# Properties

 Label 1728.2.a.l.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 $1$

# Related objects

## 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 216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +3.00000 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +3.00000 q^{7} +5.00000 q^{11} -4.00000 q^{13} +8.00000 q^{17} -2.00000 q^{19} -2.00000 q^{23} -4.00000 q^{25} +6.00000 q^{29} -7.00000 q^{31} -3.00000 q^{35} +6.00000 q^{37} +6.00000 q^{41} +2.00000 q^{43} -6.00000 q^{47} +2.00000 q^{49} +5.00000 q^{53} -5.00000 q^{55} -4.00000 q^{59} +8.00000 q^{61} +4.00000 q^{65} +10.0000 q^{67} +8.00000 q^{71} +1.00000 q^{73} +15.0000 q^{77} +16.0000 q^{79} -11.0000 q^{83} -8.00000 q^{85} -6.00000 q^{89} -12.0000 q^{91} +2.00000 q^{95} -1.00000 q^{97} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 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
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 5.00000 0.686803 0.343401 0.939189i $$-0.388421\pi$$
0.343401 + 0.939189i $$0.388421\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 10.0000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 15.0000 1.70941
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −11.0000 −1.20741 −0.603703 0.797209i $$-0.706309\pi$$
−0.603703 + 0.797209i $$0.706309\pi$$
$$84$$ 0 0
$$85$$ −8.00000 −0.867722
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 2.00000 0.186501
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −11.0000 −0.976092 −0.488046 0.872818i $$-0.662290\pi$$
−0.488046 + 0.872818i $$0.662290\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.00000 0.0873704 0.0436852 0.999045i $$-0.486090\pi$$
0.0436852 + 0.999045i $$0.486090\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −20.0000 −1.67248
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 7.00000 0.573462 0.286731 0.958011i $$-0.407431\pi$$
0.286731 + 0.958011i $$0.407431\pi$$
$$150$$ 0 0
$$151$$ 5.00000 0.406894 0.203447 0.979086i $$-0.434786\pi$$
0.203447 + 0.979086i $$0.434786\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 7.00000 0.562254
$$156$$ 0 0
$$157$$ 20.0000 1.59617 0.798087 0.602542i $$-0.205846\pi$$
0.798087 + 0.602542i $$0.205846\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ 0 0
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −21.0000 −1.59660 −0.798300 0.602260i $$-0.794267\pi$$
−0.798300 + 0.602260i $$0.794267\pi$$
$$174$$ 0 0
$$175$$ −12.0000 −0.907115
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 40.0000 2.92509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 18.0000 1.26335
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −10.0000 −0.691714
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ −21.0000 −1.42557
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −32.0000 −2.15255
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 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
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ 6.00000 0.391397
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −22.0000 −1.42306 −0.711531 0.702655i $$-0.751998\pi$$
−0.711531 + 0.702655i $$0.751998\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.0000 1.00991 0.504956 0.863145i $$-0.331509\pi$$
0.504956 + 0.863145i $$0.331509\pi$$
$$252$$ 0 0
$$253$$ −10.0000 −0.628695
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4.00000 −0.249513 −0.124757 0.992187i $$-0.539815\pi$$
−0.124757 + 0.992187i $$0.539815\pi$$
$$258$$ 0 0
$$259$$ 18.0000 1.11847
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −26.0000 −1.60323 −0.801614 0.597841i $$-0.796025\pi$$
−0.801614 + 0.597841i $$0.796025\pi$$
$$264$$ 0 0
$$265$$ −5.00000 −0.307148
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ −13.0000 −0.789694 −0.394847 0.918747i $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −20.0000 −1.20605
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −16.0000 −0.954480 −0.477240 0.878773i $$-0.658363\pi$$
−0.477240 + 0.878773i $$0.658363\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 18.0000 1.06251
$$288$$ 0 0
$$289$$ 47.0000 2.76471
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 6.00000 0.345834
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ −24.0000 −1.36975 −0.684876 0.728659i $$-0.740144\pi$$
−0.684876 + 0.728659i $$0.740144\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 21.0000 1.18699 0.593495 0.804838i $$-0.297748\pi$$
0.593495 + 0.804838i $$0.297748\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ 0 0
$$319$$ 30.0000 1.67968
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −16.0000 −0.890264
$$324$$ 0 0
$$325$$ 16.0000 0.887520
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −18.0000 −0.992372
$$330$$ 0 0
$$331$$ −14.0000 −0.769510 −0.384755 0.923019i $$-0.625714\pi$$
−0.384755 + 0.923019i $$0.625714\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −10.0000 −0.546358
$$336$$ 0 0
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −35.0000 −1.89536
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 27.0000 1.44944 0.724718 0.689046i $$-0.241970\pi$$
0.724718 + 0.689046i $$0.241970\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −22.0000 −1.17094 −0.585471 0.810693i $$-0.699090\pi$$
−0.585471 + 0.810693i $$0.699090\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.00000 −0.0523424
$$366$$ 0 0
$$367$$ −11.0000 −0.574195 −0.287098 0.957901i $$-0.592690\pi$$
−0.287098 + 0.957901i $$0.592690\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 15.0000 0.778761
$$372$$ 0 0
$$373$$ −8.00000 −0.414224 −0.207112 0.978317i $$-0.566407\pi$$
−0.207112 + 0.978317i $$0.566407\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −15.0000 −0.764471
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −1.00000 −0.0507020 −0.0253510 0.999679i $$-0.508070\pi$$
−0.0253510 + 0.999679i $$0.508070\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ 4.00000 0.200754 0.100377 0.994949i $$-0.467995\pi$$
0.100377 + 0.994949i $$0.467995\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 28.0000 1.39478
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 30.0000 1.48704
$$408$$ 0 0
$$409$$ −9.00000 −0.445021 −0.222511 0.974930i $$-0.571425\pi$$
−0.222511 + 0.974930i $$0.571425\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 11.0000 0.539969
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −32.0000 −1.55223
$$426$$ 0 0
$$427$$ 24.0000 1.16144
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.0000 −1.63772 −0.818861 0.573992i $$-0.805394\pi$$
−0.818861 + 0.573992i $$0.805394\pi$$
$$432$$ 0 0
$$433$$ 13.0000 0.624740 0.312370 0.949960i $$-0.398877\pi$$
0.312370 + 0.949960i $$0.398877\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 9.00000 0.429547 0.214773 0.976664i $$-0.431099\pi$$
0.214773 + 0.976664i $$0.431099\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 30.0000 1.41264
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ −1.00000 −0.0467780 −0.0233890 0.999726i $$-0.507446\pi$$
−0.0233890 + 0.999726i $$0.507446\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1.00000 −0.0465746 −0.0232873 0.999729i $$-0.507413\pi$$
−0.0232873 + 0.999729i $$0.507413\pi$$
$$462$$ 0 0
$$463$$ −1.00000 −0.0464739 −0.0232370 0.999730i $$-0.507397\pi$$
−0.0232370 + 0.999730i $$0.507397\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −3.00000 −0.138823 −0.0694117 0.997588i $$-0.522112\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 0 0
$$469$$ 30.0000 1.38527
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 0 0
$$475$$ 8.00000 0.367065
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 26.0000 1.18797 0.593985 0.804476i $$-0.297554\pi$$
0.593985 + 0.804476i $$0.297554\pi$$
$$480$$ 0 0
$$481$$ −24.0000 −1.09431
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.00000 0.0454077
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 48.0000 2.16181
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 0 0
$$505$$ −9.00000 −0.400495
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 3.00000 0.132712
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ −30.0000 −1.31940
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −56.0000 −2.43940
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −24.0000 −1.03956
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ −36.0000 −1.54776 −0.773880 0.633332i $$-0.781687\pi$$
−0.773880 + 0.633332i $$0.781687\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 48.0000 2.04117
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.00000 −0.0423714 −0.0211857 0.999776i $$-0.506744\pi$$
−0.0211857 + 0.999776i $$0.506744\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 11.0000 0.463595 0.231797 0.972764i $$-0.425539\pi$$
0.231797 + 0.972764i $$0.425539\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 28.0000 1.17382 0.586911 0.809652i $$-0.300344\pi$$
0.586911 + 0.809652i $$0.300344\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −33.0000 −1.36907
$$582$$ 0 0
$$583$$ 25.0000 1.03539
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −19.0000 −0.784214 −0.392107 0.919920i $$-0.628254\pi$$
−0.392107 + 0.919920i $$0.628254\pi$$
$$588$$ 0 0
$$589$$ 14.0000 0.576860
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ −24.0000 −0.983904
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −18.0000 −0.721155
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 48.0000 1.91389
$$630$$ 0 0
$$631$$ 43.0000 1.71180 0.855901 0.517139i $$-0.173003\pi$$
0.855901 + 0.517139i $$0.173003\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 11.0000 0.436522
$$636$$ 0 0
$$637$$ −8.00000 −0.316972
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 22.0000 0.868948 0.434474 0.900684i $$-0.356934\pi$$
0.434474 + 0.900684i $$0.356934\pi$$
$$642$$ 0 0
$$643$$ −12.0000 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16.0000 −0.629025 −0.314512 0.949253i $$-0.601841\pi$$
−0.314512 + 0.949253i $$0.601841\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3.00000 0.117399 0.0586995 0.998276i $$-0.481305\pi$$
0.0586995 + 0.998276i $$0.481305\pi$$
$$654$$ 0 0
$$655$$ −1.00000 −0.0390732
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −21.0000 −0.818044 −0.409022 0.912525i $$-0.634130\pi$$
−0.409022 + 0.912525i $$0.634130\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 6.00000 0.232670
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 13.0000 0.501113 0.250557 0.968102i $$-0.419386\pi$$
0.250557 + 0.968102i $$0.419386\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 0 0
$$679$$ −3.00000 −0.115129
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 0 0
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 48.0000 1.81813
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3.00000 −0.113308 −0.0566542 0.998394i $$-0.518043\pi$$
−0.0566542 + 0.998394i $$0.518043\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 27.0000 1.01544
$$708$$ 0 0
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 14.0000 0.524304
$$714$$ 0 0
$$715$$ 20.0000 0.747958
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −24.0000 −0.891338
$$726$$ 0 0
$$727$$ 3.00000 0.111264 0.0556319 0.998451i $$-0.482283\pi$$
0.0556319 + 0.998451i $$0.482283\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 50.0000 1.84177
$$738$$ 0 0
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −20.0000 −0.733729 −0.366864 0.930274i $$-0.619569\pi$$
−0.366864 + 0.930274i $$0.619569\pi$$
$$744$$ 0 0
$$745$$ −7.00000 −0.256460
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 27.0000 0.986559
$$750$$ 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
$$755$$ −5.00000 −0.181969
$$756$$ 0 0
$$757$$ 46.0000 1.67190 0.835949 0.548807i $$-0.184918\pi$$
0.835949 + 0.548807i $$0.184918\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 32.0000 1.16000 0.580000 0.814617i $$-0.303053\pi$$
0.580000 + 0.814617i $$0.303053\pi$$
$$762$$ 0 0
$$763$$ −30.0000 −1.08607
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ −7.00000 −0.252426 −0.126213 0.992003i $$-0.540282\pi$$
−0.126213 + 0.992003i $$0.540282\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 30.0000 1.07903 0.539513 0.841978i $$-0.318609\pi$$
0.539513 + 0.841978i $$0.318609\pi$$
$$774$$ 0 0
$$775$$ 28.0000 1.00579
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 40.0000 1.43131
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −20.0000 −0.713831
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ −32.0000 −1.13635
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 35.0000 1.23976 0.619882 0.784695i $$-0.287181\pi$$
0.619882 + 0.784695i $$0.287181\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 5.00000 0.176446
$$804$$ 0 0
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ 14.0000 0.491606 0.245803 0.969320i $$-0.420948\pi$$
0.245803 + 0.969320i $$0.420948\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12.0000 0.420342
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 0 0
$$823$$ −43.0000 −1.49889 −0.749443 0.662069i $$-0.769679\pi$$
−0.749443 + 0.662069i $$0.769679\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −32.0000 −1.11275 −0.556375 0.830932i $$-0.687808\pi$$
−0.556375 + 0.830932i $$0.687808\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 16.0000 0.554367
$$834$$ 0 0
$$835$$ −6.00000 −0.207639
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ 42.0000 1.44314
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 0 0
$$859$$ 24.0000 0.818869 0.409435 0.912339i $$-0.365726\pi$$
0.409435 + 0.912339i $$0.365726\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 52.0000 1.77010 0.885050 0.465495i $$-0.154124\pi$$
0.885050 + 0.465495i $$0.154124\pi$$
$$864$$ 0 0
$$865$$ 21.0000 0.714021
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 80.0000 2.71381
$$870$$ 0 0
$$871$$ −40.0000 −1.35535
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 27.0000 0.912767
$$876$$ 0 0
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −52.0000 −1.75192 −0.875962 0.482380i $$-0.839773\pi$$
−0.875962 + 0.482380i $$0.839773\pi$$
$$882$$ 0 0
$$883$$ 46.0000 1.54802 0.774012 0.633171i $$-0.218247\pi$$
0.774012 + 0.633171i $$0.218247\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −4.00000 −0.134307 −0.0671534 0.997743i $$-0.521392\pi$$
−0.0671534 + 0.997743i $$0.521392\pi$$
$$888$$ 0 0
$$889$$ −33.0000 −1.10678
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 9.00000 0.300837
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −42.0000 −1.40078
$$900$$ 0 0
$$901$$ 40.0000 1.33259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −58.0000 −1.92586 −0.962929 0.269754i $$-0.913058\pi$$
−0.962929 + 0.269754i $$0.913058\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38.0000 −1.25900 −0.629498 0.777002i $$-0.716739\pi$$
−0.629498 + 0.777002i $$0.716739\pi$$
$$912$$ 0 0
$$913$$ −55.0000 −1.82023
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 3.00000 0.0990687
$$918$$ 0 0
$$919$$ −47.0000 −1.55039 −0.775193 0.631724i $$-0.782348\pi$$
−0.775193 + 0.631724i $$0.782348\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −32.0000 −1.05329
$$924$$ 0 0
$$925$$ −24.0000 −0.789115
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −4.00000 −0.131236 −0.0656179 0.997845i $$-0.520902\pi$$
−0.0656179 + 0.997845i $$0.520902\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −40.0000 −1.30814
$$936$$ 0 0
$$937$$ −21.0000 −0.686040 −0.343020 0.939328i $$-0.611450\pi$$
−0.343020 + 0.939328i $$0.611450\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −45.0000 −1.46696 −0.733479 0.679712i $$-0.762105\pi$$
−0.733479 + 0.679712i $$0.762105\pi$$
$$942$$ 0 0
$$943$$ −12.0000 −0.390774
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3.00000 0.0974869 0.0487435 0.998811i $$-0.484478\pi$$
0.0487435 + 0.998811i $$0.484478\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 19.0000 0.611632
$$966$$ 0 0
$$967$$ 35.0000 1.12552 0.562762 0.826619i $$-0.309739\pi$$
0.562762 + 0.826619i $$0.309739\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −51.0000 −1.63667 −0.818334 0.574743i $$-0.805102\pi$$
−0.818334 + 0.574743i $$0.805102\pi$$
$$972$$ 0 0
$$973$$ −36.0000 −1.15411
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 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
$$985$$ 3.00000 0.0955879
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 19.0000 0.603555 0.301777 0.953378i $$-0.402420\pi$$
0.301777 + 0.953378i $$0.402420\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 11.0000 0.348723
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ 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.l.1.1 1
3.2 odd 2 1728.2.a.s.1.1 1
4.3 odd 2 1728.2.a.i.1.1 1
8.3 odd 2 432.2.a.f.1.1 1
8.5 even 2 216.2.a.c.1.1 yes 1
12.11 even 2 1728.2.a.r.1.1 1
24.5 odd 2 216.2.a.b.1.1 1
24.11 even 2 432.2.a.c.1.1 1
40.13 odd 4 5400.2.f.b.649.1 2
40.29 even 2 5400.2.a.e.1.1 1
40.37 odd 4 5400.2.f.b.649.2 2
72.5 odd 6 648.2.i.e.217.1 2
72.11 even 6 1296.2.i.l.433.1 2
72.13 even 6 648.2.i.c.217.1 2
72.29 odd 6 648.2.i.e.433.1 2
72.43 odd 6 1296.2.i.g.433.1 2
72.59 even 6 1296.2.i.l.865.1 2
72.61 even 6 648.2.i.c.433.1 2
72.67 odd 6 1296.2.i.g.865.1 2
120.29 odd 2 5400.2.a.h.1.1 1
120.53 even 4 5400.2.f.z.649.1 2
120.77 even 4 5400.2.f.z.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.a.b.1.1 1 24.5 odd 2
216.2.a.c.1.1 yes 1 8.5 even 2
432.2.a.c.1.1 1 24.11 even 2
432.2.a.f.1.1 1 8.3 odd 2
648.2.i.c.217.1 2 72.13 even 6
648.2.i.c.433.1 2 72.61 even 6
648.2.i.e.217.1 2 72.5 odd 6
648.2.i.e.433.1 2 72.29 odd 6
1296.2.i.g.433.1 2 72.43 odd 6
1296.2.i.g.865.1 2 72.67 odd 6
1296.2.i.l.433.1 2 72.11 even 6
1296.2.i.l.865.1 2 72.59 even 6
1728.2.a.i.1.1 1 4.3 odd 2
1728.2.a.l.1.1 1 1.1 even 1 trivial
1728.2.a.r.1.1 1 12.11 even 2
1728.2.a.s.1.1 1 3.2 odd 2
5400.2.a.e.1.1 1 40.29 even 2
5400.2.a.h.1.1 1 120.29 odd 2
5400.2.f.b.649.1 2 40.13 odd 4
5400.2.f.b.649.2 2 40.37 odd 4
5400.2.f.z.649.1 2 120.53 even 4
5400.2.f.z.649.2 2 120.77 even 4