# Properties

 Label 7728.2.a.u.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 966) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +3.00000 q^{13} +3.00000 q^{15} -1.00000 q^{21} +1.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +1.00000 q^{29} +2.00000 q^{31} -4.00000 q^{33} -3.00000 q^{35} -5.00000 q^{37} +3.00000 q^{39} +5.00000 q^{41} +7.00000 q^{43} +3.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} +2.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} +9.00000 q^{65} +12.0000 q^{67} +1.00000 q^{69} -10.0000 q^{71} +4.00000 q^{75} +4.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +1.00000 q^{87} +10.0000 q^{89} -3.00000 q^{91} +2.00000 q^{93} +19.0000 q^{97} -4.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ −12.0000 −1.61808
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 9.00000 1.11631
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 1.00000 0.107211
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ 2.00000 0.207390
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 19.0000 1.92916 0.964579 0.263795i $$-0.0849741\pi$$
0.964579 + 0.263795i $$0.0849741\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ −5.00000 −0.474579
$$112$$ 0 0
$$113$$ 7.00000 0.658505 0.329252 0.944242i $$-0.393203\pi$$
0.329252 + 0.944242i $$0.393203\pi$$
$$114$$ 0 0
$$115$$ 3.00000 0.279751
$$116$$ 0 0
$$117$$ 3.00000 0.277350
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 5.00000 0.450835
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 0 0
$$129$$ 7.00000 0.616316
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.00000 0.258199
$$136$$ 0 0
$$137$$ −17.0000 −1.45241 −0.726204 0.687479i $$-0.758717\pi$$
−0.726204 + 0.687479i $$0.758717\pi$$
$$138$$ 0 0
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 3.00000 0.249136
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −24.0000 −1.91541 −0.957704 0.287754i $$-0.907091\pi$$
−0.957704 + 0.287754i $$0.907091\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 0 0
$$165$$ −12.0000 −0.934199
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ 2.00000 0.150329
$$178$$ 0 0
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ −6.00000 −0.443533
$$184$$ 0 0
$$185$$ −15.0000 −1.10282
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ 0 0
$$195$$ 9.00000 0.644503
$$196$$ 0 0
$$197$$ −21.0000 −1.49619 −0.748094 0.663593i $$-0.769031\pi$$
−0.748094 + 0.663593i $$0.769031\pi$$
$$198$$ 0 0
$$199$$ 23.0000 1.63043 0.815213 0.579161i $$-0.196620\pi$$
0.815213 + 0.579161i $$0.196620\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ −1.00000 −0.0701862
$$204$$ 0 0
$$205$$ 15.0000 1.04765
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 0 0
$$213$$ −10.0000 −0.685189
$$214$$ 0 0
$$215$$ 21.0000 1.43219
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ 5.00000 0.331862 0.165931 0.986137i $$-0.446937\pi$$
0.165931 + 0.986137i $$0.446937\pi$$
$$228$$ 0 0
$$229$$ 16.0000 1.05731 0.528655 0.848837i $$-0.322697\pi$$
0.528655 + 0.848837i $$0.322697\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 9.00000 0.579741 0.289870 0.957066i $$-0.406388\pi$$
0.289870 + 0.957066i $$0.406388\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −19.0000 −1.19927 −0.599635 0.800274i $$-0.704687\pi$$
−0.599635 + 0.800274i $$0.704687\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ 5.00000 0.310685
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 0 0
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ 36.0000 2.21146
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ 0 0
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 0 0
$$273$$ −3.00000 −0.181568
$$274$$ 0 0
$$275$$ −16.0000 −0.964836
$$276$$ 0 0
$$277$$ −24.0000 −1.44202 −0.721010 0.692925i $$-0.756322\pi$$
−0.721010 + 0.692925i $$0.756322\pi$$
$$278$$ 0 0
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 1.00000 0.0596550 0.0298275 0.999555i $$-0.490504\pi$$
0.0298275 + 0.999555i $$0.490504\pi$$
$$282$$ 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
$$287$$ −5.00000 −0.295141
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 19.0000 1.11380
$$292$$ 0 0
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ −7.00000 −0.403473
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ −18.0000 −1.03068
$$306$$ 0 0
$$307$$ −25.0000 −1.42683 −0.713413 0.700744i $$-0.752851\pi$$
−0.713413 + 0.700744i $$0.752851\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 0 0
$$317$$ −33.0000 −1.85346 −0.926732 0.375722i $$-0.877395\pi$$
−0.926732 + 0.375722i $$0.877395\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 12.0000 0.665640
$$326$$ 0 0
$$327$$ −1.00000 −0.0553001
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ −5.00000 −0.273998
$$334$$ 0 0
$$335$$ 36.0000 1.96689
$$336$$ 0 0
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ 0 0
$$339$$ 7.00000 0.380188
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 3.00000 0.161515
$$346$$ 0 0
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ 0 0
$$353$$ −17.0000 −0.904819 −0.452409 0.891810i $$-0.649435\pi$$
−0.452409 + 0.891810i $$0.649435\pi$$
$$354$$ 0 0
$$355$$ −30.0000 −1.59223
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −9.00000 −0.475002 −0.237501 0.971387i $$-0.576328\pi$$
−0.237501 + 0.971387i $$0.576328\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −21.0000 −1.09619 −0.548096 0.836416i $$-0.684647\pi$$
−0.548096 + 0.836416i $$0.684647\pi$$
$$368$$ 0 0
$$369$$ 5.00000 0.260290
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ −7.00000 −0.358621
$$382$$ 0 0
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 0 0
$$385$$ 12.0000 0.611577
$$386$$ 0 0
$$387$$ 7.00000 0.355830
$$388$$ 0 0
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 18.0000 0.907980
$$394$$ 0 0
$$395$$ −12.0000 −0.603786
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 0 0
$$405$$ 3.00000 0.149071
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ −17.0000 −0.838548
$$412$$ 0 0
$$413$$ −2.00000 −0.0984136
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ −7.00000 −0.342791
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −1.00000 −0.0487370 −0.0243685 0.999703i $$-0.507758\pi$$
−0.0243685 + 0.999703i $$0.507758\pi$$
$$422$$ 0 0
$$423$$ 3.00000 0.145865
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6.00000 0.290360
$$428$$ 0 0
$$429$$ −12.0000 −0.579365
$$430$$ 0 0
$$431$$ −3.00000 −0.144505 −0.0722525 0.997386i $$-0.523019\pi$$
−0.0722525 + 0.997386i $$0.523019\pi$$
$$432$$ 0 0
$$433$$ −27.0000 −1.29754 −0.648769 0.760986i $$-0.724716\pi$$
−0.648769 + 0.760986i $$0.724716\pi$$
$$434$$ 0 0
$$435$$ 3.00000 0.143839
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −14.0000 −0.668184 −0.334092 0.942541i $$-0.608430\pi$$
−0.334092 + 0.942541i $$0.608430\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 29.0000 1.37783 0.688916 0.724841i $$-0.258087\pi$$
0.688916 + 0.724841i $$0.258087\pi$$
$$444$$ 0 0
$$445$$ 30.0000 1.42214
$$446$$ 0 0
$$447$$ 12.0000 0.567581
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ 0 0
$$453$$ 17.0000 0.798730
$$454$$ 0 0
$$455$$ −9.00000 −0.421927
$$456$$ 0 0
$$457$$ −4.00000 −0.187112 −0.0935561 0.995614i $$-0.529823\pi$$
−0.0935561 + 0.995614i $$0.529823\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −5.00000 −0.232370 −0.116185 0.993228i $$-0.537067\pi$$
−0.116185 + 0.993228i $$0.537067\pi$$
$$464$$ 0 0
$$465$$ 6.00000 0.278243
$$466$$ 0 0
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ −24.0000 −1.10586
$$472$$ 0 0
$$473$$ −28.0000 −1.28744
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ −15.0000 −0.683941
$$482$$ 0 0
$$483$$ −1.00000 −0.0455016
$$484$$ 0 0
$$485$$ 57.0000 2.58824
$$486$$ 0 0
$$487$$ −13.0000 −0.589086 −0.294543 0.955638i $$-0.595167\pi$$
−0.294543 + 0.955638i $$0.595167\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ 10.0000 0.448561
$$498$$ 0 0
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 2.00000 0.0891756 0.0445878 0.999005i $$-0.485803\pi$$
0.0445878 + 0.999005i $$0.485803\pi$$
$$504$$ 0 0
$$505$$ 42.0000 1.86898
$$506$$ 0 0
$$507$$ −4.00000 −0.177646
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 3.00000 0.132196
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ 34.0000 1.48957 0.744784 0.667306i $$-0.232553\pi$$
0.744784 + 0.667306i $$0.232553\pi$$
$$522$$ 0 0
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ 0 0
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 2.00000 0.0867926
$$532$$ 0 0
$$533$$ 15.0000 0.649722
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ −9.00000 −0.388379
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −24.0000 −1.03184 −0.515920 0.856637i $$-0.672550\pi$$
−0.515920 + 0.856637i $$0.672550\pi$$
$$542$$ 0 0
$$543$$ 18.0000 0.772454
$$544$$ 0 0
$$545$$ −3.00000 −0.128506
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ 0 0
$$555$$ −15.0000 −0.636715
$$556$$ 0 0
$$557$$ 36.0000 1.52537 0.762684 0.646771i $$-0.223881\pi$$
0.762684 + 0.646771i $$0.223881\pi$$
$$558$$ 0 0
$$559$$ 21.0000 0.888205
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −39.0000 −1.64365 −0.821827 0.569737i $$-0.807045\pi$$
−0.821827 + 0.569737i $$0.807045\pi$$
$$564$$ 0 0
$$565$$ 21.0000 0.883477
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −19.0000 −0.796521 −0.398261 0.917272i $$-0.630386\pi$$
−0.398261 + 0.917272i $$0.630386\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 20.0000 0.835512
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 0 0
$$579$$ −1.00000 −0.0415586
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ −48.0000 −1.98796
$$584$$ 0 0
$$585$$ 9.00000 0.372104
$$586$$ 0 0
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −21.0000 −0.863825
$$592$$ 0 0
$$593$$ 9.00000 0.369586 0.184793 0.982777i $$-0.440839\pi$$
0.184793 + 0.982777i $$0.440839\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 23.0000 0.941327
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ 0 0
$$605$$ 15.0000 0.609837
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 0 0
$$609$$ −1.00000 −0.0405220
$$610$$ 0 0
$$611$$ 9.00000 0.364101
$$612$$ 0 0
$$613$$ 21.0000 0.848182 0.424091 0.905620i $$-0.360594\pi$$
0.424091 + 0.905620i $$0.360594\pi$$
$$614$$ 0 0
$$615$$ 15.0000 0.604858
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −21.0000 −0.833360
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 0 0
$$639$$ −10.0000 −0.395594
$$640$$ 0 0
$$641$$ 1.00000 0.0394976 0.0197488 0.999805i $$-0.493713\pi$$
0.0197488 + 0.999805i $$0.493713\pi$$
$$642$$ 0 0
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 0 0
$$645$$ 21.0000 0.826874
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ 0 0
$$653$$ 13.0000 0.508729 0.254365 0.967108i $$-0.418134\pi$$
0.254365 + 0.967108i $$0.418134\pi$$
$$654$$ 0 0
$$655$$ 54.0000 2.10995
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 8.00000 0.311636 0.155818 0.987786i $$-0.450199\pi$$
0.155818 + 0.987786i $$0.450199\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.00000 0.0387202
$$668$$ 0 0
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ −19.0000 −0.729153
$$680$$ 0 0
$$681$$ 5.00000 0.191600
$$682$$ 0 0
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ −51.0000 −1.94861
$$686$$ 0 0
$$687$$ 16.0000 0.610438
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −21.0000 −0.798878 −0.399439 0.916760i $$-0.630795\pi$$
−0.399439 + 0.916760i $$0.630795\pi$$
$$692$$ 0 0
$$693$$ 4.00000 0.151947
$$694$$ 0 0
$$695$$ −21.0000 −0.796575
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ −48.0000 −1.81293 −0.906467 0.422276i $$-0.861231\pi$$
−0.906467 + 0.422276i $$0.861231\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 9.00000 0.338960
$$706$$ 0 0
$$707$$ −14.0000 −0.526524
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 2.00000 0.0749006
$$714$$ 0 0
$$715$$ −36.0000 −1.34632
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ 43.0000 1.60363 0.801815 0.597573i $$-0.203868\pi$$
0.801815 + 0.597573i $$0.203868\pi$$
$$720$$ 0 0
$$721$$ −1.00000 −0.0372419
$$722$$ 0 0
$$723$$ 9.00000 0.334714
$$724$$ 0 0
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 36.0000 1.32969 0.664845 0.746981i $$-0.268498\pi$$
0.664845 + 0.746981i $$0.268498\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 36.0000 1.31894
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ −19.0000 −0.692398
$$754$$ 0 0
$$755$$ 51.0000 1.85608
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 0 0
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 1.00000 0.0362024
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ 0 0
$$773$$ 35.0000 1.25886 0.629431 0.777056i $$-0.283288\pi$$
0.629431 + 0.777056i $$0.283288\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ 0 0
$$777$$ 5.00000 0.179374
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 40.0000 1.43131
$$782$$ 0 0
$$783$$ 1.00000 0.0357371
$$784$$ 0 0
$$785$$ −72.0000 −2.56979
$$786$$ 0 0
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 0 0
$$789$$ 21.0000 0.747620
$$790$$ 0 0
$$791$$ −7.00000 −0.248891
$$792$$ 0 0
$$793$$ −18.0000 −0.639199
$$794$$ 0 0
$$795$$ 36.0000 1.27679
$$796$$ 0 0
$$797$$ 47.0000 1.66483 0.832413 0.554156i $$-0.186959\pi$$
0.832413 + 0.554156i $$0.186959\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ 0 0
$$807$$ −2.00000 −0.0704033
$$808$$ 0 0
$$809$$ 48.0000 1.68759 0.843795 0.536666i $$-0.180316\pi$$
0.843795 + 0.536666i $$0.180316\pi$$
$$810$$ 0 0
$$811$$ −23.0000 −0.807639 −0.403820 0.914839i $$-0.632318\pi$$
−0.403820 + 0.914839i $$0.632318\pi$$
$$812$$ 0 0
$$813$$ 12.0000 0.420858
$$814$$ 0 0
$$815$$ 36.0000 1.26102
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −3.00000 −0.104828
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 0 0
$$823$$ −31.0000 −1.08059 −0.540296 0.841475i $$-0.681688\pi$$
−0.540296 + 0.841475i $$0.681688\pi$$
$$824$$ 0 0
$$825$$ −16.0000 −0.557048
$$826$$ 0 0
$$827$$ 42.0000 1.46048 0.730242 0.683189i $$-0.239408\pi$$
0.730242 + 0.683189i $$0.239408\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ −24.0000 −0.832551
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 36.0000 1.24583
$$836$$ 0 0
$$837$$ 2.00000 0.0691301
$$838$$ 0 0
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 1.00000 0.0344418
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ −5.00000 −0.171398
$$852$$ 0 0
$$853$$ 41.0000 1.40381 0.701907 0.712269i $$-0.252332\pi$$
0.701907 + 0.712269i $$0.252332\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 0 0
$$859$$ 29.0000 0.989467 0.494734 0.869045i $$-0.335266\pi$$
0.494734 + 0.869045i $$0.335266\pi$$
$$860$$ 0 0
$$861$$ −5.00000 −0.170400
$$862$$ 0 0
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ −17.0000 −0.577350
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 36.0000 1.21981
$$872$$ 0 0
$$873$$ 19.0000 0.643053
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ −20.0000 −0.675352 −0.337676 0.941262i $$-0.609641\pi$$
−0.337676 + 0.941262i $$0.609641\pi$$
$$878$$ 0 0
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 38.0000 1.28025 0.640126 0.768270i $$-0.278882\pi$$
0.640126 + 0.768270i $$0.278882\pi$$
$$882$$ 0 0
$$883$$ −6.00000 −0.201916 −0.100958 0.994891i $$-0.532191\pi$$
−0.100958 + 0.994891i $$0.532191\pi$$
$$884$$ 0 0
$$885$$ 6.00000 0.201688
$$886$$ 0 0
$$887$$ 16.0000 0.537227 0.268614 0.963248i $$-0.413434\pi$$
0.268614 + 0.963248i $$0.413434\pi$$
$$888$$ 0 0
$$889$$ 7.00000 0.234772
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −27.0000 −0.902510
$$896$$ 0 0
$$897$$ 3.00000 0.100167
$$898$$ 0 0
$$899$$ 2.00000 0.0667037
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −7.00000 −0.232945
$$904$$ 0 0
$$905$$ 54.0000 1.79502
$$906$$ 0 0
$$907$$ 49.0000 1.62702 0.813509 0.581552i $$-0.197554\pi$$
0.813509 + 0.581552i $$0.197554\pi$$
$$908$$ 0 0
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ 55.0000 1.82223 0.911116 0.412151i $$-0.135222\pi$$
0.911116 + 0.412151i $$0.135222\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 0 0
$$915$$ −18.0000 −0.595062
$$916$$ 0 0
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ −60.0000 −1.97922 −0.989609 0.143787i $$-0.954072\pi$$
−0.989609 + 0.143787i $$0.954072\pi$$
$$920$$ 0 0
$$921$$ −25.0000 −0.823778
$$922$$ 0 0
$$923$$ −30.0000 −0.987462
$$924$$ 0 0
$$925$$ −20.0000 −0.657596
$$926$$ 0 0
$$927$$ 1.00000 0.0328443
$$928$$ 0 0
$$929$$ 21.0000 0.688988 0.344494 0.938789i $$-0.388051\pi$$
0.344494 + 0.938789i $$0.388051\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 4.00000 0.130954
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −53.0000 −1.73143 −0.865717 0.500533i $$-0.833137\pi$$
−0.865717 + 0.500533i $$0.833137\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ −21.0000 −0.684580 −0.342290 0.939594i $$-0.611203\pi$$
−0.342290 + 0.939594i $$0.611203\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 0 0
$$947$$ −3.00000 −0.0974869 −0.0487435 0.998811i $$-0.515522\pi$$
−0.0487435 + 0.998811i $$0.515522\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −33.0000 −1.07010
$$952$$ 0 0
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ 0 0
$$955$$ 60.0000 1.94155
$$956$$ 0 0
$$957$$ −4.00000 −0.129302
$$958$$ 0 0
$$959$$ 17.0000 0.548959
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ 0 0
$$965$$ −3.00000 −0.0965734
$$966$$ 0 0
$$967$$ −52.0000 −1.67221 −0.836104 0.548572i $$-0.815172\pi$$
−0.836104 + 0.548572i $$0.815172\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 40.0000 1.28366 0.641831 0.766846i $$-0.278175\pi$$
0.641831 + 0.766846i $$0.278175\pi$$
$$972$$ 0 0
$$973$$ 7.00000 0.224410
$$974$$ 0 0
$$975$$ 12.0000 0.384308
$$976$$ 0 0
$$977$$ −13.0000 −0.415907 −0.207953 0.978139i $$-0.566680\pi$$
−0.207953 + 0.978139i $$0.566680\pi$$
$$978$$ 0 0
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ −1.00000 −0.0319275
$$982$$ 0 0
$$983$$ −2.00000 −0.0637901 −0.0318950 0.999491i $$-0.510154\pi$$
−0.0318950 + 0.999491i $$0.510154\pi$$
$$984$$ 0 0
$$985$$ −63.0000 −2.00735
$$986$$ 0 0
$$987$$ −3.00000 −0.0954911
$$988$$ 0 0
$$989$$ 7.00000 0.222587
$$990$$ 0 0
$$991$$ −44.0000 −1.39771 −0.698853 0.715265i $$-0.746306\pi$$
−0.698853 + 0.715265i $$0.746306\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 69.0000 2.18745
$$996$$ 0 0
$$997$$ −50.0000 −1.58352 −0.791758 0.610835i $$-0.790834\pi$$
−0.791758 + 0.610835i $$0.790834\pi$$
$$998$$ 0 0
$$999$$ −5.00000 −0.158193
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.u.1.1 1
4.3 odd 2 966.2.a.d.1.1 1
12.11 even 2 2898.2.a.k.1.1 1
28.27 even 2 6762.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.d.1.1 1 4.3 odd 2
2898.2.a.k.1.1 1 12.11 even 2
6762.2.a.l.1.1 1 28.27 even 2
7728.2.a.u.1.1 1 1.1 even 1 trivial