# Properties

 Label 7728.2.a.l.1.1 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$61.7083906820$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -2.00000 q^{13} -4.00000 q^{15} +5.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.00000 q^{31} -5.00000 q^{33} +4.00000 q^{35} +6.00000 q^{37} +2.00000 q^{39} +5.00000 q^{41} -8.00000 q^{43} +4.00000 q^{45} +9.00000 q^{47} +1.00000 q^{49} +9.00000 q^{53} +20.0000 q^{55} -5.00000 q^{57} -9.00000 q^{59} -5.00000 q^{61} +1.00000 q^{63} -8.00000 q^{65} -4.00000 q^{67} -1.00000 q^{69} -12.0000 q^{71} -11.0000 q^{75} +5.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +18.0000 q^{83} +2.00000 q^{87} +10.0000 q^{89} -2.00000 q^{91} +6.00000 q^{93} +20.0000 q^{95} -18.0000 q^{97} +5.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −4.00000 −1.03280
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 0 0
$$33$$ −5.00000 −0.870388
$$34$$ 0 0
$$35$$ 4.00000 0.676123
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 4.00000 0.596285
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 20.0000 2.69680
$$56$$ 0 0
$$57$$ −5.00000 −0.662266
$$58$$ 0 0
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 0 0
$$61$$ −5.00000 −0.640184 −0.320092 0.947386i $$-0.603714\pi$$
−0.320092 + 0.947386i $$0.603714\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ −11.0000 −1.27017
$$76$$ 0 0
$$77$$ 5.00000 0.569803
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 18.0000 1.97576 0.987878 0.155230i $$-0.0496119\pi$$
0.987878 + 0.155230i $$0.0496119\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 20.0000 2.05196
$$96$$ 0 0
$$97$$ −18.0000 −1.82762 −0.913812 0.406138i $$-0.866875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 0 0
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 5.00000 0.497519 0.248759 0.968565i $$-0.419977\pi$$
0.248759 + 0.968565i $$0.419977\pi$$
$$102$$ 0 0
$$103$$ 19.0000 1.87213 0.936063 0.351833i $$-0.114441\pi$$
0.936063 + 0.351833i $$0.114441\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −5.00000 −0.450835
$$124$$ 0 0
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ −9.00000 −0.798621 −0.399310 0.916816i $$-0.630750\pi$$
−0.399310 + 0.916816i $$0.630750\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 5.00000 0.436852 0.218426 0.975854i $$-0.429908\pi$$
0.218426 + 0.975854i $$0.429908\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 0 0
$$143$$ −10.0000 −0.836242
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ −13.0000 −1.01824 −0.509119 0.860696i $$-0.670029\pi$$
−0.509119 + 0.860696i $$0.670029\pi$$
$$164$$ 0 0
$$165$$ −20.0000 −1.55700
$$166$$ 0 0
$$167$$ −19.0000 −1.47026 −0.735132 0.677924i $$-0.762880\pi$$
−0.735132 + 0.677924i $$0.762880\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 5.00000 0.382360
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 11.0000 0.831522
$$176$$ 0 0
$$177$$ 9.00000 0.676481
$$178$$ 0 0
$$179$$ −8.00000 −0.597948 −0.298974 0.954261i $$-0.596644\pi$$
−0.298974 + 0.954261i $$0.596644\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 5.00000 0.369611
$$184$$ 0 0
$$185$$ 24.0000 1.76452
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 23.0000 1.66422 0.832111 0.554609i $$-0.187132\pi$$
0.832111 + 0.554609i $$0.187132\pi$$
$$192$$ 0 0
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 0 0
$$195$$ 8.00000 0.572892
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −3.00000 −0.212664 −0.106332 0.994331i $$-0.533911\pi$$
−0.106332 + 0.994331i $$0.533911\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 25.0000 1.72929
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ 12.0000 0.822226
$$214$$ 0 0
$$215$$ −32.0000 −2.18238
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ 11.0000 0.733333
$$226$$ 0 0
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ 0 0
$$231$$ −5.00000 −0.328976
$$232$$ 0 0
$$233$$ −12.0000 −0.786146 −0.393073 0.919507i $$-0.628588\pi$$
−0.393073 + 0.919507i $$0.628588\pi$$
$$234$$ 0 0
$$235$$ 36.0000 2.34838
$$236$$ 0 0
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 4.00000 0.255551
$$246$$ 0 0
$$247$$ −10.0000 −0.636285
$$248$$ 0 0
$$249$$ −18.0000 −1.14070
$$250$$ 0 0
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 0 0
$$253$$ 5.00000 0.314347
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 17.0000 1.06043 0.530215 0.847863i $$-0.322111\pi$$
0.530215 + 0.847863i $$0.322111\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −7.00000 −0.431638 −0.215819 0.976433i $$-0.569242\pi$$
−0.215819 + 0.976433i $$0.569242\pi$$
$$264$$ 0 0
$$265$$ 36.0000 2.21146
$$266$$ 0 0
$$267$$ −10.0000 −0.611990
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ 0 0
$$275$$ 55.0000 3.31662
$$276$$ 0 0
$$277$$ 7.00000 0.420589 0.210295 0.977638i $$-0.432558\pi$$
0.210295 + 0.977638i $$0.432558\pi$$
$$278$$ 0 0
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 0 0
$$285$$ −20.0000 −1.18470
$$286$$ 0 0
$$287$$ 5.00000 0.295141
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ 0 0
$$293$$ −4.00000 −0.233682 −0.116841 0.993151i $$-0.537277\pi$$
−0.116841 + 0.993151i $$0.537277\pi$$
$$294$$ 0 0
$$295$$ −36.0000 −2.09600
$$296$$ 0 0
$$297$$ −5.00000 −0.290129
$$298$$ 0 0
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ −5.00000 −0.287242
$$304$$ 0 0
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ −19.0000 −1.08087
$$310$$ 0 0
$$311$$ 15.0000 0.850572 0.425286 0.905059i $$-0.360174\pi$$
0.425286 + 0.905059i $$0.360174\pi$$
$$312$$ 0 0
$$313$$ −25.0000 −1.41308 −0.706542 0.707671i $$-0.749746\pi$$
−0.706542 + 0.707671i $$0.749746\pi$$
$$314$$ 0 0
$$315$$ 4.00000 0.225374
$$316$$ 0 0
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ −10.0000 −0.559893
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −22.0000 −1.22034
$$326$$ 0 0
$$327$$ 12.0000 0.663602
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ −29.0000 −1.59398 −0.796992 0.603990i $$-0.793577\pi$$
−0.796992 + 0.603990i $$0.793577\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ −16.0000 −0.874173
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −30.0000 −1.62459
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ −2.00000 −0.107366 −0.0536828 0.998558i $$-0.517096\pi$$
−0.0536828 + 0.998558i $$0.517096\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ −48.0000 −2.54758
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ 0 0
$$369$$ 5.00000 0.260290
$$370$$ 0 0
$$371$$ 9.00000 0.467257
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ −24.0000 −1.23935
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 30.0000 1.54100 0.770498 0.637442i $$-0.220007\pi$$
0.770498 + 0.637442i $$0.220007\pi$$
$$380$$ 0 0
$$381$$ 9.00000 0.461084
$$382$$ 0 0
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 20.0000 1.01929
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −5.00000 −0.252217
$$394$$ 0 0
$$395$$ 40.0000 2.01262
$$396$$ 0 0
$$397$$ 30.0000 1.50566 0.752828 0.658217i $$-0.228689\pi$$
0.752828 + 0.658217i $$0.228689\pi$$
$$398$$ 0 0
$$399$$ −5.00000 −0.250313
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 0 0
$$403$$ 12.0000 0.597763
$$404$$ 0 0
$$405$$ 4.00000 0.198762
$$406$$ 0 0
$$407$$ 30.0000 1.48704
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ 0 0
$$413$$ −9.00000 −0.442861
$$414$$ 0 0
$$415$$ 72.0000 3.53434
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ 18.0000 0.879358 0.439679 0.898155i $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 0 0
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5.00000 −0.241967
$$428$$ 0 0
$$429$$ 10.0000 0.482805
$$430$$ 0 0
$$431$$ −27.0000 −1.30054 −0.650272 0.759701i $$-0.725345\pi$$
−0.650272 + 0.759701i $$0.725345\pi$$
$$432$$ 0 0
$$433$$ −11.0000 −0.528626 −0.264313 0.964437i $$-0.585145\pi$$
−0.264313 + 0.964437i $$0.585145\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ 0 0
$$437$$ 5.00000 0.239182
$$438$$ 0 0
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ 40.0000 1.89618
$$446$$ 0 0
$$447$$ −3.00000 −0.141895
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 25.0000 1.17720
$$452$$ 0 0
$$453$$ −19.0000 −0.892698
$$454$$ 0 0
$$455$$ −8.00000 −0.375046
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 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
$$463$$ 35.0000 1.62659 0.813294 0.581853i $$-0.197672\pi$$
0.813294 + 0.581853i $$0.197672\pi$$
$$464$$ 0 0
$$465$$ 24.0000 1.11297
$$466$$ 0 0
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −7.00000 −0.322543
$$472$$ 0 0
$$473$$ −40.0000 −1.83920
$$474$$ 0 0
$$475$$ 55.0000 2.52357
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ 0 0
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ −1.00000 −0.0455016
$$484$$ 0 0
$$485$$ −72.0000 −3.26935
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 0 0
$$489$$ 13.0000 0.587880
$$490$$ 0 0
$$491$$ −14.0000 −0.631811 −0.315906 0.948791i $$-0.602308\pi$$
−0.315906 + 0.948791i $$0.602308\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 20.0000 0.898933
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 19.0000 0.848857
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 0 0
$$509$$ −19.0000 −0.842160 −0.421080 0.907023i $$-0.638349\pi$$
−0.421080 + 0.907023i $$0.638349\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ 76.0000 3.34896
$$516$$ 0 0
$$517$$ 45.0000 1.97910
$$518$$ 0 0
$$519$$ −2.00000 −0.0877903
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ −19.0000 −0.830812 −0.415406 0.909636i $$-0.636360\pi$$
−0.415406 + 0.909636i $$0.636360\pi$$
$$524$$ 0 0
$$525$$ −11.0000 −0.480079
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −9.00000 −0.390567
$$532$$ 0 0
$$533$$ −10.0000 −0.433148
$$534$$ 0 0
$$535$$ −16.0000 −0.691740
$$536$$ 0 0
$$537$$ 8.00000 0.345225
$$538$$ 0 0
$$539$$ 5.00000 0.215365
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 0 0
$$543$$ 14.0000 0.600798
$$544$$ 0 0
$$545$$ −48.0000 −2.05609
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ −5.00000 −0.213395
$$550$$ 0 0
$$551$$ −10.0000 −0.426014
$$552$$ 0 0
$$553$$ 10.0000 0.425243
$$554$$ 0 0
$$555$$ −24.0000 −1.01874
$$556$$ 0 0
$$557$$ 46.0000 1.94908 0.974541 0.224208i $$-0.0719796\pi$$
0.974541 + 0.224208i $$0.0719796\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −40.0000 −1.68580 −0.842900 0.538071i $$-0.819153\pi$$
−0.842900 + 0.538071i $$0.819153\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −17.0000 −0.712677 −0.356339 0.934357i $$-0.615975\pi$$
−0.356339 + 0.934357i $$0.615975\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 0 0
$$573$$ −23.0000 −0.960839
$$574$$ 0 0
$$575$$ 11.0000 0.458732
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 19.0000 0.789613
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ 45.0000 1.86371
$$584$$ 0 0
$$585$$ −8.00000 −0.330759
$$586$$ 0 0
$$587$$ 17.0000 0.701665 0.350833 0.936438i $$-0.385899\pi$$
0.350833 + 0.936438i $$0.385899\pi$$
$$588$$ 0 0
$$589$$ −30.0000 −1.23613
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.00000 0.122782
$$598$$ 0 0
$$599$$ −42.0000 −1.71607 −0.858037 0.513588i $$-0.828316\pi$$
−0.858037 + 0.513588i $$0.828316\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 56.0000 2.27672
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 0 0
$$615$$ −20.0000 −0.806478
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ 10.0000 0.400642
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ −25.0000 −0.998404
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 10.0000 0.398094 0.199047 0.979990i $$-0.436215\pi$$
0.199047 + 0.979990i $$0.436215\pi$$
$$632$$ 0 0
$$633$$ 13.0000 0.516704
$$634$$ 0 0
$$635$$ −36.0000 −1.42862
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ 31.0000 1.22252 0.611260 0.791430i $$-0.290663\pi$$
0.611260 + 0.791430i $$0.290663\pi$$
$$644$$ 0 0
$$645$$ 32.0000 1.26000
$$646$$ 0 0
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ −45.0000 −1.76640
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ 0 0
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ 0 0
$$655$$ 20.0000 0.781465
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −5.00000 −0.194477 −0.0972387 0.995261i $$-0.531001\pi$$
−0.0972387 + 0.995261i $$0.531001\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 20.0000 0.775567
$$666$$ 0 0
$$667$$ −2.00000 −0.0774403
$$668$$ 0 0
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ −25.0000 −0.965114
$$672$$ 0 0
$$673$$ 49.0000 1.88881 0.944406 0.328783i $$-0.106638\pi$$
0.944406 + 0.328783i $$0.106638\pi$$
$$674$$ 0 0
$$675$$ −11.0000 −0.423390
$$676$$ 0 0
$$677$$ 36.0000 1.38359 0.691796 0.722093i $$-0.256820\pi$$
0.691796 + 0.722093i $$0.256820\pi$$
$$678$$ 0 0
$$679$$ −18.0000 −0.690777
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 36.0000 1.37549
$$686$$ 0 0
$$687$$ 13.0000 0.495981
$$688$$ 0 0
$$689$$ −18.0000 −0.685745
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 0 0
$$693$$ 5.00000 0.189934
$$694$$ 0 0
$$695$$ −48.0000 −1.82074
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ −15.0000 −0.566542 −0.283271 0.959040i $$-0.591420\pi$$
−0.283271 + 0.959040i $$0.591420\pi$$
$$702$$ 0 0
$$703$$ 30.0000 1.13147
$$704$$ 0 0
$$705$$ −36.0000 −1.35584
$$706$$ 0 0
$$707$$ 5.00000 0.188044
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 0 0
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ −40.0000 −1.49592
$$716$$ 0 0
$$717$$ 12.0000 0.448148
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 19.0000 0.707597
$$722$$ 0 0
$$723$$ −17.0000 −0.632237
$$724$$ 0 0
$$725$$ −22.0000 −0.817059
$$726$$ 0 0
$$727$$ −41.0000 −1.52061 −0.760303 0.649569i $$-0.774949\pi$$
−0.760303 + 0.649569i $$0.774949\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ −4.00000 −0.147542
$$736$$ 0 0
$$737$$ −20.0000 −0.736709
$$738$$ 0 0
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ 10.0000 0.367359
$$742$$ 0 0
$$743$$ −15.0000 −0.550297 −0.275148 0.961402i $$-0.588727\pi$$
−0.275148 + 0.961402i $$0.588727\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 0 0
$$747$$ 18.0000 0.658586
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ 10.0000 0.364420
$$754$$ 0 0
$$755$$ 76.0000 2.76592
$$756$$ 0 0
$$757$$ 8.00000 0.290765 0.145382 0.989376i $$-0.453559\pi$$
0.145382 + 0.989376i $$0.453559\pi$$
$$758$$ 0 0
$$759$$ −5.00000 −0.181489
$$760$$ 0 0
$$761$$ −21.0000 −0.761249 −0.380625 0.924730i $$-0.624291\pi$$
−0.380625 + 0.924730i $$0.624291\pi$$
$$762$$ 0 0
$$763$$ −12.0000 −0.434429
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.0000 0.649942
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −17.0000 −0.612240
$$772$$ 0 0
$$773$$ 20.0000 0.719350 0.359675 0.933078i $$-0.382888\pi$$
0.359675 + 0.933078i $$0.382888\pi$$
$$774$$ 0 0
$$775$$ −66.0000 −2.37079
$$776$$ 0 0
$$777$$ −6.00000 −0.215249
$$778$$ 0 0
$$779$$ 25.0000 0.895718
$$780$$ 0 0
$$781$$ −60.0000 −2.14697
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 28.0000 0.999363
$$786$$ 0 0
$$787$$ 23.0000 0.819861 0.409931 0.912117i $$-0.365553\pi$$
0.409931 + 0.912117i $$0.365553\pi$$
$$788$$ 0 0
$$789$$ 7.00000 0.249207
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 0 0
$$795$$ −36.0000 −1.27679
$$796$$ 0 0
$$797$$ 46.0000 1.62940 0.814702 0.579880i $$-0.196901\pi$$
0.814702 + 0.579880i $$0.196901\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 4.00000 0.140981
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ −52.0000 −1.82148
$$816$$ 0 0
$$817$$ −40.0000 −1.39942
$$818$$ 0 0
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ −4.00000 −0.139601 −0.0698005 0.997561i $$-0.522236\pi$$
−0.0698005 + 0.997561i $$0.522236\pi$$
$$822$$ 0 0
$$823$$ −27.0000 −0.941161 −0.470580 0.882357i $$-0.655955\pi$$
−0.470580 + 0.882357i $$0.655955\pi$$
$$824$$ 0 0
$$825$$ −55.0000 −1.91485
$$826$$ 0 0
$$827$$ 21.0000 0.730242 0.365121 0.930960i $$-0.381028\pi$$
0.365121 + 0.930960i $$0.381028\pi$$
$$828$$ 0 0
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ −7.00000 −0.242827
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −76.0000 −2.63009
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 0 0
$$839$$ −46.0000 −1.58810 −0.794048 0.607855i $$-0.792030\pi$$
−0.794048 + 0.607855i $$0.792030\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −2.00000 −0.0688837
$$844$$ 0 0
$$845$$ −36.0000 −1.23844
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ 0 0
$$853$$ 8.00000 0.273915 0.136957 0.990577i $$-0.456268\pi$$
0.136957 + 0.990577i $$0.456268\pi$$
$$854$$ 0 0
$$855$$ 20.0000 0.683986
$$856$$ 0 0
$$857$$ 1.00000 0.0341593 0.0170797 0.999854i $$-0.494563\pi$$
0.0170797 + 0.999854i $$0.494563\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ −5.00000 −0.170400
$$862$$ 0 0
$$863$$ 30.0000 1.02121 0.510606 0.859815i $$-0.329421\pi$$
0.510606 + 0.859815i $$0.329421\pi$$
$$864$$ 0 0
$$865$$ 8.00000 0.272008
$$866$$ 0 0
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ 50.0000 1.69613
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ −18.0000 −0.609208
$$874$$ 0 0
$$875$$ 24.0000 0.811348
$$876$$ 0 0
$$877$$ −17.0000 −0.574049 −0.287025 0.957923i $$-0.592666\pi$$
−0.287025 + 0.957923i $$0.592666\pi$$
$$878$$ 0 0
$$879$$ 4.00000 0.134917
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ −32.0000 −1.07689 −0.538443 0.842662i $$-0.680987\pi$$
−0.538443 + 0.842662i $$0.680987\pi$$
$$884$$ 0 0
$$885$$ 36.0000 1.21013
$$886$$ 0 0
$$887$$ 16.0000 0.537227 0.268614 0.963248i $$-0.413434\pi$$
0.268614 + 0.963248i $$0.413434\pi$$
$$888$$ 0 0
$$889$$ −9.00000 −0.301850
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ 0 0
$$893$$ 45.0000 1.50587
$$894$$ 0 0
$$895$$ −32.0000 −1.06964
$$896$$ 0 0
$$897$$ 2.00000 0.0667781
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ −56.0000 −1.86150
$$906$$ 0 0
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ 0 0
$$909$$ 5.00000 0.165840
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 90.0000 2.97857
$$914$$ 0 0
$$915$$ 20.0000 0.661180
$$916$$ 0 0
$$917$$ 5.00000 0.165115
$$918$$ 0 0
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ 0 0
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 66.0000 2.17007
$$926$$ 0 0
$$927$$ 19.0000 0.624042
$$928$$ 0 0
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ 5.00000 0.163868
$$932$$ 0 0
$$933$$ −15.0000 −0.491078
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −27.0000 −0.882052 −0.441026 0.897494i $$-0.645385\pi$$
−0.441026 + 0.897494i $$0.645385\pi$$
$$938$$ 0 0
$$939$$ 25.0000 0.815844
$$940$$ 0 0
$$941$$ 20.0000 0.651981 0.325991 0.945373i $$-0.394302\pi$$
0.325991 + 0.945373i $$0.394302\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ −32.0000 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ −55.0000 −1.78162 −0.890812 0.454371i $$-0.849864\pi$$
−0.890812 + 0.454371i $$0.849864\pi$$
$$954$$ 0 0
$$955$$ 92.0000 2.97705
$$956$$ 0 0
$$957$$ 10.0000 0.323254
$$958$$ 0 0
$$959$$ 9.00000 0.290625
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ 0 0
$$965$$ −76.0000 −2.44653
$$966$$ 0 0
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 0 0
$$973$$ −12.0000 −0.384702
$$974$$ 0 0
$$975$$ 22.0000 0.704564
$$976$$ 0 0
$$977$$ −3.00000 −0.0959785 −0.0479893 0.998848i $$-0.515281\pi$$
−0.0479893 + 0.998848i $$0.515281\pi$$
$$978$$ 0 0
$$979$$ 50.0000 1.59801
$$980$$ 0 0
$$981$$ −12.0000 −0.383131
$$982$$ 0 0
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ 0 0
$$985$$ −8.00000 −0.254901
$$986$$ 0 0
$$987$$ −9.00000 −0.286473
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −5.00000 −0.158830 −0.0794151 0.996842i $$-0.525305\pi$$
−0.0794151 + 0.996842i $$0.525305\pi$$
$$992$$ 0 0
$$993$$ 29.0000 0.920287
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ 0 0
$$997$$ −56.0000 −1.77354 −0.886769 0.462213i $$-0.847056\pi$$
−0.886769 + 0.462213i $$0.847056\pi$$
$$998$$ 0 0
$$999$$ −6.00000 −0.189832
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 7728.2.a.l.1.1 1
4.3 odd 2 483.2.a.b.1.1 1
12.11 even 2 1449.2.a.a.1.1 1
28.27 even 2 3381.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.b.1.1 1 4.3 odd 2
1449.2.a.a.1.1 1 12.11 even 2
3381.2.a.l.1.1 1 28.27 even 2
7728.2.a.l.1.1 1 1.1 even 1 trivial