# Properties

 Label 7728.2.a.bv.1.3 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: 3.3.1509.1 Defining polynomial: $$x^{3} - x^{2} - 7x + 4$$ x^3 - x^2 - 7*x + 4 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 1.3 Root $$-2.47735$$ of defining polynomial Character $$\chi$$ $$=$$ 7728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +2.47735 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +2.47735 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.95470 q^{11} -0.137275 q^{13} +2.47735 q^{15} +4.61463 q^{17} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.13727 q^{25} +1.00000 q^{27} -1.65992 q^{29} +2.61463 q^{31} -5.95470 q^{33} -2.47735 q^{35} -11.5693 q^{37} -0.137275 q^{39} -3.68016 q^{41} -11.0920 q^{43} +2.47735 q^{45} -5.34008 q^{47} +1.00000 q^{49} +4.61463 q^{51} -5.13727 q^{53} -14.7519 q^{55} -1.00000 q^{57} -5.13727 q^{59} +4.52265 q^{61} -1.00000 q^{63} -0.340078 q^{65} -2.47735 q^{67} -1.00000 q^{69} +11.3665 q^{71} -0.340078 q^{73} +1.13727 q^{75} +5.95470 q^{77} +12.5240 q^{79} +1.00000 q^{81} -1.65992 q^{83} +11.4321 q^{85} -1.65992 q^{87} +3.86273 q^{89} +0.137275 q^{91} +2.61463 q^{93} -2.47735 q^{95} -2.88918 q^{97} -5.95470 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} + 3 q^{39} - q^{41} - 13 q^{43} - q^{45} - 11 q^{47} + 3 q^{49} + 2 q^{51} - 12 q^{53} - 29 q^{55} - 3 q^{57} - 12 q^{59} + 22 q^{61} - 3 q^{63} + 4 q^{65} + q^{67} - 3 q^{69} + 7 q^{71} + 4 q^{73} + q^{77} - 8 q^{79} + 3 q^{81} - 10 q^{83} + 9 q^{85} - 10 q^{87} + 15 q^{89} - 3 q^{91} - 4 q^{93} + q^{95} + 10 q^{97} - q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 - q^15 + 2 * q^17 - 3 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 10 * q^29 - 4 * q^31 - q^33 + q^35 - 6 * q^37 + 3 * q^39 - q^41 - 13 * q^43 - q^45 - 11 * q^47 + 3 * q^49 + 2 * q^51 - 12 * q^53 - 29 * q^55 - 3 * q^57 - 12 * q^59 + 22 * q^61 - 3 * q^63 + 4 * q^65 + q^67 - 3 * q^69 + 7 * q^71 + 4 * q^73 + q^77 - 8 * q^79 + 3 * q^81 - 10 * q^83 + 9 * q^85 - 10 * q^87 + 15 * q^89 - 3 * q^91 - 4 * q^93 + q^95 + 10 * q^97 - q^99

## 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
$$5$$ 2.47735 1.10791 0.553953 0.832548i $$-0.313119\pi$$
0.553953 + 0.832548i $$0.313119\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.95470 −1.79541 −0.897706 0.440596i $$-0.854767\pi$$
−0.897706 + 0.440596i $$0.854767\pi$$
$$12$$ 0 0
$$13$$ −0.137275 −0.0380731 −0.0190366 0.999819i $$-0.506060\pi$$
−0.0190366 + 0.999819i $$0.506060\pi$$
$$14$$ 0 0
$$15$$ 2.47735 0.639650
$$16$$ 0 0
$$17$$ 4.61463 1.11921 0.559606 0.828759i $$-0.310953\pi$$
0.559606 + 0.828759i $$0.310953\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.13727 0.227455
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.65992 −0.308240 −0.154120 0.988052i $$-0.549254\pi$$
−0.154120 + 0.988052i $$0.549254\pi$$
$$30$$ 0 0
$$31$$ 2.61463 0.469601 0.234800 0.972044i $$-0.424556\pi$$
0.234800 + 0.972044i $$0.424556\pi$$
$$32$$ 0 0
$$33$$ −5.95470 −1.03658
$$34$$ 0 0
$$35$$ −2.47735 −0.418749
$$36$$ 0 0
$$37$$ −11.5693 −1.90199 −0.950993 0.309212i $$-0.899935\pi$$
−0.950993 + 0.309212i $$0.899935\pi$$
$$38$$ 0 0
$$39$$ −0.137275 −0.0219815
$$40$$ 0 0
$$41$$ −3.68016 −0.574744 −0.287372 0.957819i $$-0.592782\pi$$
−0.287372 + 0.957819i $$0.592782\pi$$
$$42$$ 0 0
$$43$$ −11.0920 −1.69151 −0.845755 0.533571i $$-0.820850\pi$$
−0.845755 + 0.533571i $$0.820850\pi$$
$$44$$ 0 0
$$45$$ 2.47735 0.369302
$$46$$ 0 0
$$47$$ −5.34008 −0.778930 −0.389465 0.921041i $$-0.627340\pi$$
−0.389465 + 0.921041i $$0.627340\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.61463 0.646177
$$52$$ 0 0
$$53$$ −5.13727 −0.705659 −0.352829 0.935688i $$-0.614780\pi$$
−0.352829 + 0.935688i $$0.614780\pi$$
$$54$$ 0 0
$$55$$ −14.7519 −1.98915
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −5.13727 −0.668816 −0.334408 0.942428i $$-0.608536\pi$$
−0.334408 + 0.942428i $$0.608536\pi$$
$$60$$ 0 0
$$61$$ 4.52265 0.579066 0.289533 0.957168i $$-0.406500\pi$$
0.289533 + 0.957168i $$0.406500\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −0.340078 −0.0421814
$$66$$ 0 0
$$67$$ −2.47735 −0.302657 −0.151328 0.988484i $$-0.548355\pi$$
−0.151328 + 0.988484i $$0.548355\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 11.3665 1.34896 0.674479 0.738294i $$-0.264368\pi$$
0.674479 + 0.738294i $$0.264368\pi$$
$$72$$ 0 0
$$73$$ −0.340078 −0.0398031 −0.0199015 0.999802i $$-0.506335\pi$$
−0.0199015 + 0.999802i $$0.506335\pi$$
$$74$$ 0 0
$$75$$ 1.13727 0.131321
$$76$$ 0 0
$$77$$ 5.95470 0.678602
$$78$$ 0 0
$$79$$ 12.5240 1.40906 0.704532 0.709672i $$-0.251157\pi$$
0.704532 + 0.709672i $$0.251157\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −1.65992 −0.182200 −0.0911001 0.995842i $$-0.529038\pi$$
−0.0911001 + 0.995842i $$0.529038\pi$$
$$84$$ 0 0
$$85$$ 11.4321 1.23998
$$86$$ 0 0
$$87$$ −1.65992 −0.177962
$$88$$ 0 0
$$89$$ 3.86273 0.409448 0.204724 0.978820i $$-0.434370\pi$$
0.204724 + 0.978820i $$0.434370\pi$$
$$90$$ 0 0
$$91$$ 0.137275 0.0143903
$$92$$ 0 0
$$93$$ 2.61463 0.271124
$$94$$ 0 0
$$95$$ −2.47735 −0.254171
$$96$$ 0 0
$$97$$ −2.88918 −0.293351 −0.146676 0.989185i $$-0.546857\pi$$
−0.146676 + 0.989185i $$0.546857\pi$$
$$98$$ 0 0
$$99$$ −5.95470 −0.598470
$$100$$ 0 0
$$101$$ −19.3212 −1.92253 −0.961267 0.275618i $$-0.911118\pi$$
−0.961267 + 0.275618i $$0.911118\pi$$
$$102$$ 0 0
$$103$$ −6.38537 −0.629169 −0.314585 0.949229i $$-0.601865\pi$$
−0.314585 + 0.949229i $$0.601865\pi$$
$$104$$ 0 0
$$105$$ −2.47735 −0.241765
$$106$$ 0 0
$$107$$ −0.908021 −0.0877817 −0.0438908 0.999036i $$-0.513975\pi$$
−0.0438908 + 0.999036i $$0.513975\pi$$
$$108$$ 0 0
$$109$$ 6.75190 0.646715 0.323357 0.946277i $$-0.395188\pi$$
0.323357 + 0.946277i $$0.395188\pi$$
$$110$$ 0 0
$$111$$ −11.5693 −1.09811
$$112$$ 0 0
$$113$$ 18.6613 1.75551 0.877754 0.479111i $$-0.159041\pi$$
0.877754 + 0.479111i $$0.159041\pi$$
$$114$$ 0 0
$$115$$ −2.47735 −0.231014
$$116$$ 0 0
$$117$$ −0.137275 −0.0126910
$$118$$ 0 0
$$119$$ −4.61463 −0.423022
$$120$$ 0 0
$$121$$ 24.4585 2.22350
$$122$$ 0 0
$$123$$ −3.68016 −0.331828
$$124$$ 0 0
$$125$$ −9.56933 −0.855907
$$126$$ 0 0
$$127$$ 4.15751 0.368919 0.184460 0.982840i $$-0.440947\pi$$
0.184460 + 0.982840i $$0.440947\pi$$
$$128$$ 0 0
$$129$$ −11.0920 −0.976594
$$130$$ 0 0
$$131$$ −14.2293 −1.24322 −0.621608 0.783329i $$-0.713520\pi$$
−0.621608 + 0.783329i $$0.713520\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 2.47735 0.213217
$$136$$ 0 0
$$137$$ −7.27455 −0.621507 −0.310753 0.950491i $$-0.600581\pi$$
−0.310753 + 0.950491i $$0.600581\pi$$
$$138$$ 0 0
$$139$$ −15.4321 −1.30893 −0.654465 0.756092i $$-0.727106\pi$$
−0.654465 + 0.756092i $$0.727106\pi$$
$$140$$ 0 0
$$141$$ −5.34008 −0.449716
$$142$$ 0 0
$$143$$ 0.817430 0.0683569
$$144$$ 0 0
$$145$$ −4.11221 −0.341501
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −15.2495 −1.24929 −0.624643 0.780910i $$-0.714756\pi$$
−0.624643 + 0.780910i $$0.714756\pi$$
$$150$$ 0 0
$$151$$ −16.5693 −1.34839 −0.674197 0.738552i $$-0.735510\pi$$
−0.674197 + 0.738552i $$0.735510\pi$$
$$152$$ 0 0
$$153$$ 4.61463 0.373070
$$154$$ 0 0
$$155$$ 6.47735 0.520273
$$156$$ 0 0
$$157$$ −0.659922 −0.0526675 −0.0263338 0.999653i $$-0.508383\pi$$
−0.0263338 + 0.999653i $$0.508383\pi$$
$$158$$ 0 0
$$159$$ −5.13727 −0.407412
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ −23.3212 −1.82666 −0.913330 0.407220i $$-0.866498\pi$$
−0.913330 + 0.407220i $$0.866498\pi$$
$$164$$ 0 0
$$165$$ −14.7519 −1.14843
$$166$$ 0 0
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ −12.9812 −0.998550
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 15.1637 1.15288 0.576438 0.817141i $$-0.304442\pi$$
0.576438 + 0.817141i $$0.304442\pi$$
$$174$$ 0 0
$$175$$ −1.13727 −0.0859699
$$176$$ 0 0
$$177$$ −5.13727 −0.386141
$$178$$ 0 0
$$179$$ 16.3868 1.22480 0.612402 0.790546i $$-0.290203\pi$$
0.612402 + 0.790546i $$0.290203\pi$$
$$180$$ 0 0
$$181$$ 11.0202 0.819127 0.409564 0.912282i $$-0.365681\pi$$
0.409564 + 0.912282i $$0.365681\pi$$
$$182$$ 0 0
$$183$$ 4.52265 0.334324
$$184$$ 0 0
$$185$$ −28.6613 −2.10722
$$186$$ 0 0
$$187$$ −27.4787 −2.00944
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −8.02023 −0.580324 −0.290162 0.956978i $$-0.593709\pi$$
−0.290162 + 0.956978i $$0.593709\pi$$
$$192$$ 0 0
$$193$$ −3.88918 −0.279949 −0.139975 0.990155i $$-0.544702\pi$$
−0.139975 + 0.990155i $$0.544702\pi$$
$$194$$ 0 0
$$195$$ −0.340078 −0.0243535
$$196$$ 0 0
$$197$$ 1.09198 0.0778003 0.0389002 0.999243i $$-0.487615\pi$$
0.0389002 + 0.999243i $$0.487615\pi$$
$$198$$ 0 0
$$199$$ 25.2306 1.78855 0.894276 0.447515i $$-0.147691\pi$$
0.894276 + 0.447515i $$0.147691\pi$$
$$200$$ 0 0
$$201$$ −2.47735 −0.174739
$$202$$ 0 0
$$203$$ 1.65992 0.116504
$$204$$ 0 0
$$205$$ −9.11704 −0.636762
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 5.95470 0.411896
$$210$$ 0 0
$$211$$ 17.9547 1.23605 0.618026 0.786157i $$-0.287932\pi$$
0.618026 + 0.786157i $$0.287932\pi$$
$$212$$ 0 0
$$213$$ 11.3665 0.778822
$$214$$ 0 0
$$215$$ −27.4787 −1.87403
$$216$$ 0 0
$$217$$ −2.61463 −0.177492
$$218$$ 0 0
$$219$$ −0.340078 −0.0229803
$$220$$ 0 0
$$221$$ −0.633471 −0.0426119
$$222$$ 0 0
$$223$$ −28.0216 −1.87647 −0.938233 0.346003i $$-0.887539\pi$$
−0.938233 + 0.346003i $$0.887539\pi$$
$$224$$ 0 0
$$225$$ 1.13727 0.0758183
$$226$$ 0 0
$$227$$ 4.47735 0.297172 0.148586 0.988899i $$-0.452528\pi$$
0.148586 + 0.988899i $$0.452528\pi$$
$$228$$ 0 0
$$229$$ −4.49759 −0.297209 −0.148604 0.988897i $$-0.547478\pi$$
−0.148604 + 0.988897i $$0.547478\pi$$
$$230$$ 0 0
$$231$$ 5.95470 0.391791
$$232$$ 0 0
$$233$$ 11.7066 0.766925 0.383463 0.923556i $$-0.374731\pi$$
0.383463 + 0.923556i $$0.374731\pi$$
$$234$$ 0 0
$$235$$ −13.2293 −0.862981
$$236$$ 0 0
$$237$$ 12.5240 0.813524
$$238$$ 0 0
$$239$$ −2.34630 −0.151769 −0.0758846 0.997117i $$-0.524178\pi$$
−0.0758846 + 0.997117i $$0.524178\pi$$
$$240$$ 0 0
$$241$$ 1.27455 0.0821009 0.0410505 0.999157i $$-0.486930\pi$$
0.0410505 + 0.999157i $$0.486930\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 2.47735 0.158272
$$246$$ 0 0
$$247$$ 0.137275 0.00873458
$$248$$ 0 0
$$249$$ −1.65992 −0.105193
$$250$$ 0 0
$$251$$ 16.1840 1.02152 0.510761 0.859723i $$-0.329364\pi$$
0.510761 + 0.859723i $$0.329364\pi$$
$$252$$ 0 0
$$253$$ 5.95470 0.374369
$$254$$ 0 0
$$255$$ 11.4321 0.715903
$$256$$ 0 0
$$257$$ −25.1184 −1.56684 −0.783422 0.621490i $$-0.786528\pi$$
−0.783422 + 0.621490i $$0.786528\pi$$
$$258$$ 0 0
$$259$$ 11.5693 0.718883
$$260$$ 0 0
$$261$$ −1.65992 −0.102747
$$262$$ 0 0
$$263$$ 18.1387 1.11848 0.559239 0.829007i $$-0.311093\pi$$
0.559239 + 0.829007i $$0.311093\pi$$
$$264$$ 0 0
$$265$$ −12.7268 −0.781804
$$266$$ 0 0
$$267$$ 3.86273 0.236395
$$268$$ 0 0
$$269$$ 30.1401 1.83767 0.918836 0.394640i $$-0.129131\pi$$
0.918836 + 0.394640i $$0.129131\pi$$
$$270$$ 0 0
$$271$$ 0.209021 0.0126971 0.00634856 0.999980i $$-0.497979\pi$$
0.00634856 + 0.999980i $$0.497979\pi$$
$$272$$ 0 0
$$273$$ 0.137275 0.00830824
$$274$$ 0 0
$$275$$ −6.77213 −0.408375
$$276$$ 0 0
$$277$$ −10.7972 −0.648741 −0.324370 0.945930i $$-0.605152\pi$$
−0.324370 + 0.945930i $$0.605152\pi$$
$$278$$ 0 0
$$279$$ 2.61463 0.156534
$$280$$ 0 0
$$281$$ −26.2745 −1.56741 −0.783704 0.621134i $$-0.786672\pi$$
−0.783704 + 0.621134i $$0.786672\pi$$
$$282$$ 0 0
$$283$$ 11.7066 0.695886 0.347943 0.937516i $$-0.386880\pi$$
0.347943 + 0.937516i $$0.386880\pi$$
$$284$$ 0 0
$$285$$ −2.47735 −0.146746
$$286$$ 0 0
$$287$$ 3.68016 0.217233
$$288$$ 0 0
$$289$$ 4.29478 0.252634
$$290$$ 0 0
$$291$$ −2.88918 −0.169367
$$292$$ 0 0
$$293$$ −22.4585 −1.31204 −0.656020 0.754743i $$-0.727761\pi$$
−0.656020 + 0.754743i $$0.727761\pi$$
$$294$$ 0 0
$$295$$ −12.7268 −0.740985
$$296$$ 0 0
$$297$$ −5.95470 −0.345527
$$298$$ 0 0
$$299$$ 0.137275 0.00793880
$$300$$ 0 0
$$301$$ 11.0920 0.639331
$$302$$ 0 0
$$303$$ −19.3212 −1.10998
$$304$$ 0 0
$$305$$ 11.2042 0.641550
$$306$$ 0 0
$$307$$ −11.9345 −0.681136 −0.340568 0.940220i $$-0.610619\pi$$
−0.340568 + 0.940220i $$0.610619\pi$$
$$308$$ 0 0
$$309$$ −6.38537 −0.363251
$$310$$ 0 0
$$311$$ 7.75190 0.439570 0.219785 0.975548i $$-0.429464\pi$$
0.219785 + 0.975548i $$0.429464\pi$$
$$312$$ 0 0
$$313$$ 6.93447 0.391960 0.195980 0.980608i $$-0.437211\pi$$
0.195980 + 0.980608i $$0.437211\pi$$
$$314$$ 0 0
$$315$$ −2.47735 −0.139583
$$316$$ 0 0
$$317$$ −26.3868 −1.48203 −0.741014 0.671489i $$-0.765655\pi$$
−0.741014 + 0.671489i $$0.765655\pi$$
$$318$$ 0 0
$$319$$ 9.88435 0.553417
$$320$$ 0 0
$$321$$ −0.908021 −0.0506808
$$322$$ 0 0
$$323$$ −4.61463 −0.256765
$$324$$ 0 0
$$325$$ −0.156119 −0.00865992
$$326$$ 0 0
$$327$$ 6.75190 0.373381
$$328$$ 0 0
$$329$$ 5.34008 0.294408
$$330$$ 0 0
$$331$$ −7.61463 −0.418538 −0.209269 0.977858i $$-0.567108\pi$$
−0.209269 + 0.977858i $$0.567108\pi$$
$$332$$ 0 0
$$333$$ −11.5693 −0.633995
$$334$$ 0 0
$$335$$ −6.13727 −0.335315
$$336$$ 0 0
$$337$$ 17.0920 0.931059 0.465530 0.885032i $$-0.345864\pi$$
0.465530 + 0.885032i $$0.345864\pi$$
$$338$$ 0 0
$$339$$ 18.6613 1.01354
$$340$$ 0 0
$$341$$ −15.5693 −0.843127
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −2.47735 −0.133376
$$346$$ 0 0
$$347$$ 35.1136 1.88500 0.942498 0.334211i $$-0.108470\pi$$
0.942498 + 0.334211i $$0.108470\pi$$
$$348$$ 0 0
$$349$$ 24.1651 1.29353 0.646764 0.762690i $$-0.276122\pi$$
0.646764 + 0.762690i $$0.276122\pi$$
$$350$$ 0 0
$$351$$ −0.137275 −0.00732718
$$352$$ 0 0
$$353$$ −0.156119 −0.00830938 −0.00415469 0.999991i $$-0.501322\pi$$
−0.00415469 + 0.999991i $$0.501322\pi$$
$$354$$ 0 0
$$355$$ 28.1589 1.49452
$$356$$ 0 0
$$357$$ −4.61463 −0.244232
$$358$$ 0 0
$$359$$ 12.1122 0.639258 0.319629 0.947543i $$-0.396442\pi$$
0.319629 + 0.947543i $$0.396442\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 24.4585 1.28374
$$364$$ 0 0
$$365$$ −0.842492 −0.0440981
$$366$$ 0 0
$$367$$ 21.8830 1.14228 0.571141 0.820852i $$-0.306501\pi$$
0.571141 + 0.820852i $$0.306501\pi$$
$$368$$ 0 0
$$369$$ −3.68016 −0.191581
$$370$$ 0 0
$$371$$ 5.13727 0.266714
$$372$$ 0 0
$$373$$ 6.88918 0.356708 0.178354 0.983966i $$-0.442923\pi$$
0.178354 + 0.983966i $$0.442923\pi$$
$$374$$ 0 0
$$375$$ −9.56933 −0.494158
$$376$$ 0 0
$$377$$ 0.227865 0.0117357
$$378$$ 0 0
$$379$$ 29.1387 1.49675 0.748376 0.663274i $$-0.230834\pi$$
0.748376 + 0.663274i $$0.230834\pi$$
$$380$$ 0 0
$$381$$ 4.15751 0.212996
$$382$$ 0 0
$$383$$ −16.4307 −0.839568 −0.419784 0.907624i $$-0.637894\pi$$
−0.419784 + 0.907624i $$0.637894\pi$$
$$384$$ 0 0
$$385$$ 14.7519 0.751827
$$386$$ 0 0
$$387$$ −11.0920 −0.563837
$$388$$ 0 0
$$389$$ −21.3854 −1.08428 −0.542141 0.840288i $$-0.682386\pi$$
−0.542141 + 0.840288i $$0.682386\pi$$
$$390$$ 0 0
$$391$$ −4.61463 −0.233372
$$392$$ 0 0
$$393$$ −14.2293 −0.717771
$$394$$ 0 0
$$395$$ 31.0265 1.56111
$$396$$ 0 0
$$397$$ −25.4132 −1.27545 −0.637726 0.770263i $$-0.720125\pi$$
−0.637726 + 0.770263i $$0.720125\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 7.00000 0.349563 0.174782 0.984607i $$-0.444078\pi$$
0.174782 + 0.984607i $$0.444078\pi$$
$$402$$ 0 0
$$403$$ −0.358922 −0.0178792
$$404$$ 0 0
$$405$$ 2.47735 0.123101
$$406$$ 0 0
$$407$$ 68.8920 3.41485
$$408$$ 0 0
$$409$$ −6.06553 −0.299921 −0.149961 0.988692i $$-0.547915\pi$$
−0.149961 + 0.988692i $$0.547915\pi$$
$$410$$ 0 0
$$411$$ −7.27455 −0.358827
$$412$$ 0 0
$$413$$ 5.13727 0.252789
$$414$$ 0 0
$$415$$ −4.11221 −0.201861
$$416$$ 0 0
$$417$$ −15.4321 −0.755711
$$418$$ 0 0
$$419$$ 19.3415 0.944892 0.472446 0.881359i $$-0.343371\pi$$
0.472446 + 0.881359i $$0.343371\pi$$
$$420$$ 0 0
$$421$$ −2.54288 −0.123932 −0.0619662 0.998078i $$-0.519737\pi$$
−0.0619662 + 0.998078i $$0.519737\pi$$
$$422$$ 0 0
$$423$$ −5.34008 −0.259643
$$424$$ 0 0
$$425$$ 5.24810 0.254570
$$426$$ 0 0
$$427$$ −4.52265 −0.218866
$$428$$ 0 0
$$429$$ 0.817430 0.0394659
$$430$$ 0 0
$$431$$ −2.79720 −0.134736 −0.0673681 0.997728i $$-0.521460\pi$$
−0.0673681 + 0.997728i $$0.521460\pi$$
$$432$$ 0 0
$$433$$ −11.7080 −0.562650 −0.281325 0.959612i $$-0.590774\pi$$
−0.281325 + 0.959612i $$0.590774\pi$$
$$434$$ 0 0
$$435$$ −4.11221 −0.197165
$$436$$ 0 0
$$437$$ 1.00000 0.0478365
$$438$$ 0 0
$$439$$ −17.9749 −0.857897 −0.428948 0.903329i $$-0.641116\pi$$
−0.428948 + 0.903329i $$0.641116\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 7.81882 0.371483 0.185742 0.982599i $$-0.440531\pi$$
0.185742 + 0.982599i $$0.440531\pi$$
$$444$$ 0 0
$$445$$ 9.56933 0.453630
$$446$$ 0 0
$$447$$ −15.2495 −0.721276
$$448$$ 0 0
$$449$$ −25.0265 −1.18107 −0.590536 0.807012i $$-0.701083\pi$$
−0.590536 + 0.807012i $$0.701083\pi$$
$$450$$ 0 0
$$451$$ 21.9142 1.03190
$$452$$ 0 0
$$453$$ −16.5693 −0.778495
$$454$$ 0 0
$$455$$ 0.340078 0.0159431
$$456$$ 0 0
$$457$$ 26.3868 1.23432 0.617160 0.786837i $$-0.288283\pi$$
0.617160 + 0.786837i $$0.288283\pi$$
$$458$$ 0 0
$$459$$ 4.61463 0.215392
$$460$$ 0 0
$$461$$ −19.2481 −0.896473 −0.448237 0.893915i $$-0.647948\pi$$
−0.448237 + 0.893915i $$0.647948\pi$$
$$462$$ 0 0
$$463$$ −9.04047 −0.420146 −0.210073 0.977686i $$-0.567370\pi$$
−0.210073 + 0.977686i $$0.567370\pi$$
$$464$$ 0 0
$$465$$ 6.47735 0.300380
$$466$$ 0 0
$$467$$ −35.2571 −1.63150 −0.815752 0.578402i $$-0.803677\pi$$
−0.815752 + 0.578402i $$0.803677\pi$$
$$468$$ 0 0
$$469$$ 2.47735 0.114394
$$470$$ 0 0
$$471$$ −0.659922 −0.0304076
$$472$$ 0 0
$$473$$ 66.0495 3.03696
$$474$$ 0 0
$$475$$ −1.13727 −0.0521817
$$476$$ 0 0
$$477$$ −5.13727 −0.235220
$$478$$ 0 0
$$479$$ −35.5721 −1.62533 −0.812666 0.582730i $$-0.801984\pi$$
−0.812666 + 0.582730i $$0.801984\pi$$
$$480$$ 0 0
$$481$$ 1.58818 0.0724146
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ −7.15751 −0.325006
$$486$$ 0 0
$$487$$ −0.614627 −0.0278514 −0.0139257 0.999903i $$-0.504433\pi$$
−0.0139257 + 0.999903i $$0.504433\pi$$
$$488$$ 0 0
$$489$$ −23.3212 −1.05462
$$490$$ 0 0
$$491$$ 12.2683 0.553662 0.276831 0.960919i $$-0.410716\pi$$
0.276831 + 0.960919i $$0.410716\pi$$
$$492$$ 0 0
$$493$$ −7.65992 −0.344986
$$494$$ 0 0
$$495$$ −14.7519 −0.663049
$$496$$ 0 0
$$497$$ −11.3665 −0.509858
$$498$$ 0 0
$$499$$ 24.1122 1.07941 0.539705 0.841854i $$-0.318536\pi$$
0.539705 + 0.841854i $$0.318536\pi$$
$$500$$ 0 0
$$501$$ −9.00000 −0.402090
$$502$$ 0 0
$$503$$ −22.7519 −1.01446 −0.507229 0.861812i $$-0.669330\pi$$
−0.507229 + 0.861812i $$0.669330\pi$$
$$504$$ 0 0
$$505$$ −47.8655 −2.12999
$$506$$ 0 0
$$507$$ −12.9812 −0.576513
$$508$$ 0 0
$$509$$ −15.7581 −0.698466 −0.349233 0.937036i $$-0.613558\pi$$
−0.349233 + 0.937036i $$0.613558\pi$$
$$510$$ 0 0
$$511$$ 0.340078 0.0150442
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ −15.8188 −0.697060
$$516$$ 0 0
$$517$$ 31.7986 1.39850
$$518$$ 0 0
$$519$$ 15.1637 0.665614
$$520$$ 0 0
$$521$$ 9.45090 0.414052 0.207026 0.978335i $$-0.433622\pi$$
0.207026 + 0.978335i $$0.433622\pi$$
$$522$$ 0 0
$$523$$ −4.93447 −0.215769 −0.107885 0.994163i $$-0.534408\pi$$
−0.107885 + 0.994163i $$0.534408\pi$$
$$524$$ 0 0
$$525$$ −1.13727 −0.0496347
$$526$$ 0 0
$$527$$ 12.0655 0.525583
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −5.13727 −0.222939
$$532$$ 0 0
$$533$$ 0.505192 0.0218823
$$534$$ 0 0
$$535$$ −2.24949 −0.0972538
$$536$$ 0 0
$$537$$ 16.3868 0.707141
$$538$$ 0 0
$$539$$ −5.95470 −0.256487
$$540$$ 0 0
$$541$$ 43.8391 1.88479 0.942394 0.334505i $$-0.108569\pi$$
0.942394 + 0.334505i $$0.108569\pi$$
$$542$$ 0 0
$$543$$ 11.0202 0.472923
$$544$$ 0 0
$$545$$ 16.7268 0.716499
$$546$$ 0 0
$$547$$ −9.61602 −0.411151 −0.205576 0.978641i $$-0.565907\pi$$
−0.205576 + 0.978641i $$0.565907\pi$$
$$548$$ 0 0
$$549$$ 4.52265 0.193022
$$550$$ 0 0
$$551$$ 1.65992 0.0707151
$$552$$ 0 0
$$553$$ −12.5240 −0.532576
$$554$$ 0 0
$$555$$ −28.6613 −1.21660
$$556$$ 0 0
$$557$$ −1.90663 −0.0807866 −0.0403933 0.999184i $$-0.512861\pi$$
−0.0403933 + 0.999184i $$0.512861\pi$$
$$558$$ 0 0
$$559$$ 1.52265 0.0644011
$$560$$ 0 0
$$561$$ −27.4787 −1.16015
$$562$$ 0 0
$$563$$ −15.3665 −0.647622 −0.323811 0.946122i $$-0.604964\pi$$
−0.323811 + 0.946122i $$0.604964\pi$$
$$564$$ 0 0
$$565$$ 46.2306 1.94494
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −18.1637 −0.761463 −0.380731 0.924686i $$-0.624328\pi$$
−0.380731 + 0.924686i $$0.624328\pi$$
$$570$$ 0 0
$$571$$ 16.4334 0.687718 0.343859 0.939021i $$-0.388266\pi$$
0.343859 + 0.939021i $$0.388266\pi$$
$$572$$ 0 0
$$573$$ −8.02023 −0.335050
$$574$$ 0 0
$$575$$ −1.13727 −0.0474276
$$576$$ 0 0
$$577$$ 26.3679 1.09771 0.548855 0.835917i $$-0.315064\pi$$
0.548855 + 0.835917i $$0.315064\pi$$
$$578$$ 0 0
$$579$$ −3.88918 −0.161629
$$580$$ 0 0
$$581$$ 1.65992 0.0688652
$$582$$ 0 0
$$583$$ 30.5910 1.26695
$$584$$ 0 0
$$585$$ −0.340078 −0.0140605
$$586$$ 0 0
$$587$$ 3.11221 0.128455 0.0642274 0.997935i $$-0.479542\pi$$
0.0642274 + 0.997935i $$0.479542\pi$$
$$588$$ 0 0
$$589$$ −2.61463 −0.107734
$$590$$ 0 0
$$591$$ 1.09198 0.0449180
$$592$$ 0 0
$$593$$ 10.2745 0.421925 0.210963 0.977494i $$-0.432340\pi$$
0.210963 + 0.977494i $$0.432340\pi$$
$$594$$ 0 0
$$595$$ −11.4321 −0.468669
$$596$$ 0 0
$$597$$ 25.2306 1.03262
$$598$$ 0 0
$$599$$ −17.9032 −0.731505 −0.365752 0.930712i $$-0.619188\pi$$
−0.365752 + 0.930712i $$0.619188\pi$$
$$600$$ 0 0
$$601$$ −1.06692 −0.0435205 −0.0217602 0.999763i $$-0.506927\pi$$
−0.0217602 + 0.999763i $$0.506927\pi$$
$$602$$ 0 0
$$603$$ −2.47735 −0.100886
$$604$$ 0 0
$$605$$ 60.5923 2.46343
$$606$$ 0 0
$$607$$ −3.77213 −0.153106 −0.0765531 0.997066i $$-0.524391\pi$$
−0.0765531 + 0.997066i $$0.524391\pi$$
$$608$$ 0 0
$$609$$ 1.65992 0.0672634
$$610$$ 0 0
$$611$$ 0.733057 0.0296563
$$612$$ 0 0
$$613$$ −3.97494 −0.160546 −0.0802731 0.996773i $$-0.525579\pi$$
−0.0802731 + 0.996773i $$0.525579\pi$$
$$614$$ 0 0
$$615$$ −9.11704 −0.367635
$$616$$ 0 0
$$617$$ 25.2355 1.01594 0.507971 0.861374i $$-0.330396\pi$$
0.507971 + 0.861374i $$0.330396\pi$$
$$618$$ 0 0
$$619$$ −2.23064 −0.0896571 −0.0448286 0.998995i $$-0.514274\pi$$
−0.0448286 + 0.998995i $$0.514274\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ −3.86273 −0.154757
$$624$$ 0 0
$$625$$ −29.3930 −1.17572
$$626$$ 0 0
$$627$$ 5.95470 0.237808
$$628$$ 0 0
$$629$$ −53.3882 −2.12872
$$630$$ 0 0
$$631$$ 6.47113 0.257612 0.128806 0.991670i $$-0.458886\pi$$
0.128806 + 0.991670i $$0.458886\pi$$
$$632$$ 0 0
$$633$$ 17.9547 0.713635
$$634$$ 0 0
$$635$$ 10.2996 0.408728
$$636$$ 0 0
$$637$$ −0.137275 −0.00543902
$$638$$ 0 0
$$639$$ 11.3665 0.449653
$$640$$ 0 0
$$641$$ −22.7471 −0.898455 −0.449228 0.893417i $$-0.648301\pi$$
−0.449228 + 0.893417i $$0.648301\pi$$
$$642$$ 0 0
$$643$$ 19.5707 0.771794 0.385897 0.922542i $$-0.373892\pi$$
0.385897 + 0.922542i $$0.373892\pi$$
$$644$$ 0 0
$$645$$ −27.4787 −1.08197
$$646$$ 0 0
$$647$$ −23.9812 −0.942797 −0.471398 0.881920i $$-0.656251\pi$$
−0.471398 + 0.881920i $$0.656251\pi$$
$$648$$ 0 0
$$649$$ 30.5910 1.20080
$$650$$ 0 0
$$651$$ −2.61463 −0.102475
$$652$$ 0 0
$$653$$ −15.1198 −0.591684 −0.295842 0.955237i $$-0.595600\pi$$
−0.295842 + 0.955237i $$0.595600\pi$$
$$654$$ 0 0
$$655$$ −35.2509 −1.37737
$$656$$ 0 0
$$657$$ −0.340078 −0.0132677
$$658$$ 0 0
$$659$$ 17.5693 0.684404 0.342202 0.939626i $$-0.388827\pi$$
0.342202 + 0.939626i $$0.388827\pi$$
$$660$$ 0 0
$$661$$ −43.7331 −1.70102 −0.850509 0.525960i $$-0.823706\pi$$
−0.850509 + 0.525960i $$0.823706\pi$$
$$662$$ 0 0
$$663$$ −0.633471 −0.0246020
$$664$$ 0 0
$$665$$ 2.47735 0.0960676
$$666$$ 0 0
$$667$$ 1.65992 0.0642724
$$668$$ 0 0
$$669$$ −28.0216 −1.08338
$$670$$ 0 0
$$671$$ −26.9310 −1.03966
$$672$$ 0 0
$$673$$ 7.09059 0.273322 0.136661 0.990618i $$-0.456363\pi$$
0.136661 + 0.990618i $$0.456363\pi$$
$$674$$ 0 0
$$675$$ 1.13727 0.0437737
$$676$$ 0 0
$$677$$ −6.67394 −0.256500 −0.128250 0.991742i $$-0.540936\pi$$
−0.128250 + 0.991742i $$0.540936\pi$$
$$678$$ 0 0
$$679$$ 2.88918 0.110876
$$680$$ 0 0
$$681$$ 4.47735 0.171573
$$682$$ 0 0
$$683$$ 39.5721 1.51418 0.757092 0.653308i $$-0.226619\pi$$
0.757092 + 0.653308i $$0.226619\pi$$
$$684$$ 0 0
$$685$$ −18.0216 −0.688571
$$686$$ 0 0
$$687$$ −4.49759 −0.171594
$$688$$ 0 0
$$689$$ 0.705218 0.0268667
$$690$$ 0 0
$$691$$ −24.4146 −0.928775 −0.464388 0.885632i $$-0.653726\pi$$
−0.464388 + 0.885632i $$0.653726\pi$$
$$692$$ 0 0
$$693$$ 5.95470 0.226201
$$694$$ 0 0
$$695$$ −38.2306 −1.45017
$$696$$ 0 0
$$697$$ −16.9825 −0.643260
$$698$$ 0 0
$$699$$ 11.7066 0.442785
$$700$$ 0 0
$$701$$ −32.3693 −1.22257 −0.611286 0.791410i $$-0.709347\pi$$
−0.611286 + 0.791410i $$0.709347\pi$$
$$702$$ 0 0
$$703$$ 11.5693 0.436346
$$704$$ 0 0
$$705$$ −13.2293 −0.498243
$$706$$ 0 0
$$707$$ 19.3212 0.726650
$$708$$ 0 0
$$709$$ −34.5457 −1.29739 −0.648695 0.761049i $$-0.724685\pi$$
−0.648695 + 0.761049i $$0.724685\pi$$
$$710$$ 0 0
$$711$$ 12.5240 0.469688
$$712$$ 0 0
$$713$$ −2.61463 −0.0979186
$$714$$ 0 0
$$715$$ 2.02506 0.0757330
$$716$$ 0 0
$$717$$ −2.34630 −0.0876240
$$718$$ 0 0
$$719$$ −25.0202 −0.933097 −0.466549 0.884496i $$-0.654503\pi$$
−0.466549 + 0.884496i $$0.654503\pi$$
$$720$$ 0 0
$$721$$ 6.38537 0.237804
$$722$$ 0 0
$$723$$ 1.27455 0.0474010
$$724$$ 0 0
$$725$$ −1.88779 −0.0701107
$$726$$ 0 0
$$727$$ 33.5491 1.24427 0.622134 0.782911i $$-0.286266\pi$$
0.622134 + 0.782911i $$0.286266\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −51.1853 −1.89316
$$732$$ 0 0
$$733$$ −4.73306 −0.174819 −0.0874097 0.996172i $$-0.527859\pi$$
−0.0874097 + 0.996172i $$0.527859\pi$$
$$734$$ 0 0
$$735$$ 2.47735 0.0913785
$$736$$ 0 0
$$737$$ 14.7519 0.543393
$$738$$ 0 0
$$739$$ −29.0202 −1.06753 −0.533763 0.845634i $$-0.679223\pi$$
−0.533763 + 0.845634i $$0.679223\pi$$
$$740$$ 0 0
$$741$$ 0.137275 0.00504291
$$742$$ 0 0
$$743$$ 11.2279 0.411910 0.205955 0.978561i $$-0.433970\pi$$
0.205955 + 0.978561i $$0.433970\pi$$
$$744$$ 0 0
$$745$$ −37.7784 −1.38409
$$746$$ 0 0
$$747$$ −1.65992 −0.0607334
$$748$$ 0 0
$$749$$ 0.908021 0.0331784
$$750$$ 0 0
$$751$$ −19.3010 −0.704304 −0.352152 0.935943i $$-0.614550\pi$$
−0.352152 + 0.935943i $$0.614550\pi$$
$$752$$ 0 0
$$753$$ 16.1840 0.589776
$$754$$ 0 0
$$755$$ −41.0481 −1.49389
$$756$$ 0 0
$$757$$ −7.65992 −0.278405 −0.139202 0.990264i $$-0.544454\pi$$
−0.139202 + 0.990264i $$0.544454\pi$$
$$758$$ 0 0
$$759$$ 5.95470 0.216142
$$760$$ 0 0
$$761$$ 22.9345 0.831374 0.415687 0.909508i $$-0.363541\pi$$
0.415687 + 0.909508i $$0.363541\pi$$
$$762$$ 0 0
$$763$$ −6.75190 −0.244435
$$764$$ 0 0
$$765$$ 11.4321 0.413327
$$766$$ 0 0
$$767$$ 0.705218 0.0254639
$$768$$ 0 0
$$769$$ 54.6425 1.97046 0.985229 0.171243i $$-0.0547782\pi$$
0.985229 + 0.171243i $$0.0547782\pi$$
$$770$$ 0 0
$$771$$ −25.1184 −0.904618
$$772$$ 0 0
$$773$$ −8.02784 −0.288741 −0.144371 0.989524i $$-0.546116\pi$$
−0.144371 + 0.989524i $$0.546116\pi$$
$$774$$ 0 0
$$775$$ 2.97355 0.106813
$$776$$ 0 0
$$777$$ 11.5693 0.415047
$$778$$ 0 0
$$779$$ 3.68016 0.131855
$$780$$ 0 0
$$781$$ −67.6843 −2.42194
$$782$$ 0 0
$$783$$ −1.65992 −0.0593208
$$784$$ 0 0
$$785$$ −1.63486 −0.0583507
$$786$$ 0 0
$$787$$ 43.3617 1.54568 0.772839 0.634602i $$-0.218836\pi$$
0.772839 + 0.634602i $$0.218836\pi$$
$$788$$ 0 0
$$789$$ 18.1387 0.645754
$$790$$ 0 0
$$791$$ −18.6613 −0.663520
$$792$$ 0 0
$$793$$ −0.620845 −0.0220468
$$794$$ 0 0
$$795$$ −12.7268 −0.451374
$$796$$ 0 0
$$797$$ 44.8641 1.58917 0.794584 0.607154i $$-0.207689\pi$$
0.794584 + 0.607154i $$0.207689\pi$$
$$798$$ 0 0
$$799$$ −24.6425 −0.871788
$$800$$ 0 0
$$801$$ 3.86273 0.136483
$$802$$ 0 0
$$803$$ 2.02506 0.0714629
$$804$$ 0 0
$$805$$ 2.47735 0.0873152
$$806$$ 0 0
$$807$$ 30.1401 1.06098
$$808$$ 0 0
$$809$$ −5.40422 −0.190002 −0.0950011 0.995477i $$-0.530285\pi$$
−0.0950011 + 0.995477i $$0.530285\pi$$
$$810$$ 0 0
$$811$$ 32.7485 1.14995 0.574977 0.818170i $$-0.305011\pi$$
0.574977 + 0.818170i $$0.305011\pi$$
$$812$$ 0 0
$$813$$ 0.209021 0.00733068
$$814$$ 0 0
$$815$$ −57.7749 −2.02377
$$816$$ 0 0
$$817$$ 11.0920 0.388059
$$818$$ 0 0
$$819$$ 0.137275 0.00479676
$$820$$ 0 0
$$821$$ 14.9142 0.520511 0.260255 0.965540i $$-0.416193\pi$$
0.260255 + 0.965540i $$0.416193\pi$$
$$822$$ 0 0
$$823$$ 4.00139 0.139480 0.0697398 0.997565i $$-0.477783\pi$$
0.0697398 + 0.997565i $$0.477783\pi$$
$$824$$ 0 0
$$825$$ −6.77213 −0.235775
$$826$$ 0 0
$$827$$ 6.91563 0.240480 0.120240 0.992745i $$-0.461634\pi$$
0.120240 + 0.992745i $$0.461634\pi$$
$$828$$ 0 0
$$829$$ 15.7128 0.545729 0.272864 0.962052i $$-0.412029\pi$$
0.272864 + 0.962052i $$0.412029\pi$$
$$830$$ 0 0
$$831$$ −10.7972 −0.374551
$$832$$ 0 0
$$833$$ 4.61463 0.159887
$$834$$ 0 0
$$835$$ −22.2962 −0.771591
$$836$$ 0 0
$$837$$ 2.61463 0.0903747
$$838$$ 0 0
$$839$$ −37.9966 −1.31179 −0.655893 0.754853i $$-0.727708\pi$$
−0.655893 + 0.754853i $$0.727708\pi$$
$$840$$ 0 0
$$841$$ −26.2447 −0.904988
$$842$$ 0 0
$$843$$ −26.2745 −0.904944
$$844$$ 0 0
$$845$$ −32.1589 −1.10630
$$846$$ 0 0
$$847$$ −24.4585 −0.840404
$$848$$ 0 0
$$849$$ 11.7066 0.401770
$$850$$ 0 0
$$851$$ 11.5693 0.396592
$$852$$ 0 0
$$853$$ 21.7659 0.745251 0.372625 0.927982i $$-0.378458\pi$$
0.372625 + 0.927982i $$0.378458\pi$$
$$854$$ 0 0
$$855$$ −2.47735 −0.0847237
$$856$$ 0 0
$$857$$ −54.9777 −1.87800 −0.939001 0.343913i $$-0.888247\pi$$
−0.939001 + 0.343913i $$0.888247\pi$$
$$858$$ 0 0
$$859$$ 54.4084 1.85639 0.928195 0.372094i $$-0.121360\pi$$
0.928195 + 0.372094i $$0.121360\pi$$
$$860$$ 0 0
$$861$$ 3.68016 0.125419
$$862$$ 0 0
$$863$$ −7.81604 −0.266061 −0.133031 0.991112i $$-0.542471\pi$$
−0.133031 + 0.991112i $$0.542471\pi$$
$$864$$ 0 0
$$865$$ 37.5659 1.27728
$$866$$ 0 0
$$867$$ 4.29478 0.145858
$$868$$ 0 0
$$869$$ −74.5769 −2.52985
$$870$$ 0 0
$$871$$ 0.340078 0.0115231
$$872$$ 0 0
$$873$$ −2.88918 −0.0977838
$$874$$ 0 0
$$875$$ 9.56933 0.323502
$$876$$ 0 0
$$877$$ 45.2118 1.52669 0.763347 0.645989i $$-0.223555\pi$$
0.763347 + 0.645989i $$0.223555\pi$$
$$878$$ 0 0
$$879$$ −22.4585 −0.757507
$$880$$ 0 0
$$881$$ 37.5443 1.26490 0.632449 0.774602i $$-0.282050\pi$$
0.632449 + 0.774602i $$0.282050\pi$$
$$882$$ 0 0
$$883$$ −23.5910 −0.793899 −0.396949 0.917840i $$-0.629931\pi$$
−0.396949 + 0.917840i $$0.629931\pi$$
$$884$$ 0 0
$$885$$ −12.7268 −0.427808
$$886$$ 0 0
$$887$$ −42.4118 −1.42405 −0.712025 0.702154i $$-0.752222\pi$$
−0.712025 + 0.702154i $$0.752222\pi$$
$$888$$ 0 0
$$889$$ −4.15751 −0.139438
$$890$$ 0 0
$$891$$ −5.95470 −0.199490
$$892$$ 0 0
$$893$$ 5.34008 0.178699
$$894$$ 0 0
$$895$$ 40.5958 1.35697
$$896$$ 0 0
$$897$$ 0.137275 0.00458347
$$898$$ 0 0
$$899$$ −4.34008 −0.144750
$$900$$ 0 0
$$901$$ −23.7066 −0.789782
$$902$$ 0 0
$$903$$ 11.0920 0.369118
$$904$$ 0 0
$$905$$ 27.3010 0.907516
$$906$$ 0 0
$$907$$ 40.0745 1.33065 0.665326 0.746553i $$-0.268292\pi$$
0.665326 + 0.746553i $$0.268292\pi$$
$$908$$ 0 0
$$909$$ −19.3212 −0.640845
$$910$$ 0 0
$$911$$ 13.0858 0.433551 0.216775 0.976222i $$-0.430446\pi$$
0.216775 + 0.976222i $$0.430446\pi$$
$$912$$ 0 0
$$913$$ 9.88435 0.327124
$$914$$ 0 0
$$915$$ 11.2042 0.370399
$$916$$ 0 0
$$917$$ 14.2293 0.469891
$$918$$ 0 0
$$919$$ 49.6599 1.63813 0.819065 0.573701i $$-0.194493\pi$$
0.819065 + 0.573701i $$0.194493\pi$$
$$920$$ 0 0
$$921$$ −11.9345 −0.393254
$$922$$ 0 0
$$923$$ −1.56034 −0.0513591
$$924$$ 0 0
$$925$$ −13.1575 −0.432616
$$926$$ 0 0
$$927$$ −6.38537 −0.209723
$$928$$ 0 0
$$929$$ 29.3944 0.964398 0.482199 0.876062i $$-0.339838\pi$$
0.482199 + 0.876062i $$0.339838\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 7.75190 0.253786
$$934$$ 0 0
$$935$$ −68.0745 −2.22627
$$936$$ 0 0
$$937$$ 44.2697 1.44623 0.723114 0.690728i $$-0.242710\pi$$
0.723114 + 0.690728i $$0.242710\pi$$
$$938$$ 0 0
$$939$$ 6.93447 0.226298
$$940$$ 0 0
$$941$$ −15.2697 −0.497779 −0.248889 0.968532i $$-0.580066\pi$$
−0.248889 + 0.968532i $$0.580066\pi$$
$$942$$ 0 0
$$943$$ 3.68016 0.119842
$$944$$ 0 0
$$945$$ −2.47735 −0.0805883
$$946$$ 0 0
$$947$$ 25.2293 0.819841 0.409920 0.912121i $$-0.365557\pi$$
0.409920 + 0.912121i $$0.365557\pi$$
$$948$$ 0 0
$$949$$ 0.0466840 0.00151543
$$950$$ 0 0
$$951$$ −26.3868 −0.855649
$$952$$ 0 0
$$953$$ 27.1527 0.879562 0.439781 0.898105i $$-0.355056\pi$$
0.439781 + 0.898105i $$0.355056\pi$$
$$954$$ 0 0
$$955$$ −19.8689 −0.642944
$$956$$ 0 0
$$957$$ 9.88435 0.319516
$$958$$ 0 0
$$959$$ 7.27455 0.234907
$$960$$ 0 0
$$961$$ −24.1637 −0.779475
$$962$$ 0 0
$$963$$ −0.908021 −0.0292606
$$964$$ 0 0
$$965$$ −9.63486 −0.310157
$$966$$ 0 0
$$967$$ 58.9575 1.89594 0.947972 0.318353i $$-0.103130\pi$$
0.947972 + 0.318353i $$0.103130\pi$$
$$968$$ 0 0
$$969$$ −4.61463 −0.148243
$$970$$ 0 0
$$971$$ −40.2306 −1.29106 −0.645531 0.763734i $$-0.723364\pi$$
−0.645531 + 0.763734i $$0.723364\pi$$
$$972$$ 0 0
$$973$$ 15.4321 0.494729
$$974$$ 0 0
$$975$$ −0.156119 −0.00499981
$$976$$ 0 0
$$977$$ −0.0669171 −0.00214087 −0.00107043 0.999999i $$-0.500341\pi$$
−0.00107043 + 0.999999i $$0.500341\pi$$
$$978$$ 0 0
$$979$$ −23.0014 −0.735128
$$980$$ 0 0
$$981$$ 6.75190 0.215572
$$982$$ 0 0
$$983$$ 38.0278 1.21290 0.606450 0.795122i $$-0.292593\pi$$
0.606450 + 0.795122i $$0.292593\pi$$
$$984$$ 0 0
$$985$$ 2.70522 0.0861954
$$986$$ 0 0
$$987$$ 5.34008 0.169977
$$988$$ 0 0
$$989$$ 11.0920 0.352704
$$990$$ 0 0
$$991$$ −34.6565 −1.10090 −0.550450 0.834868i $$-0.685544\pi$$
−0.550450 + 0.834868i $$0.685544\pi$$
$$992$$ 0 0
$$993$$ −7.61463 −0.241643
$$994$$ 0 0
$$995$$ 62.5052 1.98155
$$996$$ 0 0
$$997$$ −21.1637 −0.670262 −0.335131 0.942172i $$-0.608781\pi$$
−0.335131 + 0.942172i $$0.608781\pi$$
$$998$$ 0 0
$$999$$ −11.5693 −0.366037
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.bv.1.3 3
4.3 odd 2 1932.2.a.i.1.3 3
12.11 even 2 5796.2.a.p.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.i.1.3 3 4.3 odd 2
5796.2.a.p.1.1 3 12.11 even 2
7728.2.a.bv.1.3 3 1.1 even 1 trivial