# Properties

 Label 7728.2.a.br.1.2 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1932) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.43163$$ of defining polynomial Character $$\chi$$ $$=$$ 7728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.43163 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.43163 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.95044 q^{13} -1.43163 q^{15} +4.51882 q^{17} -3.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -2.95044 q^{25} -1.00000 q^{27} -10.4197 q^{29} -0.344438 q^{31} +1.00000 q^{33} +1.43163 q^{35} -7.55645 q^{37} -3.95044 q^{39} +5.17438 q^{41} -7.08719 q^{43} +1.43163 q^{45} -7.51882 q^{47} +1.00000 q^{49} -4.51882 q^{51} -4.95044 q^{53} -1.43163 q^{55} +3.00000 q^{57} -4.77606 q^{59} +4.60601 q^{61} +1.00000 q^{63} +5.65556 q^{65} -10.4693 q^{67} +1.00000 q^{69} -12.1248 q^{71} +0.518817 q^{73} +2.95044 q^{75} -1.00000 q^{77} -1.38207 q^{79} +1.00000 q^{81} +6.41970 q^{83} +6.46926 q^{85} +10.4197 q^{87} -2.81370 q^{89} +3.95044 q^{91} +0.344438 q^{93} -4.29488 q^{95} -6.51882 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} - q^{13} - q^{15} + 4 q^{17} - 9 q^{19} - 3 q^{21} - 3 q^{23} + 4 q^{25} - 3 q^{27} + 4 q^{29} - 4 q^{31} + 3 q^{33} + q^{35} + 6 q^{37} + q^{39} + 3 q^{41} - 15 q^{43} + q^{45} - 13 q^{47} + 3 q^{49} - 4 q^{51} - 2 q^{53} - q^{55} + 9 q^{57} - 14 q^{59} - 2 q^{61} + 3 q^{63} + 14 q^{65} - 9 q^{67} + 3 q^{69} - 11 q^{71} - 8 q^{73} - 4 q^{75} - 3 q^{77} + 12 q^{79} + 3 q^{81} - 16 q^{83} - 3 q^{85} - 4 q^{87} + 11 q^{89} - q^{91} + 4 q^{93} - 3 q^{95} - 10 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
$$5$$ 1.43163 0.640243 0.320122 0.947376i $$-0.396276\pi$$
0.320122 + 0.947376i $$0.396276\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ 3.95044 1.09566 0.547828 0.836591i $$-0.315455\pi$$
0.547828 + 0.836591i $$0.315455\pi$$
$$14$$ 0 0
$$15$$ −1.43163 −0.369645
$$16$$ 0 0
$$17$$ 4.51882 1.09597 0.547987 0.836487i $$-0.315394\pi$$
0.547987 + 0.836487i $$0.315394\pi$$
$$18$$ 0 0
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ −2.95044 −0.590089
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −10.4197 −1.93489 −0.967445 0.253080i $$-0.918556\pi$$
−0.967445 + 0.253080i $$0.918556\pi$$
$$30$$ 0 0
$$31$$ −0.344438 −0.0618628 −0.0309314 0.999522i $$-0.509847\pi$$
−0.0309314 + 0.999522i $$0.509847\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ 1.43163 0.241989
$$36$$ 0 0
$$37$$ −7.55645 −1.24227 −0.621136 0.783703i $$-0.713329\pi$$
−0.621136 + 0.783703i $$0.713329\pi$$
$$38$$ 0 0
$$39$$ −3.95044 −0.632577
$$40$$ 0 0
$$41$$ 5.17438 0.808102 0.404051 0.914736i $$-0.367602\pi$$
0.404051 + 0.914736i $$0.367602\pi$$
$$42$$ 0 0
$$43$$ −7.08719 −1.08079 −0.540393 0.841413i $$-0.681724\pi$$
−0.540393 + 0.841413i $$0.681724\pi$$
$$44$$ 0 0
$$45$$ 1.43163 0.213414
$$46$$ 0 0
$$47$$ −7.51882 −1.09673 −0.548366 0.836238i $$-0.684750\pi$$
−0.548366 + 0.836238i $$0.684750\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −4.51882 −0.632761
$$52$$ 0 0
$$53$$ −4.95044 −0.679996 −0.339998 0.940426i $$-0.610426\pi$$
−0.339998 + 0.940426i $$0.610426\pi$$
$$54$$ 0 0
$$55$$ −1.43163 −0.193041
$$56$$ 0 0
$$57$$ 3.00000 0.397360
$$58$$ 0 0
$$59$$ −4.77606 −0.621791 −0.310895 0.950444i $$-0.600629\pi$$
−0.310895 + 0.950444i $$0.600629\pi$$
$$60$$ 0 0
$$61$$ 4.60601 0.589739 0.294869 0.955538i $$-0.404724\pi$$
0.294869 + 0.955538i $$0.404724\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 5.65556 0.701486
$$66$$ 0 0
$$67$$ −10.4693 −1.27902 −0.639512 0.768781i $$-0.720864\pi$$
−0.639512 + 0.768781i $$0.720864\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −12.1248 −1.43895 −0.719476 0.694517i $$-0.755618\pi$$
−0.719476 + 0.694517i $$0.755618\pi$$
$$72$$ 0 0
$$73$$ 0.518817 0.0607229 0.0303615 0.999539i $$-0.490334\pi$$
0.0303615 + 0.999539i $$0.490334\pi$$
$$74$$ 0 0
$$75$$ 2.95044 0.340688
$$76$$ 0 0
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ −1.38207 −0.155495 −0.0777476 0.996973i $$-0.524773\pi$$
−0.0777476 + 0.996973i $$0.524773\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.41970 0.704654 0.352327 0.935877i $$-0.385390\pi$$
0.352327 + 0.935877i $$0.385390\pi$$
$$84$$ 0 0
$$85$$ 6.46926 0.701690
$$86$$ 0 0
$$87$$ 10.4197 1.11711
$$88$$ 0 0
$$89$$ −2.81370 −0.298251 −0.149126 0.988818i $$-0.547646\pi$$
−0.149126 + 0.988818i $$0.547646\pi$$
$$90$$ 0 0
$$91$$ 3.95044 0.414119
$$92$$ 0 0
$$93$$ 0.344438 0.0357165
$$94$$ 0 0
$$95$$ −4.29488 −0.440646
$$96$$ 0 0
$$97$$ −6.51882 −0.661886 −0.330943 0.943651i $$-0.607367\pi$$
−0.330943 + 0.943651i $$0.607367\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 12.8513 1.27876 0.639378 0.768893i $$-0.279192\pi$$
0.639378 + 0.768893i $$0.279192\pi$$
$$102$$ 0 0
$$103$$ 11.4197 1.12522 0.562608 0.826723i $$-0.309798\pi$$
0.562608 + 0.826723i $$0.309798\pi$$
$$104$$ 0 0
$$105$$ −1.43163 −0.139713
$$106$$ 0 0
$$107$$ 1.95044 0.188557 0.0942783 0.995546i $$-0.469946\pi$$
0.0942783 + 0.995546i $$0.469946\pi$$
$$108$$ 0 0
$$109$$ −7.15814 −0.685625 −0.342813 0.939404i $$-0.611380\pi$$
−0.342813 + 0.939404i $$0.611380\pi$$
$$110$$ 0 0
$$111$$ 7.55645 0.717227
$$112$$ 0 0
$$113$$ −0.469261 −0.0441443 −0.0220722 0.999756i $$-0.507026\pi$$
−0.0220722 + 0.999756i $$0.507026\pi$$
$$114$$ 0 0
$$115$$ −1.43163 −0.133500
$$116$$ 0 0
$$117$$ 3.95044 0.365219
$$118$$ 0 0
$$119$$ 4.51882 0.414239
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ −5.17438 −0.466558
$$124$$ 0 0
$$125$$ −11.3821 −1.01804
$$126$$ 0 0
$$127$$ −2.70512 −0.240040 −0.120020 0.992771i $$-0.538296\pi$$
−0.120020 + 0.992771i $$0.538296\pi$$
$$128$$ 0 0
$$129$$ 7.08719 0.623992
$$130$$ 0 0
$$131$$ −10.9009 −0.952415 −0.476207 0.879333i $$-0.657989\pi$$
−0.476207 + 0.879333i $$0.657989\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ 0 0
$$135$$ −1.43163 −0.123215
$$136$$ 0 0
$$137$$ 8.31112 0.710067 0.355034 0.934854i $$-0.384469\pi$$
0.355034 + 0.934854i $$0.384469\pi$$
$$138$$ 0 0
$$139$$ −16.5445 −1.40329 −0.701644 0.712527i $$-0.747550\pi$$
−0.701644 + 0.712527i $$0.747550\pi$$
$$140$$ 0 0
$$141$$ 7.51882 0.633199
$$142$$ 0 0
$$143$$ −3.95044 −0.330353
$$144$$ 0 0
$$145$$ −14.9171 −1.23880
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 13.5941 1.11367 0.556835 0.830623i $$-0.312015\pi$$
0.556835 + 0.830623i $$0.312015\pi$$
$$150$$ 0 0
$$151$$ 10.5565 0.859072 0.429536 0.903050i $$-0.358677\pi$$
0.429536 + 0.903050i $$0.358677\pi$$
$$152$$ 0 0
$$153$$ 4.51882 0.365325
$$154$$ 0 0
$$155$$ −0.493106 −0.0396072
$$156$$ 0 0
$$157$$ 3.24533 0.259005 0.129503 0.991579i $$-0.458662\pi$$
0.129503 + 0.991579i $$0.458662\pi$$
$$158$$ 0 0
$$159$$ 4.95044 0.392596
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ 17.1248 1.34132 0.670660 0.741765i $$-0.266011\pi$$
0.670660 + 0.741765i $$0.266011\pi$$
$$164$$ 0 0
$$165$$ 1.43163 0.111452
$$166$$ 0 0
$$167$$ 7.86325 0.608477 0.304238 0.952596i $$-0.401598\pi$$
0.304238 + 0.952596i $$0.401598\pi$$
$$168$$ 0 0
$$169$$ 2.60601 0.200462
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ 0 0
$$173$$ 3.20769 0.243876 0.121938 0.992538i $$-0.461089\pi$$
0.121938 + 0.992538i $$0.461089\pi$$
$$174$$ 0 0
$$175$$ −2.95044 −0.223033
$$176$$ 0 0
$$177$$ 4.77606 0.358991
$$178$$ 0 0
$$179$$ 16.6436 1.24400 0.622002 0.783016i $$-0.286320\pi$$
0.622002 + 0.783016i $$0.286320\pi$$
$$180$$ 0 0
$$181$$ −4.79231 −0.356209 −0.178105 0.984012i $$-0.556997\pi$$
−0.178105 + 0.984012i $$0.556997\pi$$
$$182$$ 0 0
$$183$$ −4.60601 −0.340486
$$184$$ 0 0
$$185$$ −10.8180 −0.795357
$$186$$ 0 0
$$187$$ −4.51882 −0.330449
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −5.24533 −0.379538 −0.189769 0.981829i $$-0.560774\pi$$
−0.189769 + 0.981829i $$0.560774\pi$$
$$192$$ 0 0
$$193$$ −18.2830 −1.31604 −0.658018 0.753002i $$-0.728605\pi$$
−0.658018 + 0.753002i $$0.728605\pi$$
$$194$$ 0 0
$$195$$ −5.65556 −0.405003
$$196$$ 0 0
$$197$$ 5.77606 0.411528 0.205764 0.978602i $$-0.434032\pi$$
0.205764 + 0.978602i $$0.434032\pi$$
$$198$$ 0 0
$$199$$ −9.29920 −0.659203 −0.329601 0.944120i $$-0.606914\pi$$
−0.329601 + 0.944120i $$0.606914\pi$$
$$200$$ 0 0
$$201$$ 10.4693 0.738445
$$202$$ 0 0
$$203$$ −10.4197 −0.731320
$$204$$ 0 0
$$205$$ 7.40778 0.517382
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ 4.21201 0.289967 0.144983 0.989434i $$-0.453687\pi$$
0.144983 + 0.989434i $$0.453687\pi$$
$$212$$ 0 0
$$213$$ 12.1248 0.830779
$$214$$ 0 0
$$215$$ −10.1462 −0.691966
$$216$$ 0 0
$$217$$ −0.344438 −0.0233819
$$218$$ 0 0
$$219$$ −0.518817 −0.0350584
$$220$$ 0 0
$$221$$ 17.8513 1.20081
$$222$$ 0 0
$$223$$ 22.0967 1.47970 0.739851 0.672771i $$-0.234896\pi$$
0.739851 + 0.672771i $$0.234896\pi$$
$$224$$ 0 0
$$225$$ −2.95044 −0.196696
$$226$$ 0 0
$$227$$ −2.49311 −0.165473 −0.0827366 0.996571i $$-0.526366\pi$$
−0.0827366 + 0.996571i $$0.526366\pi$$
$$228$$ 0 0
$$229$$ −23.0257 −1.52158 −0.760791 0.648997i $$-0.775189\pi$$
−0.760791 + 0.648997i $$0.775189\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ 0 0
$$233$$ −20.5445 −1.34592 −0.672958 0.739680i $$-0.734977\pi$$
−0.672958 + 0.739680i $$0.734977\pi$$
$$234$$ 0 0
$$235$$ −10.7641 −0.702175
$$236$$ 0 0
$$237$$ 1.38207 0.0897752
$$238$$ 0 0
$$239$$ −13.6813 −0.884968 −0.442484 0.896776i $$-0.645903\pi$$
−0.442484 + 0.896776i $$0.645903\pi$$
$$240$$ 0 0
$$241$$ −18.6274 −1.19990 −0.599948 0.800039i $$-0.704812\pi$$
−0.599948 + 0.800039i $$0.704812\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 1.43163 0.0914633
$$246$$ 0 0
$$247$$ −11.8513 −0.754082
$$248$$ 0 0
$$249$$ −6.41970 −0.406832
$$250$$ 0 0
$$251$$ −11.6274 −0.733915 −0.366957 0.930238i $$-0.619601\pi$$
−0.366957 + 0.930238i $$0.619601\pi$$
$$252$$ 0 0
$$253$$ 1.00000 0.0628695
$$254$$ 0 0
$$255$$ −6.46926 −0.405121
$$256$$ 0 0
$$257$$ 9.24533 0.576708 0.288354 0.957524i $$-0.406892\pi$$
0.288354 + 0.957524i $$0.406892\pi$$
$$258$$ 0 0
$$259$$ −7.55645 −0.469535
$$260$$ 0 0
$$261$$ −10.4197 −0.644964
$$262$$ 0 0
$$263$$ 0.825621 0.0509100 0.0254550 0.999676i $$-0.491897\pi$$
0.0254550 + 0.999676i $$0.491897\pi$$
$$264$$ 0 0
$$265$$ −7.08719 −0.435363
$$266$$ 0 0
$$267$$ 2.81370 0.172196
$$268$$ 0 0
$$269$$ −15.2616 −0.930514 −0.465257 0.885176i $$-0.654038\pi$$
−0.465257 + 0.885176i $$0.654038\pi$$
$$270$$ 0 0
$$271$$ 23.9718 1.45619 0.728093 0.685479i $$-0.240407\pi$$
0.728093 + 0.685479i $$0.240407\pi$$
$$272$$ 0 0
$$273$$ −3.95044 −0.239092
$$274$$ 0 0
$$275$$ 2.95044 0.177918
$$276$$ 0 0
$$277$$ −5.29488 −0.318139 −0.159069 0.987267i $$-0.550849\pi$$
−0.159069 + 0.987267i $$0.550849\pi$$
$$278$$ 0 0
$$279$$ −0.344438 −0.0206209
$$280$$ 0 0
$$281$$ 16.1744 0.964883 0.482441 0.875928i $$-0.339750\pi$$
0.482441 + 0.875928i $$0.339750\pi$$
$$282$$ 0 0
$$283$$ −4.29488 −0.255304 −0.127652 0.991819i $$-0.540744\pi$$
−0.127652 + 0.991819i $$0.540744\pi$$
$$284$$ 0 0
$$285$$ 4.29488 0.254407
$$286$$ 0 0
$$287$$ 5.17438 0.305434
$$288$$ 0 0
$$289$$ 3.41970 0.201159
$$290$$ 0 0
$$291$$ 6.51882 0.382140
$$292$$ 0 0
$$293$$ −5.72651 −0.334546 −0.167273 0.985911i $$-0.553496\pi$$
−0.167273 + 0.985911i $$0.553496\pi$$
$$294$$ 0 0
$$295$$ −6.83754 −0.398097
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ −3.95044 −0.228460
$$300$$ 0 0
$$301$$ −7.08719 −0.408499
$$302$$ 0 0
$$303$$ −12.8513 −0.738290
$$304$$ 0 0
$$305$$ 6.59408 0.377576
$$306$$ 0 0
$$307$$ 15.1838 0.866588 0.433294 0.901253i $$-0.357351\pi$$
0.433294 + 0.901253i $$0.357351\pi$$
$$308$$ 0 0
$$309$$ −11.4197 −0.649644
$$310$$ 0 0
$$311$$ −7.66749 −0.434783 −0.217392 0.976084i $$-0.569755\pi$$
−0.217392 + 0.976084i $$0.569755\pi$$
$$312$$ 0 0
$$313$$ 26.8061 1.51517 0.757585 0.652736i $$-0.226379\pi$$
0.757585 + 0.652736i $$0.226379\pi$$
$$314$$ 0 0
$$315$$ 1.43163 0.0806630
$$316$$ 0 0
$$317$$ 2.05388 0.115357 0.0576786 0.998335i $$-0.481630\pi$$
0.0576786 + 0.998335i $$0.481630\pi$$
$$318$$ 0 0
$$319$$ 10.4197 0.583391
$$320$$ 0 0
$$321$$ −1.95044 −0.108863
$$322$$ 0 0
$$323$$ −13.5565 −0.754301
$$324$$ 0 0
$$325$$ −11.6556 −0.646534
$$326$$ 0 0
$$327$$ 7.15814 0.395846
$$328$$ 0 0
$$329$$ −7.51882 −0.414526
$$330$$ 0 0
$$331$$ 28.8727 1.58699 0.793494 0.608578i $$-0.208260\pi$$
0.793494 + 0.608578i $$0.208260\pi$$
$$332$$ 0 0
$$333$$ −7.55645 −0.414091
$$334$$ 0 0
$$335$$ −14.9881 −0.818886
$$336$$ 0 0
$$337$$ −13.3368 −0.726504 −0.363252 0.931691i $$-0.618334\pi$$
−0.363252 + 0.931691i $$0.618334\pi$$
$$338$$ 0 0
$$339$$ 0.469261 0.0254867
$$340$$ 0 0
$$341$$ 0.344438 0.0186523
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 1.43163 0.0770762
$$346$$ 0 0
$$347$$ −21.2830 −1.14253 −0.571265 0.820766i $$-0.693547\pi$$
−0.571265 + 0.820766i $$0.693547\pi$$
$$348$$ 0 0
$$349$$ −31.7804 −1.70117 −0.850583 0.525842i $$-0.823750\pi$$
−0.850583 + 0.525842i $$0.823750\pi$$
$$350$$ 0 0
$$351$$ −3.95044 −0.210859
$$352$$ 0 0
$$353$$ 22.3206 1.18801 0.594003 0.804463i $$-0.297547\pi$$
0.594003 + 0.804463i $$0.297547\pi$$
$$354$$ 0 0
$$355$$ −17.3582 −0.921279
$$356$$ 0 0
$$357$$ −4.51882 −0.239161
$$358$$ 0 0
$$359$$ 31.0590 1.63923 0.819616 0.572913i $$-0.194187\pi$$
0.819616 + 0.572913i $$0.194187\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 0 0
$$363$$ 10.0000 0.524864
$$364$$ 0 0
$$365$$ 0.742752 0.0388774
$$366$$ 0 0
$$367$$ −7.74275 −0.404168 −0.202084 0.979368i $$-0.564771\pi$$
−0.202084 + 0.979368i $$0.564771\pi$$
$$368$$ 0 0
$$369$$ 5.17438 0.269367
$$370$$ 0 0
$$371$$ −4.95044 −0.257014
$$372$$ 0 0
$$373$$ −15.5565 −0.805482 −0.402741 0.915314i $$-0.631943\pi$$
−0.402741 + 0.915314i $$0.631943\pi$$
$$374$$ 0 0
$$375$$ 11.3821 0.587768
$$376$$ 0 0
$$377$$ −41.1625 −2.11997
$$378$$ 0 0
$$379$$ −29.1129 −1.49543 −0.747715 0.664020i $$-0.768849\pi$$
−0.747715 + 0.664020i $$0.768849\pi$$
$$380$$ 0 0
$$381$$ 2.70512 0.138587
$$382$$ 0 0
$$383$$ 31.2591 1.59727 0.798633 0.601818i $$-0.205557\pi$$
0.798633 + 0.601818i $$0.205557\pi$$
$$384$$ 0 0
$$385$$ −1.43163 −0.0729625
$$386$$ 0 0
$$387$$ −7.08719 −0.360262
$$388$$ 0 0
$$389$$ −8.10343 −0.410860 −0.205430 0.978672i $$-0.565859\pi$$
−0.205430 + 0.978672i $$0.565859\pi$$
$$390$$ 0 0
$$391$$ −4.51882 −0.228526
$$392$$ 0 0
$$393$$ 10.9009 0.549877
$$394$$ 0 0
$$395$$ −1.97861 −0.0995547
$$396$$ 0 0
$$397$$ −1.06148 −0.0532741 −0.0266371 0.999645i $$-0.508480\pi$$
−0.0266371 + 0.999645i $$0.508480\pi$$
$$398$$ 0 0
$$399$$ 3.00000 0.150188
$$400$$ 0 0
$$401$$ −25.3915 −1.26799 −0.633996 0.773336i $$-0.718587\pi$$
−0.633996 + 0.773336i $$0.718587\pi$$
$$402$$ 0 0
$$403$$ −1.36068 −0.0677804
$$404$$ 0 0
$$405$$ 1.43163 0.0711381
$$406$$ 0 0
$$407$$ 7.55645 0.374559
$$408$$ 0 0
$$409$$ −31.6556 −1.56527 −0.782633 0.622483i $$-0.786124\pi$$
−0.782633 + 0.622483i $$0.786124\pi$$
$$410$$ 0 0
$$411$$ −8.31112 −0.409958
$$412$$ 0 0
$$413$$ −4.77606 −0.235015
$$414$$ 0 0
$$415$$ 9.19062 0.451150
$$416$$ 0 0
$$417$$ 16.5445 0.810189
$$418$$ 0 0
$$419$$ 2.84186 0.138834 0.0694171 0.997588i $$-0.477886\pi$$
0.0694171 + 0.997588i $$0.477886\pi$$
$$420$$ 0 0
$$421$$ 24.5164 1.19485 0.597427 0.801923i $$-0.296190\pi$$
0.597427 + 0.801923i $$0.296190\pi$$
$$422$$ 0 0
$$423$$ −7.51882 −0.365577
$$424$$ 0 0
$$425$$ −13.3325 −0.646722
$$426$$ 0 0
$$427$$ 4.60601 0.222900
$$428$$ 0 0
$$429$$ 3.95044 0.190729
$$430$$ 0 0
$$431$$ 3.74275 0.180282 0.0901410 0.995929i $$-0.471268\pi$$
0.0901410 + 0.995929i $$0.471268\pi$$
$$432$$ 0 0
$$433$$ 32.7070 1.57180 0.785899 0.618355i $$-0.212201\pi$$
0.785899 + 0.618355i $$0.212201\pi$$
$$434$$ 0 0
$$435$$ 14.9171 0.715222
$$436$$ 0 0
$$437$$ 3.00000 0.143509
$$438$$ 0 0
$$439$$ 38.1462 1.82062 0.910310 0.413928i $$-0.135843\pi$$
0.910310 + 0.413928i $$0.135843\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 40.1505 1.90761 0.953805 0.300427i $$-0.0971293\pi$$
0.953805 + 0.300427i $$0.0971293\pi$$
$$444$$ 0 0
$$445$$ −4.02817 −0.190953
$$446$$ 0 0
$$447$$ −13.5941 −0.642978
$$448$$ 0 0
$$449$$ 23.2334 1.09645 0.548226 0.836330i $$-0.315303\pi$$
0.548226 + 0.836330i $$0.315303\pi$$
$$450$$ 0 0
$$451$$ −5.17438 −0.243652
$$452$$ 0 0
$$453$$ −10.5565 −0.495985
$$454$$ 0 0
$$455$$ 5.65556 0.265137
$$456$$ 0 0
$$457$$ 21.9975 1.02900 0.514501 0.857490i $$-0.327977\pi$$
0.514501 + 0.857490i $$0.327977\pi$$
$$458$$ 0 0
$$459$$ −4.51882 −0.210920
$$460$$ 0 0
$$461$$ −25.0590 −1.16712 −0.583558 0.812072i $$-0.698340\pi$$
−0.583558 + 0.812072i $$0.698340\pi$$
$$462$$ 0 0
$$463$$ −29.7641 −1.38326 −0.691628 0.722253i $$-0.743106\pi$$
−0.691628 + 0.722253i $$0.743106\pi$$
$$464$$ 0 0
$$465$$ 0.493106 0.0228672
$$466$$ 0 0
$$467$$ 3.30680 0.153021 0.0765103 0.997069i $$-0.475622\pi$$
0.0765103 + 0.997069i $$0.475622\pi$$
$$468$$ 0 0
$$469$$ −10.4693 −0.483426
$$470$$ 0 0
$$471$$ −3.24533 −0.149537
$$472$$ 0 0
$$473$$ 7.08719 0.325869
$$474$$ 0 0
$$475$$ 8.85133 0.406127
$$476$$ 0 0
$$477$$ −4.95044 −0.226665
$$478$$ 0 0
$$479$$ −31.2830 −1.42935 −0.714677 0.699454i $$-0.753426\pi$$
−0.714677 + 0.699454i $$0.753426\pi$$
$$480$$ 0 0
$$481$$ −29.8513 −1.36110
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ −9.33251 −0.423768
$$486$$ 0 0
$$487$$ −21.1086 −0.956521 −0.478261 0.878218i $$-0.658733\pi$$
−0.478261 + 0.878218i $$0.658733\pi$$
$$488$$ 0 0
$$489$$ −17.1248 −0.774411
$$490$$ 0 0
$$491$$ −2.32305 −0.104838 −0.0524188 0.998625i $$-0.516693\pi$$
−0.0524188 + 0.998625i $$0.516693\pi$$
$$492$$ 0 0
$$493$$ −47.0847 −2.12059
$$494$$ 0 0
$$495$$ −1.43163 −0.0643469
$$496$$ 0 0
$$497$$ −12.1248 −0.543873
$$498$$ 0 0
$$499$$ 9.43163 0.422218 0.211109 0.977463i $$-0.432293\pi$$
0.211109 + 0.977463i $$0.432293\pi$$
$$500$$ 0 0
$$501$$ −7.86325 −0.351304
$$502$$ 0 0
$$503$$ −7.95476 −0.354685 −0.177343 0.984149i $$-0.556750\pi$$
−0.177343 + 0.984149i $$0.556750\pi$$
$$504$$ 0 0
$$505$$ 18.3983 0.818714
$$506$$ 0 0
$$507$$ −2.60601 −0.115737
$$508$$ 0 0
$$509$$ −4.48118 −0.198625 −0.0993125 0.995056i $$-0.531664\pi$$
−0.0993125 + 0.995056i $$0.531664\pi$$
$$510$$ 0 0
$$511$$ 0.518817 0.0229511
$$512$$ 0 0
$$513$$ 3.00000 0.132453
$$514$$ 0 0
$$515$$ 16.3488 0.720412
$$516$$ 0 0
$$517$$ 7.51882 0.330677
$$518$$ 0 0
$$519$$ −3.20769 −0.140802
$$520$$ 0 0
$$521$$ 32.4240 1.42052 0.710261 0.703938i $$-0.248577\pi$$
0.710261 + 0.703938i $$0.248577\pi$$
$$522$$ 0 0
$$523$$ 25.4197 1.11153 0.555763 0.831341i $$-0.312426\pi$$
0.555763 + 0.831341i $$0.312426\pi$$
$$524$$ 0 0
$$525$$ 2.95044 0.128768
$$526$$ 0 0
$$527$$ −1.55645 −0.0678000
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −4.77606 −0.207264
$$532$$ 0 0
$$533$$ 20.4411 0.885402
$$534$$ 0 0
$$535$$ 2.79231 0.120722
$$536$$ 0 0
$$537$$ −16.6436 −0.718226
$$538$$ 0 0
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −14.0095 −0.602314 −0.301157 0.953575i $$-0.597373\pi$$
−0.301157 + 0.953575i $$0.597373\pi$$
$$542$$ 0 0
$$543$$ 4.79231 0.205658
$$544$$ 0 0
$$545$$ −10.2478 −0.438967
$$546$$ 0 0
$$547$$ 3.15814 0.135032 0.0675161 0.997718i $$-0.478493\pi$$
0.0675161 + 0.997718i $$0.478493\pi$$
$$548$$ 0 0
$$549$$ 4.60601 0.196580
$$550$$ 0 0
$$551$$ 31.2591 1.33168
$$552$$ 0 0
$$553$$ −1.38207 −0.0587716
$$554$$ 0 0
$$555$$ 10.8180 0.459199
$$556$$ 0 0
$$557$$ −18.2496 −0.773262 −0.386631 0.922234i $$-0.626361\pi$$
−0.386631 + 0.922234i $$0.626361\pi$$
$$558$$ 0 0
$$559$$ −27.9975 −1.18417
$$560$$ 0 0
$$561$$ 4.51882 0.190785
$$562$$ 0 0
$$563$$ −17.8513 −0.752344 −0.376172 0.926550i $$-0.622760\pi$$
−0.376172 + 0.926550i $$0.622760\pi$$
$$564$$ 0 0
$$565$$ −0.671806 −0.0282631
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 9.69320 0.406360 0.203180 0.979141i $$-0.434872\pi$$
0.203180 + 0.979141i $$0.434872\pi$$
$$570$$ 0 0
$$571$$ −39.3821 −1.64809 −0.824044 0.566526i $$-0.808287\pi$$
−0.824044 + 0.566526i $$0.808287\pi$$
$$572$$ 0 0
$$573$$ 5.24533 0.219127
$$574$$ 0 0
$$575$$ 2.95044 0.123042
$$576$$ 0 0
$$577$$ −17.4530 −0.726579 −0.363289 0.931676i $$-0.618346\pi$$
−0.363289 + 0.931676i $$0.618346\pi$$
$$578$$ 0 0
$$579$$ 18.2830 0.759814
$$580$$ 0 0
$$581$$ 6.41970 0.266334
$$582$$ 0 0
$$583$$ 4.95044 0.205026
$$584$$ 0 0
$$585$$ 5.65556 0.233829
$$586$$ 0 0
$$587$$ −33.4693 −1.38142 −0.690712 0.723130i $$-0.742703\pi$$
−0.690712 + 0.723130i $$0.742703\pi$$
$$588$$ 0 0
$$589$$ 1.03331 0.0425769
$$590$$ 0 0
$$591$$ −5.77606 −0.237596
$$592$$ 0 0
$$593$$ −2.09911 −0.0862002 −0.0431001 0.999071i $$-0.513723\pi$$
−0.0431001 + 0.999071i $$0.513723\pi$$
$$594$$ 0 0
$$595$$ 6.46926 0.265214
$$596$$ 0 0
$$597$$ 9.29920 0.380591
$$598$$ 0 0
$$599$$ 45.2805 1.85011 0.925056 0.379832i $$-0.124018\pi$$
0.925056 + 0.379832i $$0.124018\pi$$
$$600$$ 0 0
$$601$$ −28.9924 −1.18262 −0.591312 0.806443i $$-0.701390\pi$$
−0.591312 + 0.806443i $$0.701390\pi$$
$$602$$ 0 0
$$603$$ −10.4693 −0.426341
$$604$$ 0 0
$$605$$ −14.3163 −0.582039
$$606$$ 0 0
$$607$$ −2.29920 −0.0933217 −0.0466609 0.998911i $$-0.514858\pi$$
−0.0466609 + 0.998911i $$0.514858\pi$$
$$608$$ 0 0
$$609$$ 10.4197 0.422228
$$610$$ 0 0
$$611$$ −29.7027 −1.20164
$$612$$ 0 0
$$613$$ 19.4573 0.785874 0.392937 0.919565i $$-0.371459\pi$$
0.392937 + 0.919565i $$0.371459\pi$$
$$614$$ 0 0
$$615$$ −7.40778 −0.298711
$$616$$ 0 0
$$617$$ 8.73843 0.351796 0.175898 0.984408i $$-0.443717\pi$$
0.175898 + 0.984408i $$0.443717\pi$$
$$618$$ 0 0
$$619$$ 36.0257 1.44800 0.723998 0.689802i $$-0.242303\pi$$
0.723998 + 0.689802i $$0.242303\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −2.81370 −0.112728
$$624$$ 0 0
$$625$$ −1.54266 −0.0617065
$$626$$ 0 0
$$627$$ −3.00000 −0.119808
$$628$$ 0 0
$$629$$ −34.1462 −1.36150
$$630$$ 0 0
$$631$$ −20.4435 −0.813845 −0.406922 0.913463i $$-0.633398\pi$$
−0.406922 + 0.913463i $$0.633398\pi$$
$$632$$ 0 0
$$633$$ −4.21201 −0.167412
$$634$$ 0 0
$$635$$ −3.87272 −0.153684
$$636$$ 0 0
$$637$$ 3.95044 0.156522
$$638$$ 0 0
$$639$$ −12.1248 −0.479651
$$640$$ 0 0
$$641$$ 27.8846 1.10138 0.550689 0.834711i $$-0.314365\pi$$
0.550689 + 0.834711i $$0.314365\pi$$
$$642$$ 0 0
$$643$$ −41.6864 −1.64395 −0.821976 0.569522i $$-0.807128\pi$$
−0.821976 + 0.569522i $$0.807128\pi$$
$$644$$ 0 0
$$645$$ 10.1462 0.399507
$$646$$ 0 0
$$647$$ −17.2334 −0.677515 −0.338757 0.940874i $$-0.610007\pi$$
−0.338757 + 0.940874i $$0.610007\pi$$
$$648$$ 0 0
$$649$$ 4.77606 0.187477
$$650$$ 0 0
$$651$$ 0.344438 0.0134996
$$652$$ 0 0
$$653$$ −42.2710 −1.65419 −0.827097 0.562060i $$-0.810009\pi$$
−0.827097 + 0.562060i $$0.810009\pi$$
$$654$$ 0 0
$$655$$ −15.6060 −0.609777
$$656$$ 0 0
$$657$$ 0.518817 0.0202410
$$658$$ 0 0
$$659$$ −12.6179 −0.491525 −0.245762 0.969330i $$-0.579038\pi$$
−0.245762 + 0.969330i $$0.579038\pi$$
$$660$$ 0 0
$$661$$ −40.4292 −1.57251 −0.786256 0.617901i $$-0.787983\pi$$
−0.786256 + 0.617901i $$0.787983\pi$$
$$662$$ 0 0
$$663$$ −17.8513 −0.693288
$$664$$ 0 0
$$665$$ −4.29488 −0.166548
$$666$$ 0 0
$$667$$ 10.4197 0.403453
$$668$$ 0 0
$$669$$ −22.0967 −0.854306
$$670$$ 0 0
$$671$$ −4.60601 −0.177813
$$672$$ 0 0
$$673$$ −29.2258 −1.12657 −0.563286 0.826262i $$-0.690463\pi$$
−0.563286 + 0.826262i $$0.690463\pi$$
$$674$$ 0 0
$$675$$ 2.95044 0.113563
$$676$$ 0 0
$$677$$ −2.83754 −0.109056 −0.0545278 0.998512i $$-0.517365\pi$$
−0.0545278 + 0.998512i $$0.517365\pi$$
$$678$$ 0 0
$$679$$ −6.51882 −0.250169
$$680$$ 0 0
$$681$$ 2.49311 0.0955360
$$682$$ 0 0
$$683$$ −7.63172 −0.292020 −0.146010 0.989283i $$-0.546643\pi$$
−0.146010 + 0.989283i $$0.546643\pi$$
$$684$$ 0 0
$$685$$ 11.8984 0.454616
$$686$$ 0 0
$$687$$ 23.0257 0.878486
$$688$$ 0 0
$$689$$ −19.5565 −0.745041
$$690$$ 0 0
$$691$$ 2.14867 0.0817392 0.0408696 0.999164i $$-0.486987\pi$$
0.0408696 + 0.999164i $$0.486987\pi$$
$$692$$ 0 0
$$693$$ −1.00000 −0.0379869
$$694$$ 0 0
$$695$$ −23.6856 −0.898446
$$696$$ 0 0
$$697$$ 23.3821 0.885659
$$698$$ 0 0
$$699$$ 20.5445 0.777065
$$700$$ 0 0
$$701$$ −24.5112 −0.925776 −0.462888 0.886417i $$-0.653187\pi$$
−0.462888 + 0.886417i $$0.653187\pi$$
$$702$$ 0 0
$$703$$ 22.6694 0.854991
$$704$$ 0 0
$$705$$ 10.7641 0.405401
$$706$$ 0 0
$$707$$ 12.8513 0.483324
$$708$$ 0 0
$$709$$ −44.5916 −1.67467 −0.837337 0.546687i $$-0.815889\pi$$
−0.837337 + 0.546687i $$0.815889\pi$$
$$710$$ 0 0
$$711$$ −1.38207 −0.0518317
$$712$$ 0 0
$$713$$ 0.344438 0.0128993
$$714$$ 0 0
$$715$$ −5.65556 −0.211506
$$716$$ 0 0
$$717$$ 13.6813 0.510937
$$718$$ 0 0
$$719$$ 33.1086 1.23474 0.617371 0.786672i $$-0.288198\pi$$
0.617371 + 0.786672i $$0.288198\pi$$
$$720$$ 0 0
$$721$$ 11.4197 0.425292
$$722$$ 0 0
$$723$$ 18.6274 0.692760
$$724$$ 0 0
$$725$$ 30.7428 1.14176
$$726$$ 0 0
$$727$$ 12.8770 0.477583 0.238792 0.971071i $$-0.423249\pi$$
0.238792 + 0.971071i $$0.423249\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −32.0257 −1.18451
$$732$$ 0 0
$$733$$ 4.17438 0.154184 0.0770921 0.997024i $$-0.475436\pi$$
0.0770921 + 0.997024i $$0.475436\pi$$
$$734$$ 0 0
$$735$$ −1.43163 −0.0528064
$$736$$ 0 0
$$737$$ 10.4693 0.385640
$$738$$ 0 0
$$739$$ −37.5001 −1.37946 −0.689732 0.724065i $$-0.742272\pi$$
−0.689732 + 0.724065i $$0.742272\pi$$
$$740$$ 0 0
$$741$$ 11.8513 0.435370
$$742$$ 0 0
$$743$$ 41.1010 1.50785 0.753924 0.656961i $$-0.228159\pi$$
0.753924 + 0.656961i $$0.228159\pi$$
$$744$$ 0 0
$$745$$ 19.4617 0.713020
$$746$$ 0 0
$$747$$ 6.41970 0.234885
$$748$$ 0 0
$$749$$ 1.95044 0.0712677
$$750$$ 0 0
$$751$$ −46.3701 −1.69207 −0.846035 0.533127i $$-0.821017\pi$$
−0.846035 + 0.533127i $$0.821017\pi$$
$$752$$ 0 0
$$753$$ 11.6274 0.423726
$$754$$ 0 0
$$755$$ 15.1129 0.550015
$$756$$ 0 0
$$757$$ −18.3872 −0.668295 −0.334147 0.942521i $$-0.608448\pi$$
−0.334147 + 0.942521i $$0.608448\pi$$
$$758$$ 0 0
$$759$$ −1.00000 −0.0362977
$$760$$ 0 0
$$761$$ −5.49497 −0.199193 −0.0995963 0.995028i $$-0.531755\pi$$
−0.0995963 + 0.995028i $$0.531755\pi$$
$$762$$ 0 0
$$763$$ −7.15814 −0.259142
$$764$$ 0 0
$$765$$ 6.46926 0.233897
$$766$$ 0 0
$$767$$ −18.8676 −0.681269
$$768$$ 0 0
$$769$$ −33.7027 −1.21535 −0.607675 0.794186i $$-0.707897\pi$$
−0.607675 + 0.794186i $$0.707897\pi$$
$$770$$ 0 0
$$771$$ −9.24533 −0.332962
$$772$$ 0 0
$$773$$ 30.4950 1.09683 0.548414 0.836207i $$-0.315232\pi$$
0.548414 + 0.836207i $$0.315232\pi$$
$$774$$ 0 0
$$775$$ 1.01624 0.0365045
$$776$$ 0 0
$$777$$ 7.55645 0.271086
$$778$$ 0 0
$$779$$ −15.5231 −0.556174
$$780$$ 0 0
$$781$$ 12.1248 0.433860
$$782$$ 0 0
$$783$$ 10.4197 0.372370
$$784$$ 0 0
$$785$$ 4.64610 0.165826
$$786$$ 0 0
$$787$$ −21.9128 −0.781107 −0.390554 0.920580i $$-0.627716\pi$$
−0.390554 + 0.920580i $$0.627716\pi$$
$$788$$ 0 0
$$789$$ −0.825621 −0.0293929
$$790$$ 0 0
$$791$$ −0.469261 −0.0166850
$$792$$ 0 0
$$793$$ 18.1958 0.646151
$$794$$ 0 0
$$795$$ 7.08719 0.251357
$$796$$ 0 0
$$797$$ −27.2873 −0.966565 −0.483283 0.875464i $$-0.660556\pi$$
−0.483283 + 0.875464i $$0.660556\pi$$
$$798$$ 0 0
$$799$$ −33.9762 −1.20199
$$800$$ 0 0
$$801$$ −2.81370 −0.0994171
$$802$$ 0 0
$$803$$ −0.518817 −0.0183086
$$804$$ 0 0
$$805$$ −1.43163 −0.0504582
$$806$$ 0 0
$$807$$ 15.2616 0.537233
$$808$$ 0 0
$$809$$ −16.5402 −0.581523 −0.290761 0.956796i $$-0.593909\pi$$
−0.290761 + 0.956796i $$0.593909\pi$$
$$810$$ 0 0
$$811$$ 3.89657 0.136827 0.0684135 0.997657i $$-0.478206\pi$$
0.0684135 + 0.997657i $$0.478206\pi$$
$$812$$ 0 0
$$813$$ −23.9718 −0.840729
$$814$$ 0 0
$$815$$ 24.5164 0.858771
$$816$$ 0 0
$$817$$ 21.2616 0.743848
$$818$$ 0 0
$$819$$ 3.95044 0.138040
$$820$$ 0 0
$$821$$ 19.3539 0.675456 0.337728 0.941244i $$-0.390342\pi$$
0.337728 + 0.941244i $$0.390342\pi$$
$$822$$ 0 0
$$823$$ −6.16246 −0.214810 −0.107405 0.994215i $$-0.534254\pi$$
−0.107405 + 0.994215i $$0.534254\pi$$
$$824$$ 0 0
$$825$$ −2.95044 −0.102721
$$826$$ 0 0
$$827$$ −24.2377 −0.842828 −0.421414 0.906868i $$-0.638466\pi$$
−0.421414 + 0.906868i $$0.638466\pi$$
$$828$$ 0 0
$$829$$ −5.72219 −0.198740 −0.0993699 0.995051i $$-0.531683\pi$$
−0.0993699 + 0.995051i $$0.531683\pi$$
$$830$$ 0 0
$$831$$ 5.29488 0.183677
$$832$$ 0 0
$$833$$ 4.51882 0.156568
$$834$$ 0 0
$$835$$ 11.2572 0.389573
$$836$$ 0 0
$$837$$ 0.344438 0.0119055
$$838$$ 0 0
$$839$$ 31.2052 1.07732 0.538662 0.842522i $$-0.318930\pi$$
0.538662 + 0.842522i $$0.318930\pi$$
$$840$$ 0 0
$$841$$ 79.5702 2.74380
$$842$$ 0 0
$$843$$ −16.1744 −0.557075
$$844$$ 0 0
$$845$$ 3.73083 0.128344
$$846$$ 0 0
$$847$$ −10.0000 −0.343604
$$848$$ 0 0
$$849$$ 4.29488 0.147400
$$850$$ 0 0
$$851$$ 7.55645 0.259032
$$852$$ 0 0
$$853$$ 11.7590 0.402620 0.201310 0.979528i $$-0.435480\pi$$
0.201310 + 0.979528i $$0.435480\pi$$
$$854$$ 0 0
$$855$$ −4.29488 −0.146882
$$856$$ 0 0
$$857$$ 14.1324 0.482754 0.241377 0.970431i $$-0.422401\pi$$
0.241377 + 0.970431i $$0.422401\pi$$
$$858$$ 0 0
$$859$$ 31.3111 1.06832 0.534161 0.845383i $$-0.320628\pi$$
0.534161 + 0.845383i $$0.320628\pi$$
$$860$$ 0 0
$$861$$ −5.17438 −0.176342
$$862$$ 0 0
$$863$$ −19.8256 −0.674872 −0.337436 0.941348i $$-0.609560\pi$$
−0.337436 + 0.941348i $$0.609560\pi$$
$$864$$ 0 0
$$865$$ 4.59222 0.156140
$$866$$ 0 0
$$867$$ −3.41970 −0.116139
$$868$$ 0 0
$$869$$ 1.38207 0.0468835
$$870$$ 0 0
$$871$$ −41.3582 −1.40137
$$872$$ 0 0
$$873$$ −6.51882 −0.220629
$$874$$ 0 0
$$875$$ −11.3821 −0.384784
$$876$$ 0 0
$$877$$ 6.04195 0.204022 0.102011 0.994783i $$-0.467472\pi$$
0.102011 + 0.994783i $$0.467472\pi$$
$$878$$ 0 0
$$879$$ 5.72651 0.193150
$$880$$ 0 0
$$881$$ −12.5231 −0.421915 −0.210958 0.977495i $$-0.567658\pi$$
−0.210958 + 0.977495i $$0.567658\pi$$
$$882$$ 0 0
$$883$$ 48.0771 1.61792 0.808962 0.587861i $$-0.200030\pi$$
0.808962 + 0.587861i $$0.200030\pi$$
$$884$$ 0 0
$$885$$ 6.83754 0.229842
$$886$$ 0 0
$$887$$ −2.69074 −0.0903462 −0.0451731 0.998979i $$-0.514384\pi$$
−0.0451731 + 0.998979i $$0.514384\pi$$
$$888$$ 0 0
$$889$$ −2.70512 −0.0907268
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 22.5565 0.754823
$$894$$ 0 0
$$895$$ 23.8275 0.796465
$$896$$ 0 0
$$897$$ 3.95044 0.131901
$$898$$ 0 0
$$899$$ 3.58894 0.119698
$$900$$ 0 0
$$901$$ −22.3701 −0.745258
$$902$$ 0 0
$$903$$ 7.08719 0.235847
$$904$$ 0 0
$$905$$ −6.86080 −0.228061
$$906$$ 0 0
$$907$$ −28.4931 −0.946098 −0.473049 0.881036i $$-0.656847\pi$$
−0.473049 + 0.881036i $$0.656847\pi$$
$$908$$ 0 0
$$909$$ 12.8513 0.426252
$$910$$ 0 0
$$911$$ 17.6274 0.584022 0.292011 0.956415i $$-0.405676\pi$$
0.292011 + 0.956415i $$0.405676\pi$$
$$912$$ 0 0
$$913$$ −6.41970 −0.212461
$$914$$ 0 0
$$915$$ −6.59408 −0.217994
$$916$$ 0 0
$$917$$ −10.9009 −0.359979
$$918$$ 0 0
$$919$$ 8.10343 0.267308 0.133654 0.991028i $$-0.457329\pi$$
0.133654 + 0.991028i $$0.457329\pi$$
$$920$$ 0 0
$$921$$ −15.1838 −0.500325
$$922$$ 0 0
$$923$$ −47.8984 −1.57660
$$924$$ 0 0
$$925$$ 22.2949 0.733051
$$926$$ 0 0
$$927$$ 11.4197 0.375072
$$928$$ 0 0
$$929$$ 14.0967 0.462496 0.231248 0.972895i $$-0.425719\pi$$
0.231248 + 0.972895i $$0.425719\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 0 0
$$933$$ 7.66749 0.251022
$$934$$ 0 0
$$935$$ −6.46926 −0.211567
$$936$$ 0 0
$$937$$ −36.6036 −1.19579 −0.597893 0.801576i $$-0.703995\pi$$
−0.597893 + 0.801576i $$0.703995\pi$$
$$938$$ 0 0
$$939$$ −26.8061 −0.874784
$$940$$ 0 0
$$941$$ −14.4240 −0.470210 −0.235105 0.971970i $$-0.575543\pi$$
−0.235105 + 0.971970i $$0.575543\pi$$
$$942$$ 0 0
$$943$$ −5.17438 −0.168501
$$944$$ 0 0
$$945$$ −1.43163 −0.0465708
$$946$$ 0 0
$$947$$ 17.7352 0.576315 0.288157 0.957583i $$-0.406957\pi$$
0.288157 + 0.957583i $$0.406957\pi$$
$$948$$ 0 0
$$949$$ 2.04956 0.0665314
$$950$$ 0 0
$$951$$ −2.05388 −0.0666015
$$952$$ 0 0
$$953$$ 44.1104 1.42888 0.714439 0.699698i $$-0.246682\pi$$
0.714439 + 0.699698i $$0.246682\pi$$
$$954$$ 0 0
$$955$$ −7.50935 −0.242997
$$956$$ 0 0
$$957$$ −10.4197 −0.336821
$$958$$ 0 0
$$959$$ 8.31112 0.268380
$$960$$ 0 0
$$961$$ −30.8814 −0.996173
$$962$$ 0 0
$$963$$ 1.95044 0.0628522
$$964$$ 0 0
$$965$$ −26.1744 −0.842583
$$966$$ 0 0
$$967$$ −16.6889 −0.536678 −0.268339 0.963325i $$-0.586475\pi$$
−0.268339 + 0.963325i $$0.586475\pi$$
$$968$$ 0 0
$$969$$ 13.5565 0.435496
$$970$$ 0 0
$$971$$ 44.1438 1.41664 0.708320 0.705891i $$-0.249453\pi$$
0.708320 + 0.705891i $$0.249453\pi$$
$$972$$ 0 0
$$973$$ −16.5445 −0.530393
$$974$$ 0 0
$$975$$ 11.6556 0.373277
$$976$$ 0 0
$$977$$ 25.7428 0.823584 0.411792 0.911278i $$-0.364903\pi$$
0.411792 + 0.911278i $$0.364903\pi$$
$$978$$ 0 0
$$979$$ 2.81370 0.0899262
$$980$$ 0 0
$$981$$ −7.15814 −0.228542
$$982$$ 0 0
$$983$$ −48.8437 −1.55787 −0.778937 0.627103i $$-0.784241\pi$$
−0.778937 + 0.627103i $$0.784241\pi$$
$$984$$ 0 0
$$985$$ 8.26917 0.263478
$$986$$ 0 0
$$987$$ 7.51882 0.239327
$$988$$ 0 0
$$989$$ 7.08719 0.225360
$$990$$ 0 0
$$991$$ −40.1343 −1.27491 −0.637454 0.770489i $$-0.720012\pi$$
−0.637454 + 0.770489i $$0.720012\pi$$
$$992$$ 0 0
$$993$$ −28.8727 −0.916248
$$994$$ 0 0
$$995$$ −13.3130 −0.422050
$$996$$ 0 0
$$997$$ 11.8727 0.376013 0.188006 0.982168i $$-0.439797\pi$$
0.188006 + 0.982168i $$0.439797\pi$$
$$998$$ 0 0
$$999$$ 7.55645 0.239076
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.br.1.2 3
4.3 odd 2 1932.2.a.j.1.2 3
12.11 even 2 5796.2.a.o.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.j.1.2 3 4.3 odd 2
5796.2.a.o.1.2 3 12.11 even 2
7728.2.a.br.1.2 3 1.1 even 1 trivial