# Properties

 Label 8036.2.a.b.1.1 Level $8036$ Weight $2$ Character 8036.1 Self dual yes Analytic conductor $64.168$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$8036 = 2^{2} \cdot 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8036.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1677830643$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1148) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8036.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +3.00000 q^{11} +2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} -3.00000 q^{19} -5.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} -2.00000 q^{29} +5.00000 q^{31} -3.00000 q^{33} +7.00000 q^{37} -2.00000 q^{39} +1.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} -3.00000 q^{47} -1.00000 q^{51} -3.00000 q^{53} -3.00000 q^{55} +3.00000 q^{57} +5.00000 q^{59} +3.00000 q^{61} -2.00000 q^{65} -13.0000 q^{67} +5.00000 q^{69} -1.00000 q^{73} +4.00000 q^{75} -11.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} -1.00000 q^{85} +2.00000 q^{87} +5.00000 q^{89} -5.00000 q^{93} +3.00000 q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 0 0
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −5.00000 −1.04257 −0.521286 0.853382i $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ −3.00000 −0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 1.00000 0.156174
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 3.00000 0.397360
$$58$$ 0 0
$$59$$ 5.00000 0.650945 0.325472 0.945552i $$-0.394477\pi$$
0.325472 + 0.945552i $$0.394477\pi$$
$$60$$ 0 0
$$61$$ 3.00000 0.384111 0.192055 0.981384i $$-0.438485\pi$$
0.192055 + 0.981384i $$0.438485\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −13.0000 −1.58820 −0.794101 0.607785i $$-0.792058\pi$$
−0.794101 + 0.607785i $$0.792058\pi$$
$$68$$ 0 0
$$69$$ 5.00000 0.601929
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 5.00000 0.529999 0.264999 0.964249i $$-0.414628\pi$$
0.264999 + 0.964249i $$0.414628\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.00000 −0.518476
$$94$$ 0 0
$$95$$ 3.00000 0.307794
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 7.00000 0.689730 0.344865 0.938652i $$-0.387925\pi$$
0.344865 + 0.938652i $$0.387925\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1.00000 −0.0966736 −0.0483368 0.998831i $$-0.515392\pi$$
−0.0483368 + 0.998831i $$0.515392\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 5.00000 0.466252
$$116$$ 0 0
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ −1.00000 −0.0901670
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 7.00000 0.611593 0.305796 0.952097i $$-0.401077\pi$$
0.305796 + 0.952097i $$0.401077\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ 5.00000 0.427179 0.213589 0.976924i $$-0.431485\pi$$
0.213589 + 0.976924i $$0.431485\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ 3.00000 0.244137 0.122068 0.992522i $$-0.461047\pi$$
0.122068 + 0.992522i $$0.461047\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ −5.00000 −0.401610
$$156$$ 0 0
$$157$$ −3.00000 −0.239426 −0.119713 0.992809i $$-0.538197\pi$$
−0.119713 + 0.992809i $$0.538197\pi$$
$$158$$ 0 0
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1.00000 −0.0783260 −0.0391630 0.999233i $$-0.512469\pi$$
−0.0391630 + 0.999233i $$0.512469\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ 0 0
$$173$$ 11.0000 0.836315 0.418157 0.908375i $$-0.362676\pi$$
0.418157 + 0.908375i $$0.362676\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −5.00000 −0.375823
$$178$$ 0 0
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ −3.00000 −0.221766
$$184$$ 0 0
$$185$$ −7.00000 −0.514650
$$186$$ 0 0
$$187$$ 3.00000 0.219382
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −7.00000 −0.506502 −0.253251 0.967401i $$-0.581500\pi$$
−0.253251 + 0.967401i $$0.581500\pi$$
$$192$$ 0 0
$$193$$ −7.00000 −0.503871 −0.251936 0.967744i $$-0.581067\pi$$
−0.251936 + 0.967744i $$0.581067\pi$$
$$194$$ 0 0
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 19.0000 1.34687 0.673437 0.739244i $$-0.264817\pi$$
0.673437 + 0.739244i $$0.264817\pi$$
$$200$$ 0 0
$$201$$ 13.0000 0.916949
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1.00000 −0.0698430
$$206$$ 0 0
$$207$$ 10.0000 0.695048
$$208$$ 0 0
$$209$$ −9.00000 −0.622543
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1.00000 0.0675737
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ 0 0
$$225$$ 8.00000 0.533333
$$226$$ 0 0
$$227$$ 27.0000 1.79205 0.896026 0.444001i $$-0.146441\pi$$
0.896026 + 0.444001i $$0.146441\pi$$
$$228$$ 0 0
$$229$$ 1.00000 0.0660819 0.0330409 0.999454i $$-0.489481\pi$$
0.0330409 + 0.999454i $$0.489481\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ 0 0
$$235$$ 3.00000 0.195698
$$236$$ 0 0
$$237$$ 11.0000 0.714527
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 3.00000 0.193247 0.0966235 0.995321i $$-0.469196\pi$$
0.0966235 + 0.995321i $$0.469196\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −8.00000 −0.504956 −0.252478 0.967603i $$-0.581245\pi$$
−0.252478 + 0.967603i $$0.581245\pi$$
$$252$$ 0 0
$$253$$ −15.0000 −0.943042
$$254$$ 0 0
$$255$$ 1.00000 0.0626224
$$256$$ 0 0
$$257$$ 1.00000 0.0623783 0.0311891 0.999514i $$-0.490071\pi$$
0.0311891 + 0.999514i $$0.490071\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 0 0
$$263$$ −25.0000 −1.54157 −0.770783 0.637098i $$-0.780135\pi$$
−0.770783 + 0.637098i $$0.780135\pi$$
$$264$$ 0 0
$$265$$ 3.00000 0.184289
$$266$$ 0 0
$$267$$ −5.00000 −0.305995
$$268$$ 0 0
$$269$$ −9.00000 −0.548740 −0.274370 0.961624i $$-0.588469\pi$$
−0.274370 + 0.961624i $$0.588469\pi$$
$$270$$ 0 0
$$271$$ 27.0000 1.64013 0.820067 0.572268i $$-0.193936\pi$$
0.820067 + 0.572268i $$0.193936\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ −17.0000 −1.02143 −0.510716 0.859750i $$-0.670619\pi$$
−0.510716 + 0.859750i $$0.670619\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ 9.00000 0.534994 0.267497 0.963559i $$-0.413803\pi$$
0.267497 + 0.963559i $$0.413803\pi$$
$$284$$ 0 0
$$285$$ −3.00000 −0.177705
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 0 0
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ 0 0
$$295$$ −5.00000 −0.291111
$$296$$ 0 0
$$297$$ 15.0000 0.870388
$$298$$ 0 0
$$299$$ −10.0000 −0.578315
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 3.00000 0.172345
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −7.00000 −0.398216
$$310$$ 0 0
$$311$$ −13.0000 −0.737162 −0.368581 0.929596i $$-0.620156\pi$$
−0.368581 + 0.929596i $$0.620156\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.00000 0.280828 0.140414 0.990093i $$-0.455157\pi$$
0.140414 + 0.990093i $$0.455157\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ 1.00000 0.0558146
$$322$$ 0 0
$$323$$ −3.00000 −0.166924
$$324$$ 0 0
$$325$$ −8.00000 −0.443760
$$326$$ 0 0
$$327$$ 7.00000 0.387101
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 21.0000 1.15426 0.577132 0.816651i $$-0.304172\pi$$
0.577132 + 0.816651i $$0.304172\pi$$
$$332$$ 0 0
$$333$$ −14.0000 −0.767195
$$334$$ 0 0
$$335$$ 13.0000 0.710266
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −5.00000 −0.269191
$$346$$ 0 0
$$347$$ 7.00000 0.375780 0.187890 0.982190i $$-0.439835\pi$$
0.187890 + 0.982190i $$0.439835\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 10.0000 0.533761
$$352$$ 0 0
$$353$$ −13.0000 −0.691920 −0.345960 0.938249i $$-0.612447\pi$$
−0.345960 + 0.938249i $$0.612447\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −25.0000 −1.31945 −0.659725 0.751507i $$-0.729327\pi$$
−0.659725 + 0.751507i $$0.729327\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 0 0
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ 1.00000 0.0523424
$$366$$ 0 0
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ 0 0
$$375$$ −9.00000 −0.464758
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 9.00000 0.459879 0.229939 0.973205i $$-0.426147\pi$$
0.229939 + 0.973205i $$0.426147\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ −25.0000 −1.26755 −0.633775 0.773517i $$-0.718496\pi$$
−0.633775 + 0.773517i $$0.718496\pi$$
$$390$$ 0 0
$$391$$ −5.00000 −0.252861
$$392$$ 0 0
$$393$$ −7.00000 −0.353103
$$394$$ 0 0
$$395$$ 11.0000 0.553470
$$396$$ 0 0
$$397$$ −15.0000 −0.752828 −0.376414 0.926451i $$-0.622843\pi$$
−0.376414 + 0.926451i $$0.622843\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −13.0000 −0.649189 −0.324595 0.945853i $$-0.605228\pi$$
−0.324595 + 0.945853i $$0.605228\pi$$
$$402$$ 0 0
$$403$$ 10.0000 0.498135
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 21.0000 1.04093
$$408$$ 0 0
$$409$$ −9.00000 −0.445021 −0.222511 0.974930i $$-0.571425\pi$$
−0.222511 + 0.974930i $$0.571425\pi$$
$$410$$ 0 0
$$411$$ −5.00000 −0.246632
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ −19.0000 −0.915198 −0.457599 0.889159i $$-0.651290\pi$$
−0.457599 + 0.889159i $$0.651290\pi$$
$$432$$ 0 0
$$433$$ 6.00000 0.288342 0.144171 0.989553i $$-0.453949\pi$$
0.144171 + 0.989553i $$0.453949\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ 0 0
$$437$$ 15.0000 0.717547
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9.00000 −0.427603 −0.213801 0.976877i $$-0.568585\pi$$
−0.213801 + 0.976877i $$0.568585\pi$$
$$444$$ 0 0
$$445$$ −5.00000 −0.237023
$$446$$ 0 0
$$447$$ −21.0000 −0.993266
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 3.00000 0.141264
$$452$$ 0 0
$$453$$ −3.00000 −0.140952
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.00000 −0.140334 −0.0701670 0.997535i $$-0.522353\pi$$
−0.0701670 + 0.997535i $$0.522353\pi$$
$$458$$ 0 0
$$459$$ 5.00000 0.233380
$$460$$ 0 0
$$461$$ 26.0000 1.21094 0.605470 0.795868i $$-0.292985\pi$$
0.605470 + 0.795868i $$0.292985\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 0 0
$$465$$ 5.00000 0.231869
$$466$$ 0 0
$$467$$ −33.0000 −1.52706 −0.763529 0.645774i $$-0.776535\pi$$
−0.763529 + 0.645774i $$0.776535\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 3.00000 0.138233
$$472$$ 0 0
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ 12.0000 0.550598
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 19.0000 0.868132 0.434066 0.900881i $$-0.357078\pi$$
0.434066 + 0.900881i $$0.357078\pi$$
$$480$$ 0 0
$$481$$ 14.0000 0.638345
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ −11.0000 −0.498458 −0.249229 0.968445i $$-0.580177\pi$$
−0.249229 + 0.968445i $$0.580177\pi$$
$$488$$ 0 0
$$489$$ 1.00000 0.0452216
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ −2.00000 −0.0900755
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −7.00000 −0.313363 −0.156682 0.987649i $$-0.550080\pi$$
−0.156682 + 0.987649i $$0.550080\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 3.00000 0.133498
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 0 0
$$509$$ −11.0000 −0.487566 −0.243783 0.969830i $$-0.578389\pi$$
−0.243783 + 0.969830i $$0.578389\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −15.0000 −0.662266
$$514$$ 0 0
$$515$$ −7.00000 −0.308457
$$516$$ 0 0
$$517$$ −9.00000 −0.395820
$$518$$ 0 0
$$519$$ −11.0000 −0.482846
$$520$$ 0 0
$$521$$ 13.0000 0.569540 0.284770 0.958596i $$-0.408083\pi$$
0.284770 + 0.958596i $$0.408083\pi$$
$$522$$ 0 0
$$523$$ 11.0000 0.480996 0.240498 0.970650i $$-0.422689\pi$$
0.240498 + 0.970650i $$0.422689\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 5.00000 0.217803
$$528$$ 0 0
$$529$$ 2.00000 0.0869565
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 0 0
$$533$$ 2.00000 0.0866296
$$534$$ 0 0
$$535$$ 1.00000 0.0432338
$$536$$ 0 0
$$537$$ −3.00000 −0.129460
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.00000 −0.0429934 −0.0214967 0.999769i $$-0.506843\pi$$
−0.0214967 + 0.999769i $$0.506843\pi$$
$$542$$ 0 0
$$543$$ 22.0000 0.944110
$$544$$ 0 0
$$545$$ 7.00000 0.299847
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 7.00000 0.297133
$$556$$ 0 0
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −3.00000 −0.126660
$$562$$ 0 0
$$563$$ −21.0000 −0.885044 −0.442522 0.896758i $$-0.645916\pi$$
−0.442522 + 0.896758i $$0.645916\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −45.0000 −1.88650 −0.943249 0.332086i $$-0.892248\pi$$
−0.943249 + 0.332086i $$0.892248\pi$$
$$570$$ 0 0
$$571$$ −5.00000 −0.209243 −0.104622 0.994512i $$-0.533363\pi$$
−0.104622 + 0.994512i $$0.533363\pi$$
$$572$$ 0 0
$$573$$ 7.00000 0.292429
$$574$$ 0 0
$$575$$ 20.0000 0.834058
$$576$$ 0 0
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ 0 0
$$579$$ 7.00000 0.290910
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ 4.00000 0.165380
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ −15.0000 −0.618064
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 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
$$597$$ −19.0000 −0.777618
$$598$$ 0 0
$$599$$ 17.0000 0.694601 0.347301 0.937754i $$-0.387098\pi$$
0.347301 + 0.937754i $$0.387098\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 0 0
$$603$$ 26.0000 1.05880
$$604$$ 0 0
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ −29.0000 −1.17130 −0.585649 0.810564i $$-0.699160\pi$$
−0.585649 + 0.810564i $$0.699160\pi$$
$$614$$ 0 0
$$615$$ 1.00000 0.0403239
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ 1.00000 0.0401934 0.0200967 0.999798i $$-0.493603\pi$$
0.0200967 + 0.999798i $$0.493603\pi$$
$$620$$ 0 0
$$621$$ −25.0000 −1.00322
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 9.00000 0.359425
$$628$$ 0 0
$$629$$ 7.00000 0.279108
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −35.0000 −1.38242 −0.691208 0.722655i $$-0.742921\pi$$
−0.691208 + 0.722655i $$0.742921\pi$$
$$642$$ 0 0
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ 0 0
$$647$$ 9.00000 0.353827 0.176913 0.984226i $$-0.443389\pi$$
0.176913 + 0.984226i $$0.443389\pi$$
$$648$$ 0 0
$$649$$ 15.0000 0.588802
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −43.0000 −1.68272 −0.841360 0.540475i $$-0.818245\pi$$
−0.841360 + 0.540475i $$0.818245\pi$$
$$654$$ 0 0
$$655$$ −7.00000 −0.273513
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −45.0000 −1.75030 −0.875149 0.483854i $$-0.839236\pi$$
−0.875149 + 0.483854i $$0.839236\pi$$
$$662$$ 0 0
$$663$$ −2.00000 −0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.0000 0.387202
$$668$$ 0 0
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ 9.00000 0.347441
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ 0 0
$$677$$ 27.0000 1.03769 0.518847 0.854867i $$-0.326361\pi$$
0.518847 + 0.854867i $$0.326361\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −27.0000 −1.03464
$$682$$ 0 0
$$683$$ −5.00000 −0.191320 −0.0956598 0.995414i $$-0.530496\pi$$
−0.0956598 + 0.995414i $$0.530496\pi$$
$$684$$ 0 0
$$685$$ −5.00000 −0.191040
$$686$$ 0 0
$$687$$ −1.00000 −0.0381524
$$688$$ 0 0
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ −11.0000 −0.418460 −0.209230 0.977866i $$-0.567096\pi$$
−0.209230 + 0.977866i $$0.567096\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 1.00000 0.0378777
$$698$$ 0 0
$$699$$ 27.0000 1.02123
$$700$$ 0 0
$$701$$ −14.0000 −0.528773 −0.264386 0.964417i $$-0.585169\pi$$
−0.264386 + 0.964417i $$0.585169\pi$$
$$702$$ 0 0
$$703$$ −21.0000 −0.792030
$$704$$ 0 0
$$705$$ −3.00000 −0.112987
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 5.00000 0.187779 0.0938895 0.995583i $$-0.470070\pi$$
0.0938895 + 0.995583i $$0.470070\pi$$
$$710$$ 0 0
$$711$$ 22.0000 0.825064
$$712$$ 0 0
$$713$$ −25.0000 −0.936257
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ 0 0
$$717$$ 16.0000 0.597531
$$718$$ 0 0
$$719$$ 45.0000 1.67822 0.839108 0.543964i $$-0.183077\pi$$
0.839108 + 0.543964i $$0.183077\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −3.00000 −0.111571
$$724$$ 0 0
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ 7.00000 0.258551 0.129275 0.991609i $$-0.458735\pi$$
0.129275 + 0.991609i $$0.458735\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −39.0000 −1.43658
$$738$$ 0 0
$$739$$ −23.0000 −0.846069 −0.423034 0.906114i $$-0.639035\pi$$
−0.423034 + 0.906114i $$0.639035\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ −21.0000 −0.769380
$$746$$ 0 0
$$747$$ −8.00000 −0.292705
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ −3.00000 −0.109181
$$756$$ 0 0
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ 0 0
$$759$$ 15.0000 0.544466
$$760$$ 0 0
$$761$$ −13.0000 −0.471250 −0.235625 0.971844i $$-0.575714\pi$$
−0.235625 + 0.971844i $$0.575714\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ 10.0000 0.361079
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −1.00000 −0.0360141
$$772$$ 0 0
$$773$$ −3.00000 −0.107903 −0.0539513 0.998544i $$-0.517182\pi$$
−0.0539513 + 0.998544i $$0.517182\pi$$
$$774$$ 0 0
$$775$$ −20.0000 −0.718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3.00000 −0.107486
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 0 0
$$785$$ 3.00000 0.107075
$$786$$ 0 0
$$787$$ −11.0000 −0.392108 −0.196054 0.980593i $$-0.562813\pi$$
−0.196054 + 0.980593i $$0.562813\pi$$
$$788$$ 0 0
$$789$$ 25.0000 0.890024
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6.00000 0.213066
$$794$$ 0 0
$$795$$ −3.00000 −0.106399
$$796$$ 0 0
$$797$$ −34.0000 −1.20434 −0.602171 0.798367i $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 0 0
$$799$$ −3.00000 −0.106132
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 0 0
$$803$$ −3.00000 −0.105868
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 9.00000 0.316815
$$808$$ 0 0
$$809$$ 41.0000 1.44148 0.720742 0.693204i $$-0.243801\pi$$
0.720742 + 0.693204i $$0.243801\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ 0 0
$$813$$ −27.0000 −0.946931
$$814$$ 0 0
$$815$$ 1.00000 0.0350285
$$816$$ 0 0
$$817$$ −12.0000 −0.419827
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31.0000 1.08191 0.540954 0.841052i $$-0.318063\pi$$
0.540954 + 0.841052i $$0.318063\pi$$
$$822$$ 0 0
$$823$$ −13.0000 −0.453152 −0.226576 0.973994i $$-0.572753\pi$$
−0.226576 + 0.973994i $$0.572753\pi$$
$$824$$ 0 0
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ 40.0000 1.39094 0.695468 0.718557i $$-0.255197\pi$$
0.695468 + 0.718557i $$0.255197\pi$$
$$828$$ 0 0
$$829$$ 7.00000 0.243120 0.121560 0.992584i $$-0.461210\pi$$
0.121560 + 0.992584i $$0.461210\pi$$
$$830$$ 0 0
$$831$$ 17.0000 0.589723
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 25.0000 0.864126
$$838$$ 0 0
$$839$$ −28.0000 −0.966667 −0.483334 0.875436i $$-0.660574\pi$$
−0.483334 + 0.875436i $$0.660574\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 22.0000 0.757720
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9.00000 −0.308879
$$850$$ 0 0
$$851$$ −35.0000 −1.19978
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ −6.00000 −0.205196
$$856$$ 0 0
$$857$$ −9.00000 −0.307434 −0.153717 0.988115i $$-0.549124\pi$$
−0.153717 + 0.988115i $$0.549124\pi$$
$$858$$ 0 0
$$859$$ −25.0000 −0.852989 −0.426494 0.904490i $$-0.640252\pi$$
−0.426494 + 0.904490i $$0.640252\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 27.0000 0.919091 0.459545 0.888154i $$-0.348012\pi$$
0.459545 + 0.888154i $$0.348012\pi$$
$$864$$ 0 0
$$865$$ −11.0000 −0.374011
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ −33.0000 −1.11945
$$870$$ 0 0
$$871$$ −26.0000 −0.880976
$$872$$ 0 0
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −5.00000 −0.168838 −0.0844190 0.996430i $$-0.526903\pi$$
−0.0844190 + 0.996430i $$0.526903\pi$$
$$878$$ 0 0
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ 38.0000 1.28025 0.640126 0.768270i $$-0.278882\pi$$
0.640126 + 0.768270i $$0.278882\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ 5.00000 0.168073
$$886$$ 0 0
$$887$$ −15.0000 −0.503651 −0.251825 0.967773i $$-0.581031\pi$$
−0.251825 + 0.967773i $$0.581031\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 0 0
$$893$$ 9.00000 0.301174
$$894$$ 0 0
$$895$$ −3.00000 −0.100279
$$896$$ 0 0
$$897$$ 10.0000 0.333890
$$898$$ 0 0
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 22.0000 0.731305
$$906$$ 0 0
$$907$$ 13.0000 0.431658 0.215829 0.976431i $$-0.430755\pi$$
0.215829 + 0.976431i $$0.430755\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 3.00000 0.0991769
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 0 0
$$921$$ 16.0000 0.527218
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −28.0000 −0.920634
$$926$$ 0 0
$$927$$ −14.0000 −0.459820
$$928$$ 0 0
$$929$$ −39.0000 −1.27955 −0.639774 0.768563i $$-0.720972\pi$$
−0.639774 + 0.768563i $$0.720972\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 13.0000 0.425601
$$934$$ 0 0
$$935$$ −3.00000 −0.0981105
$$936$$ 0 0
$$937$$ −58.0000 −1.89478 −0.947389 0.320085i $$-0.896288\pi$$
−0.947389 + 0.320085i $$0.896288\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ −49.0000 −1.59735 −0.798677 0.601760i $$-0.794466\pi$$
−0.798677 + 0.601760i $$0.794466\pi$$
$$942$$ 0 0
$$943$$ −5.00000 −0.162822
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 35.0000 1.13735 0.568674 0.822563i $$-0.307457\pi$$
0.568674 + 0.822563i $$0.307457\pi$$
$$948$$ 0 0
$$949$$ −2.00000 −0.0649227
$$950$$ 0 0
$$951$$ −5.00000 −0.162136
$$952$$ 0 0
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ 7.00000 0.226515
$$956$$ 0 0
$$957$$ 6.00000 0.193952
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ 0 0
$$965$$ 7.00000 0.225338
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ 57.0000 1.82922 0.914609 0.404341i $$-0.132499\pi$$
0.914609 + 0.404341i $$0.132499\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 8.00000 0.256205
$$976$$ 0 0
$$977$$ 57.0000 1.82359 0.911796 0.410644i $$-0.134696\pi$$
0.911796 + 0.410644i $$0.134696\pi$$
$$978$$ 0 0
$$979$$ 15.0000 0.479402
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 0 0
$$983$$ 45.0000 1.43528 0.717639 0.696416i $$-0.245223\pi$$
0.717639 + 0.696416i $$0.245223\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −20.0000 −0.635963
$$990$$ 0 0
$$991$$ −57.0000 −1.81066 −0.905332 0.424704i $$-0.860378\pi$$
−0.905332 + 0.424704i $$0.860378\pi$$
$$992$$ 0 0
$$993$$ −21.0000 −0.666415
$$994$$ 0 0
$$995$$ −19.0000 −0.602340
$$996$$ 0 0
$$997$$ 13.0000 0.411714 0.205857 0.978582i $$-0.434002\pi$$
0.205857 + 0.978582i $$0.434002\pi$$
$$998$$ 0 0
$$999$$ 35.0000 1.10735
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 8036.2.a.b.1.1 1
7.2 even 3 1148.2.i.c.165.1 2
7.4 even 3 1148.2.i.c.821.1 yes 2
7.6 odd 2 8036.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.i.c.165.1 2 7.2 even 3
1148.2.i.c.821.1 yes 2 7.4 even 3
8036.2.a.b.1.1 1 1.1 even 1 trivial
8036.2.a.f.1.1 1 7.6 odd 2