$409$$ 21.0000 1.03838 0.519192 0.854658i $$-0.326233\pi$$
0.519192 + 0.854658i $$0.326233\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −18.0000 −0.885722
$$414$$ 0 0
$$415$$ −24.0000 −1.17811
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ 15.0000 0.731055 0.365528 0.930800i $$-0.380889\pi$$
0.365528 + 0.930800i $$0.380889\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 3.00000 0.145180
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 18.0000 0.861057
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 20.0000 0.948091
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −22.0000 −1.03824 −0.519122 0.854700i $$-0.673741\pi$$
−0.519122 + 0.854700i $$0.673741\pi$$
$$450$$ 0 0
$$451$$ −48.0000 −2.26023
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −18.0000 −0.843853
$$456$$ 0 0
$$457$$ −30.0000 −1.40334 −0.701670 0.712502i $$-0.747562\pi$$
−0.701670 + 0.712502i $$0.747562\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ 15.0000 0.697109 0.348555 0.937288i $$-0.386673\pi$$
0.348555 + 0.937288i $$0.386673\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −72.0000 −3.31056
$$474$$ 0 0
$$475$$ −3.00000 −0.137649
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ −21.0000 −0.957518
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −18.0000 −0.817338
$$486$$ 0 0
$$487$$ 15.0000 0.679715 0.339857 0.940477i $$-0.389621\pi$$
0.339857 + 0.940477i $$0.389621\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −42.0000 −1.89543 −0.947717 0.319113i $$-0.896615\pi$$
−0.947717 + 0.319113i $$0.896615\pi$$
$$492$$ 0 0
$$493$$ −16.0000 −0.720604
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 36.0000 1.61482
$$498$$ 0 0
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ 0 0
$$505$$ −32.0000 −1.42398
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ −45.0000 −1.99068
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −30.0000 −1.32196
$$516$$ 0 0
$$517$$ −36.0000 −1.58328
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ −9.00000 −0.393543 −0.196771 0.980449i $$-0.563046\pi$$
−0.196771 + 0.980449i $$0.563046\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 24.0000 1.03956
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 12.0000 0.514024
$$546$$ 0 0
$$547$$ 9.00000 0.384812 0.192406 0.981315i $$-0.438371\pi$$
0.192406 + 0.981315i $$0.438371\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 27.0000 1.14816
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ 36.0000 1.52264
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ −28.0000 −1.17797
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 46.0000 1.92842 0.964210 0.265139i $$-0.0854179\pi$$
0.964210 + 0.265139i $$0.0854179\pi$$
$$570$$ 0 0
$$571$$ 3.00000 0.125546 0.0627730 0.998028i $$-0.480006\pi$$
0.0627730 + 0.998028i $$0.480006\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 36.0000 1.49353
$$582$$ 0 0
$$583$$ 24.0000 0.993978
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 28.0000 1.14982 0.574911 0.818216i $$-0.305037\pi$$
0.574911 + 0.818216i $$0.305037\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 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
$$605$$ −50.0000 −2.03279
$$606$$ 0 0
$$607$$ 3.00000 0.121766 0.0608831 0.998145i $$-0.480608\pi$$
0.0608831 + 0.998145i $$0.480608\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ −11.0000 −0.444286 −0.222143 0.975014i $$-0.571305\pi$$
−0.222143 + 0.975014i $$0.571305\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ −9.00000 −0.361741 −0.180870 0.983507i $$-0.557891\pi$$
−0.180870 + 0.983507i $$0.557891\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −30.0000 −1.20192
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 14.0000 0.558217
$$630$$ 0 0
$$631$$ −27.0000 −1.07485 −0.537427 0.843311i $$-0.680603\pi$$
−0.537427 + 0.843311i $$0.680603\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 24.0000 0.952411
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4.00000 0.157991 0.0789953 0.996875i $$-0.474829\pi$$
0.0789953 + 0.996875i $$0.474829\pi$$
$$642$$ 0 0
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 28.0000 1.09572 0.547862 0.836569i $$-0.315442\pi$$
0.547862 + 0.836569i $$0.315442\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 5.00000 0.194477 0.0972387 0.995261i $$-0.468999\pi$$
0.0972387 + 0.995261i $$0.468999\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −18.0000 −0.698010
$$666$$ 0 0
$$667$$ 48.0000 1.85857
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.00000 −0.231627
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ 27.0000 1.03616
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 28.0000 1.06983
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −42.0000 −1.59315
$$696$$ 0 0
$$697$$ −16.0000 −0.606043
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −46.0000 −1.73740 −0.868698 0.495342i $$-0.835043\pi$$
−0.868698 + 0.495342i $$0.835043\pi$$
$$702$$ 0 0
$$703$$ −21.0000 −0.792030
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 48.0000 1.80523
$$708$$ 0 0
$$709$$ 51.0000 1.91535 0.957673 0.287860i $$-0.0929437\pi$$
0.957673 + 0.287860i $$0.0929437\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 45.0000 1.67589
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ 24.0000 0.890111 0.445055 0.895503i $$-0.353184\pi$$
0.445055 + 0.895503i $$0.353184\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 18.0000 0.664845 0.332423 0.943131i $$-0.392134\pi$$
0.332423 + 0.943131i $$0.392134\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −18.0000 −0.663039
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 18.0000 0.657706
$$750$$ 0 0
$$751$$ 45.0000 1.64207 0.821037 0.570875i $$-0.193396\pi$$
0.821037 + 0.570875i $$0.193396\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 6.00000 0.218362
$$756$$ 0 0
$$757$$ 27.0000 0.981332 0.490666 0.871348i $$-0.336754\pi$$
0.490666 + 0.871348i $$0.336754\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34.0000 −1.23250 −0.616250 0.787551i $$-0.711349\pi$$
−0.616250 + 0.787551i $$0.711349\pi$$
$$762$$ 0 0
$$763$$ −18.0000 −0.651644
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −18.0000 −0.649942
$$768$$ 0 0
$$769$$ −49.0000 −1.76699 −0.883493 0.468445i $$-0.844814\pi$$
−0.883493 + 0.468445i $$0.844814\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 20.0000 0.719350 0.359675 0.933078i $$-0.382888\pi$$
0.359675 + 0.933078i $$0.382888\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ −72.0000 −2.57636
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −28.0000 −0.999363
$$786$$ 0 0
$$787$$ 27.0000 0.962446 0.481223 0.876598i $$-0.340193\pi$$
0.481223 + 0.876598i $$0.340193\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 42.0000 1.49335
$$792$$ 0 0
$$793$$ 3.00000 0.106533
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 40.0000 1.41687 0.708436 0.705775i $$-0.249401\pi$$
0.708436 + 0.705775i $$0.249401\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 90.0000 3.17603
$$804$$ 0 0
$$805$$ −36.0000 −1.26883
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 44.0000 1.54696 0.773479 0.633822i $$-0.218515\pi$$
0.773479 + 0.633822i $$0.218515\pi$$
$$810$$ 0 0
$$811$$ −36.0000 −1.26413 −0.632065 0.774915i $$-0.717793\pi$$
−0.632065 + 0.774915i $$0.717793\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 6.00000 0.210171
$$816$$ 0 0
$$817$$ 36.0000 1.25948
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 0 0
$$823$$ 9.00000 0.313720 0.156860 0.987621i $$-0.449863\pi$$
0.156860 + 0.987621i $$0.449863\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.00000 −0.208640 −0.104320 0.994544i $$-0.533267\pi$$
−0.104320 + 0.994544i $$0.533267\pi$$
$$828$$ 0 0
$$829$$ −45.0000 −1.56291 −0.781457 0.623959i $$-0.785523\pi$$
−0.781457 + 0.623959i $$0.785523\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ 36.0000 1.24583
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 8.00000 0.275208
$$846$$ 0 0
$$847$$ 75.0000 2.57703
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −42.0000 −1.43974
$$852$$ 0 0
$$853$$ 53.0000 1.81469 0.907343 0.420392i $$-0.138107\pi$$
0.907343 + 0.420392i $$0.138107\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −4.00000 −0.136637 −0.0683187 0.997664i $$-0.521763\pi$$
−0.0683187 + 0.997664i $$0.521763\pi$$
$$858$$ 0 0
$$859$$ −27.0000 −0.921228 −0.460614 0.887601i $$-0.652371\pi$$
−0.460614 + 0.887601i $$0.652371\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −42.0000 −1.42970 −0.714848 0.699280i $$-0.753504\pi$$
−0.714848 + 0.699280i $$0.753504\pi$$
$$864$$ 0 0
$$865$$ 16.0000 0.544016
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −54.0000 −1.83182
$$870$$ 0 0
$$871$$ 9.00000 0.304953
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 36.0000 1.21702
$$876$$ 0 0
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 0 0
$$883$$ 9.00000 0.302874 0.151437 0.988467i $$-0.451610\pi$$
0.151437 + 0.988467i $$0.451610\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ −36.0000 −1.20740
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 18.0000 0.602347
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.0000 0.598340
$$906$$ 0 0
$$907$$ 39.0000 1.29497 0.647487 0.762077i $$-0.275820\pi$$
0.647487 + 0.762077i $$0.275820\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ −72.0000 −2.38285
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 36.0000 1.18495
$$924$$ 0 0
$$925$$ 7.00000 0.230159
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 28.0000 0.918650 0.459325 0.888268i $$-0.348091\pi$$
0.459325 + 0.888268i $$0.348091\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −24.0000 −0.784884
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ −45.0000 −1.46076
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −42.0000 −1.35625
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 10.0000 0.321911
$$966$$ 0 0
$$967$$ 33.0000 1.06121 0.530604 0.847620i $$-0.321965\pi$$
0.530604 + 0.847620i $$0.321965\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ 0 0
$$973$$ 63.0000 2.01969
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 44.0000 1.40768 0.703842 0.710356i $$-0.251466\pi$$
0.703842 + 0.710356i $$0.251466\pi$$
$$978$$ 0 0
$$979$$ 60.0000 1.91761
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ 4.00000 0.127451
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 72.0000 2.28947
$$990$$ 0 0
$$991$$ −21.0000 −0.667087 −0.333543 0.942735i $$-0.608244\pi$$
−0.333543 + 0.942735i $$0.608244\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −30.0000 −0.951064
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 0 0
$$999$$ 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 1728.2.a.h.1.1 1
3.2 odd 2 1728.2.a.x.1.1 1
4.3 odd 2 1728.2.a.e.1.1 1
8.3 odd 2 864.2.a.i.1.1 yes 1
8.5 even 2 864.2.a.l.1.1 yes 1
12.11 even 2 1728.2.a.u.1.1 1
24.5 odd 2 864.2.a.d.1.1 yes 1
24.11 even 2 864.2.a.a.1.1 1
72.5 odd 6 2592.2.i.r.865.1 2
72.11 even 6 2592.2.i.v.1729.1 2
72.13 even 6 2592.2.i.c.865.1 2
72.29 odd 6 2592.2.i.r.1729.1 2
72.43 odd 6 2592.2.i.g.1729.1 2
72.59 even 6 2592.2.i.v.865.1 2
72.61 even 6 2592.2.i.c.1729.1 2
72.67 odd 6 2592.2.i.g.865.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.a.1.1 1 24.11 even 2
864.2.a.d.1.1 yes 1 24.5 odd 2
864.2.a.i.1.1 yes 1 8.3 odd 2
864.2.a.l.1.1 yes 1 8.5 even 2
1728.2.a.e.1.1 1 4.3 odd 2
1728.2.a.h.1.1 1 1.1 even 1 trivial
1728.2.a.u.1.1 1 12.11 even 2
1728.2.a.x.1.1 1 3.2 odd 2
2592.2.i.c.865.1 2 72.13 even 6
2592.2.i.c.1729.1 2 72.61 even 6
2592.2.i.g.865.1 2 72.67 odd 6
2592.2.i.g.1729.1 2 72.43 odd 6
2592.2.i.r.865.1 2 72.5 odd 6
2592.2.i.r.1729.1 2 72.29 odd 6
2592.2.i.v.865.1 2 72.59 even 6
2592.2.i.v.1729.1 2 72.11 even 6