# Properties

 Label 7728.2.a.j.1.1 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -3.00000 q^{13} -3.00000 q^{15} -4.00000 q^{17} -1.00000 q^{21} -1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} +3.00000 q^{29} +6.00000 q^{31} +4.00000 q^{33} +3.00000 q^{35} -9.00000 q^{37} +3.00000 q^{39} +9.00000 q^{41} +3.00000 q^{43} +3.00000 q^{45} +7.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} -4.00000 q^{53} -12.0000 q^{55} -6.00000 q^{59} +10.0000 q^{61} +1.00000 q^{63} -9.00000 q^{65} -4.00000 q^{67} +1.00000 q^{69} +6.00000 q^{71} -8.00000 q^{73} -4.00000 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -12.0000 q^{85} -3.00000 q^{87} -14.0000 q^{89} -3.00000 q^{91} -6.00000 q^{93} -7.00000 q^{97} -4.00000 q^{99} +O(q^{100})$$

## 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 $$-0.636577\pi$$
−0.416025 + 0.909353i $$0.636577\pi$$
$$14$$ 0 0
$$15$$ −3.00000 −0.774597
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 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
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ −9.00000 −1.47959 −0.739795 0.672832i $$-0.765078\pi$$
−0.739795 + 0.672832i $$0.765078\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ 3.00000 0.457496 0.228748 0.973486i $$-0.426537\pi$$
0.228748 + 0.973486i $$0.426537\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ −12.0000 −1.61808
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −9.00000 −1.11631
$$66$$ 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
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 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
$$85$$ −12.0000 −1.30158
$$86$$ 0 0
$$87$$ −3.00000 −0.321634
$$88$$ 0 0
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ −6.00000 −0.622171
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ −5.00000 −0.492665 −0.246332 0.969185i $$-0.579225\pi$$
−0.246332 + 0.969185i $$0.579225\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ 3.00000 0.287348 0.143674 0.989625i $$-0.454108\pi$$
0.143674 + 0.989625i $$0.454108\pi$$
$$110$$ 0 0
$$111$$ 9.00000 0.854242
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 0 0
$$117$$ −3.00000 −0.277350
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −9.00000 −0.811503
$$124$$ 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
$$129$$ −3.00000 −0.264135
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −3.00000 −0.258199
$$136$$ 0 0
$$137$$ −15.0000 −1.28154 −0.640768 0.767734i $$-0.721384\pi$$
−0.640768 + 0.767734i $$0.721384\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ −7.00000 −0.589506
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 16.0000 1.31077 0.655386 0.755295i $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ 0 0
$$151$$ −15.0000 −1.22068 −0.610341 0.792139i $$-0.708968\pi$$
−0.610341 + 0.792139i $$0.708968\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 18.0000 1.44579
$$156$$ 0 0
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 12.0000 0.934199
$$166$$ 0 0
$$167$$ 20.0000 1.54765 0.773823 0.633402i $$-0.218342\pi$$
0.773823 + 0.633402i $$0.218342\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 6.00000 0.450988
$$178$$ 0 0
$$179$$ −19.0000 −1.42013 −0.710063 0.704138i $$-0.751334\pi$$
−0.710063 + 0.704138i $$0.751334\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ −27.0000 −1.98508
$$186$$ 0 0
$$187$$ 16.0000 1.17004
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −17.0000 −1.22369 −0.611843 0.790979i $$-0.709572\pi$$
−0.611843 + 0.790979i $$0.709572\pi$$
$$194$$ 0 0
$$195$$ 9.00000 0.644503
$$196$$ 0 0
$$197$$ −23.0000 −1.63868 −0.819341 0.573306i $$-0.805660\pi$$
−0.819341 + 0.573306i $$0.805660\pi$$
$$198$$ 0 0
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ 27.0000 1.88576
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ −6.00000 −0.411113
$$214$$ 0 0
$$215$$ 9.00000 0.613795
$$216$$ 0 0
$$217$$ 6.00000 0.407307
$$218$$ 0 0
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ −11.0000 −0.730096 −0.365048 0.930989i $$-0.618947\pi$$
−0.365048 + 0.930989i $$0.618947\pi$$
$$228$$ 0 0
$$229$$ 12.0000 0.792982 0.396491 0.918039i $$-0.370228\pi$$
0.396491 + 0.918039i $$0.370228\pi$$
$$230$$ 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
$$235$$ 21.0000 1.36989
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −5.00000 −0.322078 −0.161039 0.986948i $$-0.551485\pi$$
−0.161039 + 0.986948i $$0.551485\pi$$
$$242$$ 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
$$255$$ 12.0000 0.751469
$$256$$ 0 0
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ 0 0
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 0 0
$$263$$ −21.0000 −1.29492 −0.647458 0.762101i $$-0.724168\pi$$
−0.647458 + 0.762101i $$0.724168\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 14.0000 0.856786
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 3.00000 0.181568
$$274$$ 0 0
$$275$$ −16.0000 −0.964836
$$276$$ 0 0
$$277$$ 20.0000 1.20168 0.600842 0.799368i $$-0.294832\pi$$
0.600842 + 0.799368i $$0.294832\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 23.0000 1.37206 0.686032 0.727571i $$-0.259351\pi$$
0.686032 + 0.727571i $$0.259351\pi$$
$$282$$ 0 0
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ −18.0000 −1.04800
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ 3.00000 0.172917
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ 30.0000 1.71780
$$306$$ 0 0
$$307$$ −15.0000 −0.856095 −0.428048 0.903756i $$-0.640798\pi$$
−0.428048 + 0.903756i $$0.640798\pi$$
$$308$$ 0 0
$$309$$ 5.00000 0.284440
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ 0 0
$$313$$ 34.0000 1.92179 0.960897 0.276907i $$-0.0893093\pi$$
0.960897 + 0.276907i $$0.0893093\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ 5.00000 0.280828 0.140414 0.990093i $$-0.455157\pi$$
0.140414 + 0.990093i $$0.455157\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −12.0000 −0.665640
$$326$$ 0 0
$$327$$ −3.00000 −0.165900
$$328$$ 0 0
$$329$$ 7.00000 0.385922
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 0 0
$$333$$ −9.00000 −0.493197
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ 0 0
$$339$$ −9.00000 −0.488813
$$340$$ 0 0
$$341$$ −24.0000 −1.29967
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 3.00000 0.161515
$$346$$ 0 0
$$347$$ −23.0000 −1.23470 −0.617352 0.786687i $$-0.711795\pi$$
−0.617352 + 0.786687i $$0.711795\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ 0 0
$$353$$ −29.0000 −1.54351 −0.771757 0.635917i $$-0.780622\pi$$
−0.771757 + 0.635917i $$0.780622\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ 0 0
$$357$$ 4.00000 0.211702
$$358$$ 0 0
$$359$$ 1.00000 0.0527780 0.0263890 0.999652i $$-0.491599\pi$$
0.0263890 + 0.999652i $$0.491599\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ −24.0000 −1.25622
$$366$$ 0 0
$$367$$ −31.0000 −1.61819 −0.809093 0.587680i $$-0.800041\pi$$
−0.809093 + 0.587680i $$0.800041\pi$$
$$368$$ 0 0
$$369$$ 9.00000 0.468521
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ 3.00000 0.154919
$$376$$ 0 0
$$377$$ −9.00000 −0.463524
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 7.00000 0.358621
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −12.0000 −0.611577
$$386$$ 0 0
$$387$$ 3.00000 0.152499
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ −24.0000 −1.20757
$$396$$ 0 0
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 38.0000 1.89763 0.948815 0.315833i $$-0.102284\pi$$
0.948815 + 0.315833i $$0.102284\pi$$
$$402$$ 0 0
$$403$$ −18.0000 −0.896644
$$404$$ 0 0
$$405$$ 3.00000 0.149071
$$406$$ 0 0
$$407$$ 36.0000 1.78445
$$408$$ 0 0
$$409$$ 16.0000 0.791149 0.395575 0.918434i $$-0.370545\pi$$
0.395575 + 0.918434i $$0.370545\pi$$
$$410$$ 0 0
$$411$$ 15.0000 0.739895
$$412$$ 0 0
$$413$$ −6.00000 −0.295241
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 9.00000 0.440732
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −21.0000 −1.02348 −0.511739 0.859141i $$-0.670998\pi$$
−0.511739 + 0.859141i $$0.670998\pi$$
$$422$$ 0 0
$$423$$ 7.00000 0.340352
$$424$$ 0 0
$$425$$ −16.0000 −0.776114
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ 0 0
$$429$$ −12.0000 −0.579365
$$430$$ 0 0
$$431$$ −21.0000 −1.01153 −0.505767 0.862670i $$-0.668791\pi$$
−0.505767 + 0.862670i $$0.668791\pi$$
$$432$$ 0 0
$$433$$ 23.0000 1.10531 0.552655 0.833410i $$-0.313615\pi$$
0.552655 + 0.833410i $$0.313615\pi$$
$$434$$ 0 0
$$435$$ −9.00000 −0.431517
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −25.0000 −1.18779 −0.593893 0.804544i $$-0.702410\pi$$
−0.593893 + 0.804544i $$0.702410\pi$$
$$444$$ 0 0
$$445$$ −42.0000 −1.99099
$$446$$ 0 0
$$447$$ −16.0000 −0.756774
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −36.0000 −1.69517
$$452$$ 0 0
$$453$$ 15.0000 0.704761
$$454$$ 0 0
$$455$$ −9.00000 −0.421927
$$456$$ 0 0
$$457$$ 24.0000 1.12267 0.561336 0.827588i $$-0.310287\pi$$
0.561336 + 0.827588i $$0.310287\pi$$
$$458$$ 0 0
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ −22.0000 −1.02464 −0.512321 0.858794i $$-0.671214\pi$$
−0.512321 + 0.858794i $$0.671214\pi$$
$$462$$ 0 0
$$463$$ −13.0000 −0.604161 −0.302081 0.953282i $$-0.597681\pi$$
−0.302081 + 0.953282i $$0.597681\pi$$
$$464$$ 0 0
$$465$$ −18.0000 −0.834730
$$466$$ 0 0
$$467$$ −13.0000 −0.601568 −0.300784 0.953692i $$-0.597248\pi$$
−0.300784 + 0.953692i $$0.597248\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −4.00000 −0.183147
$$478$$ 0 0
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 27.0000 1.23109
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ −21.0000 −0.953561
$$486$$ 0 0
$$487$$ −13.0000 −0.589086 −0.294543 0.955638i $$-0.595167\pi$$
−0.294543 + 0.955638i $$0.595167\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ −34.0000 −1.52205 −0.761025 0.648723i $$-0.775303\pi$$
−0.761025 + 0.648723i $$0.775303\pi$$
$$500$$ 0 0
$$501$$ −20.0000 −0.893534
$$502$$ 0 0
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ −42.0000 −1.86898
$$506$$ 0 0
$$507$$ 4.00000 0.177646
$$508$$ 0 0
$$509$$ 44.0000 1.95027 0.975133 0.221621i $$-0.0711348\pi$$
0.975133 + 0.221621i $$0.0711348\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −15.0000 −0.660979
$$516$$ 0 0
$$517$$ −28.0000 −1.23144
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ −42.0000 −1.83653 −0.918266 0.395964i $$-0.870410\pi$$
−0.918266 + 0.395964i $$0.870410\pi$$
$$524$$ 0 0
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ −27.0000 −1.16950
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ 0 0
$$537$$ 19.0000 0.819911
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ 0 0
$$543$$ 2.00000 0.0858282
$$544$$ 0 0
$$545$$ 9.00000 0.385518
$$546$$ 0 0
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ 27.0000 1.14609
$$556$$ 0 0
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ 0 0
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ −15.0000 −0.632175 −0.316087 0.948730i $$-0.602369\pi$$
−0.316087 + 0.948730i $$0.602369\pi$$
$$564$$ 0 0
$$565$$ 27.0000 1.13590
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −13.0000 −0.544988 −0.272494 0.962157i $$-0.587849\pi$$
−0.272494 + 0.962157i $$0.587849\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ 0 0
$$579$$ 17.0000 0.706496
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 16.0000 0.662652
$$584$$ 0 0
$$585$$ −9.00000 −0.372104
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 23.0000 0.946094
$$592$$ 0 0
$$593$$ 29.0000 1.19089 0.595444 0.803397i $$-0.296976\pi$$
0.595444 + 0.803397i $$0.296976\pi$$
$$594$$ 0 0
$$595$$ −12.0000 −0.491952
$$596$$ 0 0
$$597$$ −5.00000 −0.204636
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 15.0000 0.609837
$$606$$ 0 0
$$607$$ −30.0000 −1.21766 −0.608831 0.793300i $$-0.708361\pi$$
−0.608831 + 0.793300i $$0.708361\pi$$
$$608$$ 0 0
$$609$$ −3.00000 −0.121566
$$610$$ 0 0
$$611$$ −21.0000 −0.849569
$$612$$ 0 0
$$613$$ 25.0000 1.00974 0.504870 0.863195i $$-0.331540\pi$$
0.504870 + 0.863195i $$0.331540\pi$$
$$614$$ 0 0
$$615$$ −27.0000 −1.08875
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −2.00000 −0.0803868 −0.0401934 0.999192i $$-0.512797\pi$$
−0.0401934 + 0.999192i $$0.512797\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −14.0000 −0.560898
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 0 0
$$635$$ −21.0000 −0.833360
$$636$$ 0 0
$$637$$ −3.00000 −0.118864
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ −9.00000 −0.354375
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ −6.00000 −0.235159
$$652$$ 0 0
$$653$$ 31.0000 1.21312 0.606562 0.795036i $$-0.292548\pi$$
0.606562 + 0.795036i $$0.292548\pi$$
$$654$$ 0 0
$$655$$ 18.0000 0.703318
$$656$$ 0 0
$$657$$ −8.00000 −0.312110
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 0 0
$$663$$ −12.0000 −0.466041
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3.00000 −0.116160
$$668$$ 0 0
$$669$$ −14.0000 −0.541271
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 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
$$677$$ 46.0000 1.76792 0.883962 0.467559i $$-0.154866\pi$$
0.883962 + 0.467559i $$0.154866\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 11.0000 0.421521
$$682$$ 0 0
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ −45.0000 −1.71936
$$686$$ 0 0
$$687$$ −12.0000 −0.457829
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 5.00000 0.190209 0.0951045 0.995467i $$-0.469681\pi$$
0.0951045 + 0.995467i $$0.469681\pi$$
$$692$$ 0 0
$$693$$ −4.00000 −0.151947
$$694$$ 0 0
$$695$$ −27.0000 −1.02417
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −21.0000 −0.790906
$$706$$ 0 0
$$707$$ −14.0000 −0.526524
$$708$$ 0 0
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ 0 0
$$717$$ −12.0000 −0.448148
$$718$$ 0 0
$$719$$ 7.00000 0.261056 0.130528 0.991445i $$-0.458333\pi$$
0.130528 + 0.991445i $$0.458333\pi$$
$$720$$ 0 0
$$721$$ −5.00000 −0.186210
$$722$$ 0 0
$$723$$ 5.00000 0.185952
$$724$$ 0 0
$$725$$ 12.0000 0.445669
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ −16.0000 −0.590973 −0.295487 0.955347i $$-0.595482\pi$$
−0.295487 + 0.955347i $$0.595482\pi$$
$$734$$ 0 0
$$735$$ −3.00000 −0.110657
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 0 0
$$745$$ 48.0000 1.75858
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 19.0000 0.692398
$$754$$ 0 0
$$755$$ −45.0000 −1.63772
$$756$$ 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
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ 3.00000 0.108607
$$764$$ 0 0
$$765$$ −12.0000 −0.433861
$$766$$ 0 0
$$767$$ 18.0000 0.649942
$$768$$ 0 0
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 0 0
$$771$$ 26.0000 0.936367
$$772$$ 0 0
$$773$$ 19.0000 0.683383 0.341691 0.939812i $$-0.389000\pi$$
0.341691 + 0.939812i $$0.389000\pi$$
$$774$$ 0 0
$$775$$ 24.0000 0.862105
$$776$$ 0 0
$$777$$ 9.00000 0.322873
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ 0 0
$$783$$ −3.00000 −0.107211
$$784$$ 0 0
$$785$$ −24.0000 −0.856597
$$786$$ 0 0
$$787$$ 40.0000 1.42585 0.712923 0.701242i $$-0.247371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ 0 0
$$789$$ 21.0000 0.747620
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ −30.0000 −1.06533
$$794$$ 0 0
$$795$$ 12.0000 0.425596
$$796$$ 0 0
$$797$$ −49.0000 −1.73567 −0.867835 0.496853i $$-0.834489\pi$$
−0.867835 + 0.496853i $$0.834489\pi$$
$$798$$ 0 0
$$799$$ −28.0000 −0.990569
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ 0 0
$$803$$ 32.0000 1.12926
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ 0 0
$$807$$ 6.00000 0.211210
$$808$$ 0 0
$$809$$ −16.0000 −0.562530 −0.281265 0.959630i $$-0.590754\pi$$
−0.281265 + 0.959630i $$0.590754\pi$$
$$810$$ 0 0
$$811$$ −33.0000 −1.15879 −0.579393 0.815048i $$-0.696710\pi$$
−0.579393 + 0.815048i $$0.696710\pi$$
$$812$$ 0 0
$$813$$ 20.0000 0.701431
$$814$$ 0 0
$$815$$ −48.0000 −1.68137
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −3.00000 −0.104828
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ −7.00000 −0.244005 −0.122002 0.992530i $$-0.538932\pi$$
−0.122002 + 0.992530i $$0.538932\pi$$
$$824$$ 0 0
$$825$$ 16.0000 0.557048
$$826$$ 0 0
$$827$$ 30.0000 1.04320 0.521601 0.853189i $$-0.325335\pi$$
0.521601 + 0.853189i $$0.325335\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ −20.0000 −0.693792
$$832$$ 0 0
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ 60.0000 2.07639
$$836$$ 0 0
$$837$$ −6.00000 −0.207390
$$838$$ 0 0
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ −23.0000 −0.792162
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ 0 0
$$849$$ −6.00000 −0.205919
$$850$$ 0 0
$$851$$ 9.00000 0.308516
$$852$$ 0 0
$$853$$ 7.00000 0.239675 0.119838 0.992793i $$-0.461763\pi$$
0.119838 + 0.992793i $$0.461763\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 33.0000 1.12726 0.563629 0.826028i $$-0.309405\pi$$
0.563629 + 0.826028i $$0.309405\pi$$
$$858$$ 0 0
$$859$$ −13.0000 −0.443554 −0.221777 0.975097i $$-0.571186\pi$$
−0.221777 + 0.975097i $$0.571186\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ 0 0
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ 0 0
$$873$$ −7.00000 −0.236914
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −36.0000 −1.21563 −0.607817 0.794077i $$-0.707955\pi$$
−0.607817 + 0.794077i $$0.707955\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ −54.0000 −1.81724 −0.908622 0.417619i $$-0.862865\pi$$
−0.908622 + 0.417619i $$0.862865\pi$$
$$884$$ 0 0
$$885$$ 18.0000 0.605063
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −7.00000 −0.234772
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −57.0000 −1.90530
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ 0 0
$$899$$ 18.0000 0.600334
$$900$$ 0 0
$$901$$ 16.0000 0.533037
$$902$$ 0 0
$$903$$ −3.00000 −0.0998337
$$904$$ 0 0
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ 53.0000 1.75984 0.879918 0.475125i $$-0.157597\pi$$
0.879918 + 0.475125i $$0.157597\pi$$
$$908$$ 0 0
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 49.0000 1.62344 0.811721 0.584045i $$-0.198531\pi$$
0.811721 + 0.584045i $$0.198531\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 0 0
$$915$$ −30.0000 −0.991769
$$916$$ 0 0
$$917$$ 6.00000 0.198137
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 15.0000 0.494267
$$922$$ 0 0
$$923$$ −18.0000 −0.592477
$$924$$ 0 0
$$925$$ −36.0000 −1.18367
$$926$$ 0 0
$$927$$ −5.00000 −0.164222
$$928$$ 0 0
$$929$$ 9.00000 0.295280 0.147640 0.989041i $$-0.452832\pi$$
0.147640 + 0.989041i $$0.452832\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −28.0000 −0.916679
$$934$$ 0 0
$$935$$ 48.0000 1.56977
$$936$$ 0 0
$$937$$ 25.0000 0.816714 0.408357 0.912822i $$-0.366102\pi$$
0.408357 + 0.912822i $$0.366102\pi$$
$$938$$ 0 0
$$939$$ −34.0000 −1.10955
$$940$$ 0 0
$$941$$ 3.00000 0.0977972 0.0488986 0.998804i $$-0.484429\pi$$
0.0488986 + 0.998804i $$0.484429\pi$$
$$942$$ 0 0
$$943$$ −9.00000 −0.293080
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 0 0
$$947$$ 31.0000 1.00736 0.503682 0.863889i $$-0.331978\pi$$
0.503682 + 0.863889i $$0.331978\pi$$
$$948$$ 0 0
$$949$$ 24.0000 0.779073
$$950$$ 0 0
$$951$$ −5.00000 −0.162136
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 0 0
$$955$$ 36.0000 1.16493
$$956$$ 0 0
$$957$$ 12.0000 0.387905
$$958$$ 0 0
$$959$$ −15.0000 −0.484375
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ 0 0
$$965$$ −51.0000 −1.64175
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ −9.00000 −0.288527
$$974$$ 0 0
$$975$$ 12.0000 0.384308
$$976$$ 0 0
$$977$$ 13.0000 0.415907 0.207953 0.978139i $$-0.433320\pi$$
0.207953 + 0.978139i $$0.433320\pi$$
$$978$$ 0 0
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ 3.00000 0.0957826
$$982$$ 0 0
$$983$$ 54.0000 1.72233 0.861166 0.508323i $$-0.169735\pi$$
0.861166 + 0.508323i $$0.169735\pi$$
$$984$$ 0 0
$$985$$ −69.0000 −2.19852
$$986$$ 0 0
$$987$$ −7.00000 −0.222812
$$988$$ 0 0
$$989$$ −3.00000 −0.0953945
$$990$$ 0 0
$$991$$ 36.0000 1.14358 0.571789 0.820401i $$-0.306250\pi$$
0.571789 + 0.820401i $$0.306250\pi$$
$$992$$ 0 0
$$993$$ 8.00000 0.253872
$$994$$ 0 0
$$995$$ 15.0000 0.475532
$$996$$ 0 0
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ 0 0
$$999$$ 9.00000 0.284747
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.j.1.1 1
4.3 odd 2 966.2.a.k.1.1 1
12.11 even 2 2898.2.a.a.1.1 1
28.27 even 2 6762.2.a.y.1.1 1

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