# Properties

 Label 7448.2.a.x.1.1 Level $7448$ Weight $2$ Character 7448.1 Self dual yes Analytic conductor $59.473$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7448 = 2^{3} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4725794254$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 7448.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} +O(q^{10})$$ $$q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} -3.24264 q^{11} +3.41421 q^{13} -3.41421 q^{15} +6.82843 q^{17} +1.00000 q^{19} +4.41421 q^{23} -4.00000 q^{25} -19.3137 q^{27} -9.41421 q^{29} -4.58579 q^{31} +11.0711 q^{33} -6.58579 q^{37} -11.6569 q^{39} +5.65685 q^{41} -3.24264 q^{43} +8.65685 q^{45} +1.24264 q^{47} -23.3137 q^{51} -5.41421 q^{53} -3.24264 q^{55} -3.41421 q^{57} +0.828427 q^{59} +5.00000 q^{61} +3.41421 q^{65} +3.17157 q^{67} -15.0711 q^{69} -7.75736 q^{71} +7.00000 q^{73} +13.6569 q^{75} +3.89949 q^{79} +39.9706 q^{81} -14.8995 q^{83} +6.82843 q^{85} +32.1421 q^{87} -13.5563 q^{89} +15.6569 q^{93} +1.00000 q^{95} +5.65685 q^{97} -28.0711 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + O(q^{10})$$ $$2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99} + O(q^{100})$$

## 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
$$3$$ −3.41421 −1.97120 −0.985599 0.169102i $$-0.945913\pi$$
−0.985599 + 0.169102i $$0.945913\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 8.65685 2.88562
$$10$$ 0 0
$$11$$ −3.24264 −0.977693 −0.488846 0.872370i $$-0.662582\pi$$
−0.488846 + 0.872370i $$0.662582\pi$$
$$12$$ 0 0
$$13$$ 3.41421 0.946932 0.473466 0.880812i $$-0.343003\pi$$
0.473466 + 0.880812i $$0.343003\pi$$
$$14$$ 0 0
$$15$$ −3.41421 −0.881546
$$16$$ 0 0
$$17$$ 6.82843 1.65614 0.828068 0.560627i $$-0.189440\pi$$
0.828068 + 0.560627i $$0.189440\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.41421 0.920427 0.460214 0.887808i $$-0.347773\pi$$
0.460214 + 0.887808i $$0.347773\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −19.3137 −3.71692
$$28$$ 0 0
$$29$$ −9.41421 −1.74818 −0.874088 0.485768i $$-0.838540\pi$$
−0.874088 + 0.485768i $$0.838540\pi$$
$$30$$ 0 0
$$31$$ −4.58579 −0.823632 −0.411816 0.911267i $$-0.635105\pi$$
−0.411816 + 0.911267i $$0.635105\pi$$
$$32$$ 0 0
$$33$$ 11.0711 1.92723
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.58579 −1.08270 −0.541348 0.840798i $$-0.682086\pi$$
−0.541348 + 0.840798i $$0.682086\pi$$
$$38$$ 0 0
$$39$$ −11.6569 −1.86659
$$40$$ 0 0
$$41$$ 5.65685 0.883452 0.441726 0.897150i $$-0.354366\pi$$
0.441726 + 0.897150i $$0.354366\pi$$
$$42$$ 0 0
$$43$$ −3.24264 −0.494498 −0.247249 0.968952i $$-0.579527\pi$$
−0.247249 + 0.968952i $$0.579527\pi$$
$$44$$ 0 0
$$45$$ 8.65685 1.29049
$$46$$ 0 0
$$47$$ 1.24264 0.181258 0.0906289 0.995885i $$-0.471112\pi$$
0.0906289 + 0.995885i $$0.471112\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −23.3137 −3.26457
$$52$$ 0 0
$$53$$ −5.41421 −0.743699 −0.371850 0.928293i $$-0.621276\pi$$
−0.371850 + 0.928293i $$0.621276\pi$$
$$54$$ 0 0
$$55$$ −3.24264 −0.437238
$$56$$ 0 0
$$57$$ −3.41421 −0.452224
$$58$$ 0 0
$$59$$ 0.828427 0.107852 0.0539260 0.998545i $$-0.482826\pi$$
0.0539260 + 0.998545i $$0.482826\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.41421 0.423481
$$66$$ 0 0
$$67$$ 3.17157 0.387469 0.193735 0.981054i $$-0.437940\pi$$
0.193735 + 0.981054i $$0.437940\pi$$
$$68$$ 0 0
$$69$$ −15.0711 −1.81434
$$70$$ 0 0
$$71$$ −7.75736 −0.920629 −0.460315 0.887756i $$-0.652263\pi$$
−0.460315 + 0.887756i $$0.652263\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 0 0
$$75$$ 13.6569 1.57696
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 3.89949 0.438727 0.219364 0.975643i $$-0.429602\pi$$
0.219364 + 0.975643i $$0.429602\pi$$
$$80$$ 0 0
$$81$$ 39.9706 4.44117
$$82$$ 0 0
$$83$$ −14.8995 −1.63543 −0.817716 0.575622i $$-0.804760\pi$$
−0.817716 + 0.575622i $$0.804760\pi$$
$$84$$ 0 0
$$85$$ 6.82843 0.740647
$$86$$ 0 0
$$87$$ 32.1421 3.44600
$$88$$ 0 0
$$89$$ −13.5563 −1.43697 −0.718485 0.695542i $$-0.755164\pi$$
−0.718485 + 0.695542i $$0.755164\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 15.6569 1.62354
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 5.65685 0.574367 0.287183 0.957876i $$-0.407281\pi$$
0.287183 + 0.957876i $$0.407281\pi$$
$$98$$ 0 0
$$99$$ −28.0711 −2.82125
$$100$$ 0 0
$$101$$ −1.48528 −0.147791 −0.0738955 0.997266i $$-0.523543\pi$$
−0.0738955 + 0.997266i $$0.523543\pi$$
$$102$$ 0 0
$$103$$ −2.34315 −0.230877 −0.115439 0.993315i $$-0.536827\pi$$
−0.115439 + 0.993315i $$0.536827\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 17.0711 1.65032 0.825161 0.564897i $$-0.191084\pi$$
0.825161 + 0.564897i $$0.191084\pi$$
$$108$$ 0 0
$$109$$ −19.3137 −1.84992 −0.924959 0.380067i $$-0.875901\pi$$
−0.924959 + 0.380067i $$0.875901\pi$$
$$110$$ 0 0
$$111$$ 22.4853 2.13421
$$112$$ 0 0
$$113$$ 11.0711 1.04148 0.520739 0.853716i $$-0.325656\pi$$
0.520739 + 0.853716i $$0.325656\pi$$
$$114$$ 0 0
$$115$$ 4.41421 0.411628
$$116$$ 0 0
$$117$$ 29.5563 2.73249
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −0.485281 −0.0441165
$$122$$ 0 0
$$123$$ −19.3137 −1.74146
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ −19.5563 −1.73535 −0.867673 0.497136i $$-0.834385\pi$$
−0.867673 + 0.497136i $$0.834385\pi$$
$$128$$ 0 0
$$129$$ 11.0711 0.974753
$$130$$ 0 0
$$131$$ −1.65685 −0.144760 −0.0723800 0.997377i $$-0.523059\pi$$
−0.0723800 + 0.997377i $$0.523059\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −19.3137 −1.66226
$$136$$ 0 0
$$137$$ 0.656854 0.0561188 0.0280594 0.999606i $$-0.491067\pi$$
0.0280594 + 0.999606i $$0.491067\pi$$
$$138$$ 0 0
$$139$$ 18.5563 1.57393 0.786964 0.616998i $$-0.211651\pi$$
0.786964 + 0.616998i $$0.211651\pi$$
$$140$$ 0 0
$$141$$ −4.24264 −0.357295
$$142$$ 0 0
$$143$$ −11.0711 −0.925809
$$144$$ 0 0
$$145$$ −9.41421 −0.781808
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.65685 −0.545351 −0.272675 0.962106i $$-0.587908\pi$$
−0.272675 + 0.962106i $$0.587908\pi$$
$$150$$ 0 0
$$151$$ 5.65685 0.460348 0.230174 0.973149i $$-0.426070\pi$$
0.230174 + 0.973149i $$0.426070\pi$$
$$152$$ 0 0
$$153$$ 59.1127 4.77898
$$154$$ 0 0
$$155$$ −4.58579 −0.368339
$$156$$ 0 0
$$157$$ 1.68629 0.134581 0.0672904 0.997733i $$-0.478565\pi$$
0.0672904 + 0.997733i $$0.478565\pi$$
$$158$$ 0 0
$$159$$ 18.4853 1.46598
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 14.0711 1.10213 0.551066 0.834462i $$-0.314221\pi$$
0.551066 + 0.834462i $$0.314221\pi$$
$$164$$ 0 0
$$165$$ 11.0711 0.861881
$$166$$ 0 0
$$167$$ −18.5858 −1.43821 −0.719106 0.694901i $$-0.755448\pi$$
−0.719106 + 0.694901i $$0.755448\pi$$
$$168$$ 0 0
$$169$$ −1.34315 −0.103319
$$170$$ 0 0
$$171$$ 8.65685 0.662006
$$172$$ 0 0
$$173$$ 9.65685 0.734197 0.367099 0.930182i $$-0.380351\pi$$
0.367099 + 0.930182i $$0.380351\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2.82843 −0.212598
$$178$$ 0 0
$$179$$ 6.48528 0.484733 0.242366 0.970185i $$-0.422076\pi$$
0.242366 + 0.970185i $$0.422076\pi$$
$$180$$ 0 0
$$181$$ 14.7279 1.09472 0.547359 0.836898i $$-0.315633\pi$$
0.547359 + 0.836898i $$0.315633\pi$$
$$182$$ 0 0
$$183$$ −17.0711 −1.26193
$$184$$ 0 0
$$185$$ −6.58579 −0.484197
$$186$$ 0 0
$$187$$ −22.1421 −1.61919
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 9.24264 0.668774 0.334387 0.942436i $$-0.391471\pi$$
0.334387 + 0.942436i $$0.391471\pi$$
$$192$$ 0 0
$$193$$ 2.24264 0.161429 0.0807144 0.996737i $$-0.474280\pi$$
0.0807144 + 0.996737i $$0.474280\pi$$
$$194$$ 0 0
$$195$$ −11.6569 −0.834765
$$196$$ 0 0
$$197$$ −0.514719 −0.0366722 −0.0183361 0.999832i $$-0.505837\pi$$
−0.0183361 + 0.999832i $$0.505837\pi$$
$$198$$ 0 0
$$199$$ −2.07107 −0.146814 −0.0734071 0.997302i $$-0.523387\pi$$
−0.0734071 + 0.997302i $$0.523387\pi$$
$$200$$ 0 0
$$201$$ −10.8284 −0.763778
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 5.65685 0.395092
$$206$$ 0 0
$$207$$ 38.2132 2.65600
$$208$$ 0 0
$$209$$ −3.24264 −0.224298
$$210$$ 0 0
$$211$$ 12.1421 0.835899 0.417950 0.908470i $$-0.362749\pi$$
0.417950 + 0.908470i $$0.362749\pi$$
$$212$$ 0 0
$$213$$ 26.4853 1.81474
$$214$$ 0 0
$$215$$ −3.24264 −0.221146
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −23.8995 −1.61498
$$220$$ 0 0
$$221$$ 23.3137 1.56825
$$222$$ 0 0
$$223$$ 16.2426 1.08769 0.543844 0.839186i $$-0.316968\pi$$
0.543844 + 0.839186i $$0.316968\pi$$
$$224$$ 0 0
$$225$$ −34.6274 −2.30849
$$226$$ 0 0
$$227$$ −1.41421 −0.0938647 −0.0469323 0.998898i $$-0.514945\pi$$
−0.0469323 + 0.998898i $$0.514945\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.34315 −0.415553 −0.207777 0.978176i $$-0.566623\pi$$
−0.207777 + 0.978176i $$0.566623\pi$$
$$234$$ 0 0
$$235$$ 1.24264 0.0810609
$$236$$ 0 0
$$237$$ −13.3137 −0.864818
$$238$$ 0 0
$$239$$ −4.34315 −0.280935 −0.140467 0.990085i $$-0.544861\pi$$
−0.140467 + 0.990085i $$0.544861\pi$$
$$240$$ 0 0
$$241$$ −8.72792 −0.562215 −0.281107 0.959676i $$-0.590702\pi$$
−0.281107 + 0.959676i $$0.590702\pi$$
$$242$$ 0 0
$$243$$ −78.5269 −5.03750
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.41421 0.217241
$$248$$ 0 0
$$249$$ 50.8701 3.22376
$$250$$ 0 0
$$251$$ 22.2132 1.40208 0.701042 0.713120i $$-0.252718\pi$$
0.701042 + 0.713120i $$0.252718\pi$$
$$252$$ 0 0
$$253$$ −14.3137 −0.899895
$$254$$ 0 0
$$255$$ −23.3137 −1.45996
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −81.4975 −5.04457
$$262$$ 0 0
$$263$$ −21.1716 −1.30550 −0.652748 0.757575i $$-0.726384\pi$$
−0.652748 + 0.757575i $$0.726384\pi$$
$$264$$ 0 0
$$265$$ −5.41421 −0.332592
$$266$$ 0 0
$$267$$ 46.2843 2.83255
$$268$$ 0 0
$$269$$ 16.0000 0.975537 0.487769 0.872973i $$-0.337811\pi$$
0.487769 + 0.872973i $$0.337811\pi$$
$$270$$ 0 0
$$271$$ −2.27208 −0.138019 −0.0690095 0.997616i $$-0.521984\pi$$
−0.0690095 + 0.997616i $$0.521984\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 12.9706 0.782154
$$276$$ 0 0
$$277$$ −27.9706 −1.68059 −0.840294 0.542131i $$-0.817618\pi$$
−0.840294 + 0.542131i $$0.817618\pi$$
$$278$$ 0 0
$$279$$ −39.6985 −2.37669
$$280$$ 0 0
$$281$$ −17.8995 −1.06779 −0.533897 0.845549i $$-0.679273\pi$$
−0.533897 + 0.845549i $$0.679273\pi$$
$$282$$ 0 0
$$283$$ −14.5563 −0.865285 −0.432643 0.901566i $$-0.642419\pi$$
−0.432643 + 0.901566i $$0.642419\pi$$
$$284$$ 0 0
$$285$$ −3.41421 −0.202241
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 29.6274 1.74279
$$290$$ 0 0
$$291$$ −19.3137 −1.13219
$$292$$ 0 0
$$293$$ −9.65685 −0.564159 −0.282080 0.959391i $$-0.591024\pi$$
−0.282080 + 0.959391i $$0.591024\pi$$
$$294$$ 0 0
$$295$$ 0.828427 0.0482329
$$296$$ 0 0
$$297$$ 62.6274 3.63401
$$298$$ 0 0
$$299$$ 15.0711 0.871582
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 5.07107 0.291325
$$304$$ 0 0
$$305$$ 5.00000 0.286299
$$306$$ 0 0
$$307$$ −18.6274 −1.06312 −0.531561 0.847020i $$-0.678395\pi$$
−0.531561 + 0.847020i $$0.678395\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ −1.14214 −0.0645573 −0.0322787 0.999479i $$-0.510276\pi$$
−0.0322787 + 0.999479i $$0.510276\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10.2426 −0.575284 −0.287642 0.957738i $$-0.592871\pi$$
−0.287642 + 0.957738i $$0.592871\pi$$
$$318$$ 0 0
$$319$$ 30.5269 1.70918
$$320$$ 0 0
$$321$$ −58.2843 −3.25311
$$322$$ 0 0
$$323$$ 6.82843 0.379944
$$324$$ 0 0
$$325$$ −13.6569 −0.757546
$$326$$ 0 0
$$327$$ 65.9411 3.64655
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −18.4853 −1.01604 −0.508021 0.861344i $$-0.669623\pi$$
−0.508021 + 0.861344i $$0.669623\pi$$
$$332$$ 0 0
$$333$$ −57.0122 −3.12425
$$334$$ 0 0
$$335$$ 3.17157 0.173282
$$336$$ 0 0
$$337$$ 5.07107 0.276239 0.138119 0.990416i $$-0.455894\pi$$
0.138119 + 0.990416i $$0.455894\pi$$
$$338$$ 0 0
$$339$$ −37.7990 −2.05296
$$340$$ 0 0
$$341$$ 14.8701 0.805259
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −15.0711 −0.811399
$$346$$ 0 0
$$347$$ 12.4142 0.666430 0.333215 0.942851i $$-0.391867\pi$$
0.333215 + 0.942851i $$0.391867\pi$$
$$348$$ 0 0
$$349$$ −5.65685 −0.302804 −0.151402 0.988472i $$-0.548379\pi$$
−0.151402 + 0.988472i $$0.548379\pi$$
$$350$$ 0 0
$$351$$ −65.9411 −3.51968
$$352$$ 0 0
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ −7.75736 −0.411718
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 36.6985 1.93687 0.968436 0.249262i $$-0.0801882\pi$$
0.968436 + 0.249262i $$0.0801882\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 1.65685 0.0869623
$$364$$ 0 0
$$365$$ 7.00000 0.366397
$$366$$ 0 0
$$367$$ −16.6274 −0.867944 −0.433972 0.900926i $$-0.642888\pi$$
−0.433972 + 0.900926i $$0.642888\pi$$
$$368$$ 0 0
$$369$$ 48.9706 2.54931
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6.34315 −0.328436 −0.164218 0.986424i $$-0.552510\pi$$
−0.164218 + 0.986424i $$0.552510\pi$$
$$374$$ 0 0
$$375$$ 30.7279 1.58678
$$376$$ 0 0
$$377$$ −32.1421 −1.65540
$$378$$ 0 0
$$379$$ 12.5858 0.646488 0.323244 0.946316i $$-0.395226\pi$$
0.323244 + 0.946316i $$0.395226\pi$$
$$380$$ 0 0
$$381$$ 66.7696 3.42071
$$382$$ 0 0
$$383$$ 3.75736 0.191992 0.0959960 0.995382i $$-0.469396\pi$$
0.0959960 + 0.995382i $$0.469396\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −28.0711 −1.42693
$$388$$ 0 0
$$389$$ −15.3137 −0.776436 −0.388218 0.921568i $$-0.626909\pi$$
−0.388218 + 0.921568i $$0.626909\pi$$
$$390$$ 0 0
$$391$$ 30.1421 1.52435
$$392$$ 0 0
$$393$$ 5.65685 0.285351
$$394$$ 0 0
$$395$$ 3.89949 0.196205
$$396$$ 0 0
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8.62742 −0.430833 −0.215416 0.976522i $$-0.569111\pi$$
−0.215416 + 0.976522i $$0.569111\pi$$
$$402$$ 0 0
$$403$$ −15.6569 −0.779923
$$404$$ 0 0
$$405$$ 39.9706 1.98615
$$406$$ 0 0
$$407$$ 21.3553 1.05854
$$408$$ 0 0
$$409$$ 29.3137 1.44947 0.724735 0.689028i $$-0.241962\pi$$
0.724735 + 0.689028i $$0.241962\pi$$
$$410$$ 0 0
$$411$$ −2.24264 −0.110621
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −14.8995 −0.731387
$$416$$ 0 0
$$417$$ −63.3553 −3.10252
$$418$$ 0 0
$$419$$ 30.6985 1.49972 0.749860 0.661597i $$-0.230121\pi$$
0.749860 + 0.661597i $$0.230121\pi$$
$$420$$ 0 0
$$421$$ −24.5858 −1.19824 −0.599119 0.800660i $$-0.704482\pi$$
−0.599119 + 0.800660i $$0.704482\pi$$
$$422$$ 0 0
$$423$$ 10.7574 0.523041
$$424$$ 0 0
$$425$$ −27.3137 −1.32491
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 37.7990 1.82495
$$430$$ 0 0
$$431$$ −13.8995 −0.669515 −0.334758 0.942304i $$-0.608654\pi$$
−0.334758 + 0.942304i $$0.608654\pi$$
$$432$$ 0 0
$$433$$ −12.0000 −0.576683 −0.288342 0.957528i $$-0.593104\pi$$
−0.288342 + 0.957528i $$0.593104\pi$$
$$434$$ 0 0
$$435$$ 32.1421 1.54110
$$436$$ 0 0
$$437$$ 4.41421 0.211160
$$438$$ 0 0
$$439$$ −26.4853 −1.26407 −0.632037 0.774938i $$-0.717781\pi$$
−0.632037 + 0.774938i $$0.717781\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16.6863 −0.792790 −0.396395 0.918080i $$-0.629739\pi$$
−0.396395 + 0.918080i $$0.629739\pi$$
$$444$$ 0 0
$$445$$ −13.5563 −0.642633
$$446$$ 0 0
$$447$$ 22.7279 1.07499
$$448$$ 0 0
$$449$$ 30.6274 1.44540 0.722699 0.691163i $$-0.242901\pi$$
0.722699 + 0.691163i $$0.242901\pi$$
$$450$$ 0 0
$$451$$ −18.3431 −0.863745
$$452$$ 0 0
$$453$$ −19.3137 −0.907437
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.82843 0.272642 0.136321 0.990665i $$-0.456472\pi$$
0.136321 + 0.990665i $$0.456472\pi$$
$$458$$ 0 0
$$459$$ −131.882 −6.15574
$$460$$ 0 0
$$461$$ −10.7990 −0.502959 −0.251480 0.967863i $$-0.580917\pi$$
−0.251480 + 0.967863i $$0.580917\pi$$
$$462$$ 0 0
$$463$$ −27.7279 −1.28863 −0.644313 0.764762i $$-0.722857\pi$$
−0.644313 + 0.764762i $$0.722857\pi$$
$$464$$ 0 0
$$465$$ 15.6569 0.726069
$$466$$ 0 0
$$467$$ 7.72792 0.357606 0.178803 0.983885i $$-0.442778\pi$$
0.178803 + 0.983885i $$0.442778\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −5.75736 −0.265285
$$472$$ 0 0
$$473$$ 10.5147 0.483467
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −46.8701 −2.14603
$$478$$ 0 0
$$479$$ 9.92893 0.453664 0.226832 0.973934i $$-0.427163\pi$$
0.226832 + 0.973934i $$0.427163\pi$$
$$480$$ 0 0
$$481$$ −22.4853 −1.02524
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5.65685 0.256865
$$486$$ 0 0
$$487$$ −11.8579 −0.537331 −0.268666 0.963234i $$-0.586583\pi$$
−0.268666 + 0.963234i $$0.586583\pi$$
$$488$$ 0 0
$$489$$ −48.0416 −2.17252
$$490$$ 0 0
$$491$$ −18.6985 −0.843851 −0.421925 0.906631i $$-0.638646\pi$$
−0.421925 + 0.906631i $$0.638646\pi$$
$$492$$ 0 0
$$493$$ −64.2843 −2.89522
$$494$$ 0 0
$$495$$ −28.0711 −1.26170
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −0.757359 −0.0339041 −0.0169520 0.999856i $$-0.505396\pi$$
−0.0169520 + 0.999856i $$0.505396\pi$$
$$500$$ 0 0
$$501$$ 63.4558 2.83500
$$502$$ 0 0
$$503$$ −21.7279 −0.968800 −0.484400 0.874847i $$-0.660962\pi$$
−0.484400 + 0.874847i $$0.660962\pi$$
$$504$$ 0 0
$$505$$ −1.48528 −0.0660942
$$506$$ 0 0
$$507$$ 4.58579 0.203662
$$508$$ 0 0
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −19.3137 −0.852721
$$514$$ 0 0
$$515$$ −2.34315 −0.103251
$$516$$ 0 0
$$517$$ −4.02944 −0.177214
$$518$$ 0 0
$$519$$ −32.9706 −1.44725
$$520$$ 0 0
$$521$$ 21.8995 0.959434 0.479717 0.877423i $$-0.340739\pi$$
0.479717 + 0.877423i $$0.340739\pi$$
$$522$$ 0 0
$$523$$ −7.31371 −0.319806 −0.159903 0.987133i $$-0.551118\pi$$
−0.159903 + 0.987133i $$0.551118\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −31.3137 −1.36405
$$528$$ 0 0
$$529$$ −3.51472 −0.152814
$$530$$ 0 0
$$531$$ 7.17157 0.311220
$$532$$ 0 0
$$533$$ 19.3137 0.836570
$$534$$ 0 0
$$535$$ 17.0711 0.738047
$$536$$ 0 0
$$537$$ −22.1421 −0.955504
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −29.1421 −1.25292 −0.626459 0.779454i $$-0.715496\pi$$
−0.626459 + 0.779454i $$0.715496\pi$$
$$542$$ 0 0
$$543$$ −50.2843 −2.15790
$$544$$ 0 0
$$545$$ −19.3137 −0.827308
$$546$$ 0 0
$$547$$ −21.5563 −0.921683 −0.460841 0.887482i $$-0.652452\pi$$
−0.460841 + 0.887482i $$0.652452\pi$$
$$548$$ 0 0
$$549$$ 43.2843 1.84733
$$550$$ 0 0
$$551$$ −9.41421 −0.401059
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 22.4853 0.954447
$$556$$ 0 0
$$557$$ 35.4853 1.50356 0.751780 0.659414i $$-0.229196\pi$$
0.751780 + 0.659414i $$0.229196\pi$$
$$558$$ 0 0
$$559$$ −11.0711 −0.468256
$$560$$ 0 0
$$561$$ 75.5980 3.19175
$$562$$ 0 0
$$563$$ −42.8701 −1.80676 −0.903379 0.428844i $$-0.858921\pi$$
−0.903379 + 0.428844i $$0.858921\pi$$
$$564$$ 0 0
$$565$$ 11.0711 0.465763
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.31371 −0.222762 −0.111381 0.993778i $$-0.535527\pi$$
−0.111381 + 0.993778i $$0.535527\pi$$
$$570$$ 0 0
$$571$$ −42.6985 −1.78688 −0.893438 0.449187i $$-0.851714\pi$$
−0.893438 + 0.449187i $$0.851714\pi$$
$$572$$ 0 0
$$573$$ −31.5563 −1.31829
$$574$$ 0 0
$$575$$ −17.6569 −0.736342
$$576$$ 0 0
$$577$$ −28.1716 −1.17280 −0.586399 0.810022i $$-0.699455\pi$$
−0.586399 + 0.810022i $$0.699455\pi$$
$$578$$ 0 0
$$579$$ −7.65685 −0.318208
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 17.5563 0.727110
$$584$$ 0 0
$$585$$ 29.5563 1.22200
$$586$$ 0 0
$$587$$ 34.0000 1.40333 0.701665 0.712507i $$-0.252440\pi$$
0.701665 + 0.712507i $$0.252440\pi$$
$$588$$ 0 0
$$589$$ −4.58579 −0.188954
$$590$$ 0 0
$$591$$ 1.75736 0.0722881
$$592$$ 0 0
$$593$$ −4.31371 −0.177143 −0.0885714 0.996070i $$-0.528230\pi$$
−0.0885714 + 0.996070i $$0.528230\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 7.07107 0.289400
$$598$$ 0 0
$$599$$ 21.2132 0.866748 0.433374 0.901214i $$-0.357323\pi$$
0.433374 + 0.901214i $$0.357323\pi$$
$$600$$ 0 0
$$601$$ 16.5269 0.674147 0.337073 0.941478i $$-0.390563\pi$$
0.337073 + 0.941478i $$0.390563\pi$$
$$602$$ 0 0
$$603$$ 27.4558 1.11809
$$604$$ 0 0
$$605$$ −0.485281 −0.0197295
$$606$$ 0 0
$$607$$ −15.8995 −0.645341 −0.322670 0.946511i $$-0.604581\pi$$
−0.322670 + 0.946511i $$0.604581\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4.24264 0.171639
$$612$$ 0 0
$$613$$ −32.4853 −1.31207 −0.656034 0.754731i $$-0.727767\pi$$
−0.656034 + 0.754731i $$0.727767\pi$$
$$614$$ 0 0
$$615$$ −19.3137 −0.778804
$$616$$ 0 0
$$617$$ 27.2843 1.09842 0.549212 0.835683i $$-0.314928\pi$$
0.549212 + 0.835683i $$0.314928\pi$$
$$618$$ 0 0
$$619$$ −9.10051 −0.365780 −0.182890 0.983133i $$-0.558545\pi$$
−0.182890 + 0.983133i $$0.558545\pi$$
$$620$$ 0 0
$$621$$ −85.2548 −3.42116
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 11.0711 0.442136
$$628$$ 0 0
$$629$$ −44.9706 −1.79309
$$630$$ 0 0
$$631$$ −17.3848 −0.692077 −0.346039 0.938220i $$-0.612473\pi$$
−0.346039 + 0.938220i $$0.612473\pi$$
$$632$$ 0 0
$$633$$ −41.4558 −1.64772
$$634$$ 0 0
$$635$$ −19.5563 −0.776070
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −67.1543 −2.65658
$$640$$ 0 0
$$641$$ 10.6863 0.422083 0.211042 0.977477i $$-0.432314\pi$$
0.211042 + 0.977477i $$0.432314\pi$$
$$642$$ 0 0
$$643$$ −27.5147 −1.08507 −0.542537 0.840032i $$-0.682536\pi$$
−0.542537 + 0.840032i $$0.682536\pi$$
$$644$$ 0 0
$$645$$ 11.0711 0.435923
$$646$$ 0 0
$$647$$ −27.9289 −1.09800 −0.549000 0.835822i $$-0.684991\pi$$
−0.549000 + 0.835822i $$0.684991\pi$$
$$648$$ 0 0
$$649$$ −2.68629 −0.105446
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.1421 −1.17955 −0.589776 0.807567i $$-0.700784\pi$$
−0.589776 + 0.807567i $$0.700784\pi$$
$$654$$ 0 0
$$655$$ −1.65685 −0.0647387
$$656$$ 0 0
$$657$$ 60.5980 2.36415
$$658$$ 0 0
$$659$$ 24.3848 0.949896 0.474948 0.880014i $$-0.342467\pi$$
0.474948 + 0.880014i $$0.342467\pi$$
$$660$$ 0 0
$$661$$ −8.87006 −0.345005 −0.172503 0.985009i $$-0.555185\pi$$
−0.172503 + 0.985009i $$0.555185\pi$$
$$662$$ 0 0
$$663$$ −79.5980 −3.09133
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −41.5563 −1.60907
$$668$$ 0 0
$$669$$ −55.4558 −2.14405
$$670$$ 0 0
$$671$$ −16.2132 −0.625904
$$672$$ 0 0
$$673$$ −40.6274 −1.56607 −0.783036 0.621976i $$-0.786330\pi$$
−0.783036 + 0.621976i $$0.786330\pi$$
$$674$$ 0 0
$$675$$ 77.2548 2.97354
$$676$$ 0 0
$$677$$ −11.3137 −0.434821 −0.217411 0.976080i $$-0.569761\pi$$
−0.217411 + 0.976080i $$0.569761\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.82843 0.185026
$$682$$ 0 0
$$683$$ −24.1421 −0.923773 −0.461887 0.886939i $$-0.652827\pi$$
−0.461887 + 0.886939i $$0.652827\pi$$
$$684$$ 0 0
$$685$$ 0.656854 0.0250971
$$686$$ 0 0
$$687$$ 54.6274 2.08417
$$688$$ 0 0
$$689$$ −18.4853 −0.704233
$$690$$ 0 0
$$691$$ 16.6274 0.632537 0.316268 0.948670i $$-0.397570\pi$$
0.316268 + 0.948670i $$0.397570\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 18.5563 0.703882
$$696$$ 0 0
$$697$$ 38.6274 1.46312
$$698$$ 0 0
$$699$$ 21.6569 0.819137
$$700$$ 0 0
$$701$$ −20.7990 −0.785567 −0.392784 0.919631i $$-0.628488\pi$$
−0.392784 + 0.919631i $$0.628488\pi$$
$$702$$ 0 0
$$703$$ −6.58579 −0.248388
$$704$$ 0 0
$$705$$ −4.24264 −0.159787
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 52.5980 1.97536 0.987679 0.156492i $$-0.0500184\pi$$
0.987679 + 0.156492i $$0.0500184\pi$$
$$710$$ 0 0
$$711$$ 33.7574 1.26600
$$712$$ 0 0
$$713$$ −20.2426 −0.758093
$$714$$ 0 0
$$715$$ −11.0711 −0.414034
$$716$$ 0 0
$$717$$ 14.8284 0.553778
$$718$$ 0 0
$$719$$ −3.31371 −0.123580 −0.0617902 0.998089i $$-0.519681\pi$$
−0.0617902 + 0.998089i $$0.519681\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 29.7990 1.10824
$$724$$ 0 0
$$725$$ 37.6569 1.39854
$$726$$ 0 0
$$727$$ −3.24264 −0.120263 −0.0601314 0.998190i $$-0.519152\pi$$
−0.0601314 + 0.998190i $$0.519152\pi$$
$$728$$ 0 0
$$729$$ 148.196 5.48874
$$730$$ 0 0
$$731$$ −22.1421 −0.818956
$$732$$ 0 0
$$733$$ 51.3137 1.89532 0.947658 0.319289i $$-0.103444\pi$$
0.947658 + 0.319289i $$0.103444\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.2843 −0.378826
$$738$$ 0 0
$$739$$ 44.6274 1.64165 0.820823 0.571183i $$-0.193515\pi$$
0.820823 + 0.571183i $$0.193515\pi$$
$$740$$ 0 0
$$741$$ −11.6569 −0.428225
$$742$$ 0 0
$$743$$ 21.2721 0.780397 0.390198 0.920731i $$-0.372406\pi$$
0.390198 + 0.920731i $$0.372406\pi$$
$$744$$ 0 0
$$745$$ −6.65685 −0.243888
$$746$$ 0 0
$$747$$ −128.983 −4.71923
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 5.51472 0.201235 0.100617 0.994925i $$-0.467918\pi$$
0.100617 + 0.994925i $$0.467918\pi$$
$$752$$ 0 0
$$753$$ −75.8406 −2.76379
$$754$$ 0 0
$$755$$ 5.65685 0.205874
$$756$$ 0 0
$$757$$ −31.6863 −1.15166 −0.575829 0.817570i $$-0.695321\pi$$
−0.575829 + 0.817570i $$0.695321\pi$$
$$758$$ 0 0
$$759$$ 48.8701 1.77387
$$760$$ 0 0
$$761$$ 13.1421 0.476402 0.238201 0.971216i $$-0.423442\pi$$
0.238201 + 0.971216i $$0.423442\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 59.1127 2.13722
$$766$$ 0 0
$$767$$ 2.82843 0.102129
$$768$$ 0 0
$$769$$ 3.54416 0.127806 0.0639028 0.997956i $$-0.479645\pi$$
0.0639028 + 0.997956i $$0.479645\pi$$
$$770$$ 0 0
$$771$$ 47.7990 1.72144
$$772$$ 0 0
$$773$$ 0.928932 0.0334114 0.0167057 0.999860i $$-0.494682\pi$$
0.0167057 + 0.999860i $$0.494682\pi$$
$$774$$ 0 0
$$775$$ 18.3431 0.658905
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.65685 0.202678
$$780$$ 0 0
$$781$$ 25.1543 0.900093
$$782$$ 0 0
$$783$$ 181.823 6.49784
$$784$$ 0 0
$$785$$ 1.68629 0.0601863
$$786$$ 0 0
$$787$$ −11.7574 −0.419105 −0.209552 0.977797i $$-0.567201\pi$$
−0.209552 + 0.977797i $$0.567201\pi$$
$$788$$ 0 0
$$789$$ 72.2843 2.57339
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 17.0711 0.606211
$$794$$ 0 0
$$795$$ 18.4853 0.655605
$$796$$ 0 0
$$797$$ 45.4975 1.61160 0.805802 0.592186i $$-0.201735\pi$$
0.805802 + 0.592186i $$0.201735\pi$$
$$798$$ 0 0
$$799$$ 8.48528 0.300188
$$800$$ 0 0
$$801$$ −117.355 −4.14655
$$802$$ 0 0
$$803$$ −22.6985 −0.801012
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −54.6274 −1.92298
$$808$$ 0 0
$$809$$ −27.9706 −0.983393 −0.491696 0.870767i $$-0.663623\pi$$
−0.491696 + 0.870767i $$0.663623\pi$$
$$810$$ 0 0
$$811$$ −32.4264 −1.13865 −0.569323 0.822114i $$-0.692794\pi$$
−0.569323 + 0.822114i $$0.692794\pi$$
$$812$$ 0 0
$$813$$ 7.75736 0.272062
$$814$$ 0 0
$$815$$ 14.0711 0.492888
$$816$$ 0 0
$$817$$ −3.24264 −0.113446
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.02944 0.280229 0.140115 0.990135i $$-0.455253\pi$$
0.140115 + 0.990135i $$0.455253\pi$$
$$822$$ 0 0
$$823$$ 9.72792 0.339094 0.169547 0.985522i $$-0.445770\pi$$
0.169547 + 0.985522i $$0.445770\pi$$
$$824$$ 0 0
$$825$$ −44.2843 −1.54178
$$826$$ 0 0
$$827$$ −0.686292 −0.0238647 −0.0119323 0.999929i $$-0.503798\pi$$
−0.0119323 + 0.999929i $$0.503798\pi$$
$$828$$ 0 0
$$829$$ −39.3137 −1.36542 −0.682711 0.730689i $$-0.739199\pi$$
−0.682711 + 0.730689i $$0.739199\pi$$
$$830$$ 0 0
$$831$$ 95.4975 3.31277
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −18.5858 −0.643188
$$836$$ 0 0
$$837$$ 88.5685 3.06138
$$838$$ 0 0
$$839$$ −30.4853 −1.05247 −0.526234 0.850340i $$-0.676397\pi$$
−0.526234 + 0.850340i $$0.676397\pi$$
$$840$$ 0 0
$$841$$ 59.6274 2.05612
$$842$$ 0 0
$$843$$ 61.1127 2.10483
$$844$$ 0 0
$$845$$ −1.34315 −0.0462056
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 49.6985 1.70565
$$850$$ 0 0
$$851$$ −29.0711 −0.996543
$$852$$ 0 0
$$853$$ 52.4558 1.79605 0.898027 0.439940i $$-0.145000\pi$$
0.898027 + 0.439940i $$0.145000\pi$$
$$854$$ 0 0
$$855$$ 8.65685 0.296058
$$856$$ 0 0
$$857$$ −7.02944 −0.240121 −0.120061 0.992767i $$-0.538309\pi$$
−0.120061 + 0.992767i $$0.538309\pi$$
$$858$$ 0 0
$$859$$ 27.1838 0.927498 0.463749 0.885967i $$-0.346504\pi$$
0.463749 + 0.885967i $$0.346504\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −24.8701 −0.846587 −0.423293 0.905993i $$-0.639126\pi$$
−0.423293 + 0.905993i $$0.639126\pi$$
$$864$$ 0 0
$$865$$ 9.65685 0.328343
$$866$$ 0 0
$$867$$ −101.154 −3.43538
$$868$$ 0 0
$$869$$ −12.6447 −0.428941
$$870$$ 0 0
$$871$$ 10.8284 0.366907
$$872$$ 0 0
$$873$$ 48.9706 1.65740
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 36.8701 1.24501 0.622507 0.782614i $$-0.286114\pi$$
0.622507 + 0.782614i $$0.286114\pi$$
$$878$$ 0 0
$$879$$ 32.9706 1.11207
$$880$$ 0 0
$$881$$ −7.31371 −0.246405 −0.123203 0.992382i $$-0.539316\pi$$
−0.123203 + 0.992382i $$0.539316\pi$$
$$882$$ 0 0
$$883$$ −25.1716 −0.847091 −0.423545 0.905875i $$-0.639215\pi$$
−0.423545 + 0.905875i $$0.639215\pi$$
$$884$$ 0 0
$$885$$ −2.82843 −0.0950765
$$886$$ 0 0
$$887$$ 28.2843 0.949693 0.474846 0.880069i $$-0.342504\pi$$
0.474846 + 0.880069i $$0.342504\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −129.610 −4.34210
$$892$$ 0 0
$$893$$ 1.24264 0.0415834
$$894$$ 0 0
$$895$$ 6.48528 0.216779
$$896$$ 0 0
$$897$$ −51.4558 −1.71806
$$898$$ 0 0
$$899$$ 43.1716 1.43985
$$900$$ 0 0
$$901$$ −36.9706 −1.23167
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 14.7279 0.489573
$$906$$ 0 0
$$907$$ −49.5563 −1.64549 −0.822746 0.568410i $$-0.807559\pi$$
−0.822746 + 0.568410i $$0.807559\pi$$
$$908$$ 0 0
$$909$$ −12.8579 −0.426468
$$910$$ 0 0
$$911$$ −25.5563 −0.846720 −0.423360 0.905962i $$-0.639149\pi$$
−0.423360 + 0.905962i $$0.639149\pi$$
$$912$$ 0 0
$$913$$ 48.3137 1.59895
$$914$$ 0 0
$$915$$ −17.0711 −0.564352
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 5.58579 0.184258 0.0921290 0.995747i $$-0.470633\pi$$
0.0921290 + 0.995747i $$0.470633\pi$$
$$920$$ 0 0
$$921$$ 63.5980 2.09562
$$922$$ 0 0
$$923$$ −26.4853 −0.871774
$$924$$ 0 0
$$925$$ 26.3431 0.866157
$$926$$ 0 0
$$927$$ −20.2843 −0.666223
$$928$$ 0 0
$$929$$ 35.6274 1.16890 0.584449 0.811431i $$-0.301311\pi$$
0.584449 + 0.811431i $$0.301311\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 27.3137 0.894211
$$934$$ 0 0
$$935$$ −22.1421 −0.724125
$$936$$ 0 0
$$937$$ −30.5147 −0.996872 −0.498436 0.866926i $$-0.666092\pi$$
−0.498436 + 0.866926i $$0.666092\pi$$
$$938$$ 0 0
$$939$$ 3.89949 0.127255
$$940$$ 0 0
$$941$$ −27.4142 −0.893678 −0.446839 0.894614i $$-0.647450\pi$$
−0.446839 + 0.894614i $$0.647450\pi$$
$$942$$ 0 0
$$943$$ 24.9706 0.813153
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −26.0000 −0.844886 −0.422443 0.906389i $$-0.638827\pi$$
−0.422443 + 0.906389i $$0.638827\pi$$
$$948$$ 0 0
$$949$$ 23.8995 0.775810
$$950$$ 0 0
$$951$$ 34.9706 1.13400
$$952$$ 0 0
$$953$$ −59.1543 −1.91620 −0.958098 0.286439i $$-0.907528\pi$$
−0.958098 + 0.286439i $$0.907528\pi$$
$$954$$ 0 0
$$955$$ 9.24264 0.299085
$$956$$ 0 0
$$957$$ −104.225 −3.36913
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −9.97056 −0.321631
$$962$$ 0 0
$$963$$ 147.782 4.76220
$$964$$ 0 0
$$965$$ 2.24264 0.0721932
$$966$$ 0 0
$$967$$ 16.9706 0.545737 0.272868 0.962051i $$-0.412028\pi$$
0.272868 + 0.962051i $$0.412028\pi$$
$$968$$ 0 0
$$969$$ −23.3137 −0.748944
$$970$$ 0 0
$$971$$ −48.8701 −1.56831 −0.784157 0.620562i $$-0.786905\pi$$
−0.784157 + 0.620562i $$0.786905\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 46.6274 1.49327
$$976$$ 0 0
$$977$$ 17.3137 0.553915 0.276957 0.960882i $$-0.410674\pi$$
0.276957 + 0.960882i $$0.410674\pi$$
$$978$$ 0 0
$$979$$ 43.9584 1.40492
$$980$$ 0 0
$$981$$ −167.196 −5.33816
$$982$$ 0 0
$$983$$ 49.9411 1.59287 0.796437 0.604721i $$-0.206715\pi$$
0.796437 + 0.604721i $$0.206715\pi$$
$$984$$ 0 0
$$985$$ −0.514719 −0.0164003
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −14.3137 −0.455149
$$990$$ 0 0
$$991$$ −43.4558 −1.38042 −0.690210 0.723609i $$-0.742482\pi$$
−0.690210 + 0.723609i $$0.742482\pi$$
$$992$$ 0 0
$$993$$ 63.1127 2.00282
$$994$$ 0 0
$$995$$ −2.07107 −0.0656573
$$996$$ 0 0
$$997$$ −19.5980 −0.620674 −0.310337 0.950627i $$-0.600442\pi$$
−0.310337 + 0.950627i $$0.600442\pi$$
$$998$$ 0 0
$$999$$ 127.196 4.02430
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 7448.2.a.x.1.1 2
7.3 odd 6 1064.2.q.k.457.1 yes 4
7.5 odd 6 1064.2.q.k.305.1 4
7.6 odd 2 7448.2.a.bd.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.k.305.1 4 7.5 odd 6
1064.2.q.k.457.1 yes 4 7.3 odd 6
7448.2.a.x.1.1 2 1.1 even 1 trivial
7448.2.a.bd.1.2 2 7.6 odd 2