# Properties

 Label 2475.2.a.w.1.2 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.41421 q^{2} +3.82843 q^{4} -0.414214 q^{7} +4.41421 q^{8} +O(q^{10})$$ $$q+2.41421 q^{2} +3.82843 q^{4} -0.414214 q^{7} +4.41421 q^{8} +1.00000 q^{11} +2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} +2.41421 q^{17} +6.41421 q^{19} +2.41421 q^{22} +1.00000 q^{23} +6.82843 q^{26} -1.58579 q^{28} -1.17157 q^{29} -8.48528 q^{31} -1.58579 q^{32} +5.82843 q^{34} +0.171573 q^{37} +15.4853 q^{38} +10.8995 q^{41} +11.6569 q^{43} +3.82843 q^{44} +2.41421 q^{46} -7.48528 q^{47} -6.82843 q^{49} +10.8284 q^{52} +7.65685 q^{53} -1.82843 q^{56} -2.82843 q^{58} -11.0000 q^{59} +8.82843 q^{61} -20.4853 q^{62} -9.82843 q^{64} +0.343146 q^{67} +9.24264 q^{68} -7.82843 q^{71} +8.82843 q^{73} +0.414214 q^{74} +24.5563 q^{76} -0.414214 q^{77} +13.2426 q^{79} +26.3137 q^{82} -4.48528 q^{83} +28.1421 q^{86} +4.41421 q^{88} -3.65685 q^{89} -1.17157 q^{91} +3.82843 q^{92} -18.0711 q^{94} -5.82843 q^{97} -16.4853 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 6 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} + 2 q^{17} + 10 q^{19} + 2 q^{22} + 2 q^{23} + 8 q^{26} - 6 q^{28} - 8 q^{29} - 6 q^{32} + 6 q^{34} + 6 q^{37} + 14 q^{38} + 2 q^{41} + 12 q^{43} + 2 q^{44} + 2 q^{46} + 2 q^{47} - 8 q^{49} + 16 q^{52} + 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} - 24 q^{62} - 14 q^{64} + 12 q^{67} + 10 q^{68} - 10 q^{71} + 12 q^{73} - 2 q^{74} + 18 q^{76} + 2 q^{77} + 18 q^{79} + 30 q^{82} + 8 q^{83} + 28 q^{86} + 6 q^{88} + 4 q^{89} - 8 q^{91} + 2 q^{92} - 22 q^{94} - 6 q^{97} - 16 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41421 1.70711 0.853553 0.521005i $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 0 0
$$4$$ 3.82843 1.91421
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.414214 −0.156558 −0.0782790 0.996931i $$-0.524942\pi$$
−0.0782790 + 0.996931i $$0.524942\pi$$
$$8$$ 4.41421 1.56066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 2.82843 0.784465 0.392232 0.919866i $$-0.371703\pi$$
0.392232 + 0.919866i $$0.371703\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 2.41421 0.585533 0.292766 0.956184i $$-0.405424\pi$$
0.292766 + 0.956184i $$0.405424\pi$$
$$18$$ 0 0
$$19$$ 6.41421 1.47152 0.735761 0.677242i $$-0.236825\pi$$
0.735761 + 0.677242i $$0.236825\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.41421 0.514712
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.82843 1.33916
$$27$$ 0 0
$$28$$ −1.58579 −0.299685
$$29$$ −1.17157 −0.217556 −0.108778 0.994066i $$-0.534694\pi$$
−0.108778 + 0.994066i $$0.534694\pi$$
$$30$$ 0 0
$$31$$ −8.48528 −1.52400 −0.762001 0.647576i $$-0.775783\pi$$
−0.762001 + 0.647576i $$0.775783\pi$$
$$32$$ −1.58579 −0.280330
$$33$$ 0 0
$$34$$ 5.82843 0.999567
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.171573 0.0282064 0.0141032 0.999901i $$-0.495511\pi$$
0.0141032 + 0.999901i $$0.495511\pi$$
$$38$$ 15.4853 2.51204
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.8995 1.70222 0.851108 0.524991i $$-0.175931\pi$$
0.851108 + 0.524991i $$0.175931\pi$$
$$42$$ 0 0
$$43$$ 11.6569 1.77765 0.888827 0.458243i $$-0.151521\pi$$
0.888827 + 0.458243i $$0.151521\pi$$
$$44$$ 3.82843 0.577157
$$45$$ 0 0
$$46$$ 2.41421 0.355956
$$47$$ −7.48528 −1.09184 −0.545920 0.837837i $$-0.683820\pi$$
−0.545920 + 0.837837i $$0.683820\pi$$
$$48$$ 0 0
$$49$$ −6.82843 −0.975490
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 10.8284 1.50163
$$53$$ 7.65685 1.05175 0.525875 0.850562i $$-0.323738\pi$$
0.525875 + 0.850562i $$0.323738\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.82843 −0.244334
$$57$$ 0 0
$$58$$ −2.82843 −0.371391
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 0 0
$$61$$ 8.82843 1.13036 0.565182 0.824966i $$-0.308806\pi$$
0.565182 + 0.824966i $$0.308806\pi$$
$$62$$ −20.4853 −2.60163
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.343146 0.0419219 0.0209610 0.999780i $$-0.493327\pi$$
0.0209610 + 0.999780i $$0.493327\pi$$
$$68$$ 9.24264 1.12083
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −7.82843 −0.929063 −0.464532 0.885556i $$-0.653777\pi$$
−0.464532 + 0.885556i $$0.653777\pi$$
$$72$$ 0 0
$$73$$ 8.82843 1.03329 0.516645 0.856200i $$-0.327181\pi$$
0.516645 + 0.856200i $$0.327181\pi$$
$$74$$ 0.414214 0.0481513
$$75$$ 0 0
$$76$$ 24.5563 2.81681
$$77$$ −0.414214 −0.0472040
$$78$$ 0 0
$$79$$ 13.2426 1.48991 0.744957 0.667113i $$-0.232470\pi$$
0.744957 + 0.667113i $$0.232470\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 26.3137 2.90586
$$83$$ −4.48528 −0.492324 −0.246162 0.969229i $$-0.579169\pi$$
−0.246162 + 0.969229i $$0.579169\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 28.1421 3.03464
$$87$$ 0 0
$$88$$ 4.41421 0.470557
$$89$$ −3.65685 −0.387626 −0.193813 0.981039i $$-0.562085\pi$$
−0.193813 + 0.981039i $$0.562085\pi$$
$$90$$ 0 0
$$91$$ −1.17157 −0.122814
$$92$$ 3.82843 0.399141
$$93$$ 0 0
$$94$$ −18.0711 −1.86389
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.82843 −0.591787 −0.295894 0.955221i $$-0.595617\pi$$
−0.295894 + 0.955221i $$0.595617\pi$$
$$98$$ −16.4853 −1.66526
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −14.8995 −1.48256 −0.741278 0.671199i $$-0.765780\pi$$
−0.741278 + 0.671199i $$0.765780\pi$$
$$102$$ 0 0
$$103$$ 13.6569 1.34565 0.672825 0.739802i $$-0.265081\pi$$
0.672825 + 0.739802i $$0.265081\pi$$
$$104$$ 12.4853 1.22428
$$105$$ 0 0
$$106$$ 18.4853 1.79545
$$107$$ −5.31371 −0.513696 −0.256848 0.966452i $$-0.582684\pi$$
−0.256848 + 0.966452i $$0.582684\pi$$
$$108$$ 0 0
$$109$$ 5.31371 0.508961 0.254480 0.967078i $$-0.418096\pi$$
0.254480 + 0.967078i $$0.418096\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.24264 −0.117419
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.48528 −0.416448
$$117$$ 0 0
$$118$$ −26.5563 −2.44471
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 21.3137 1.92965
$$123$$ 0 0
$$124$$ −32.4853 −2.91726
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.24264 −0.642680 −0.321340 0.946964i $$-0.604133\pi$$
−0.321340 + 0.946964i $$0.604133\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −6.82843 −0.596602 −0.298301 0.954472i $$-0.596420\pi$$
−0.298301 + 0.954472i $$0.596420\pi$$
$$132$$ 0 0
$$133$$ −2.65685 −0.230378
$$134$$ 0.828427 0.0715652
$$135$$ 0 0
$$136$$ 10.6569 0.913818
$$137$$ −12.1421 −1.03737 −0.518686 0.854965i $$-0.673579\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$138$$ 0 0
$$139$$ −18.9706 −1.60906 −0.804531 0.593911i $$-0.797583\pi$$
−0.804531 + 0.593911i $$0.797583\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −18.8995 −1.58601
$$143$$ 2.82843 0.236525
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 21.3137 1.76394
$$147$$ 0 0
$$148$$ 0.656854 0.0539931
$$149$$ −7.72792 −0.633096 −0.316548 0.948576i $$-0.602524\pi$$
−0.316548 + 0.948576i $$0.602524\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ 28.3137 2.29655
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ 31.9706 2.54344
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.414214 −0.0326446
$$162$$ 0 0
$$163$$ −15.7990 −1.23747 −0.618736 0.785599i $$-0.712355\pi$$
−0.618736 + 0.785599i $$0.712355\pi$$
$$164$$ 41.7279 3.25840
$$165$$ 0 0
$$166$$ −10.8284 −0.840449
$$167$$ 21.7990 1.68686 0.843428 0.537242i $$-0.180534\pi$$
0.843428 + 0.537242i $$0.180534\pi$$
$$168$$ 0 0
$$169$$ −5.00000 −0.384615
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 44.6274 3.40281
$$173$$ −12.5563 −0.954642 −0.477321 0.878729i $$-0.658392\pi$$
−0.477321 + 0.878729i $$0.658392\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −8.82843 −0.661719
$$179$$ −16.7990 −1.25562 −0.627808 0.778368i $$-0.716048\pi$$
−0.627808 + 0.778368i $$0.716048\pi$$
$$180$$ 0 0
$$181$$ −21.9706 −1.63306 −0.816530 0.577304i $$-0.804105\pi$$
−0.816530 + 0.577304i $$0.804105\pi$$
$$182$$ −2.82843 −0.209657
$$183$$ 0 0
$$184$$ 4.41421 0.325420
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.41421 0.176545
$$188$$ −28.6569 −2.09002
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.17157 −0.446559 −0.223280 0.974754i $$-0.571676\pi$$
−0.223280 + 0.974754i $$0.571676\pi$$
$$192$$ 0 0
$$193$$ −3.31371 −0.238526 −0.119263 0.992863i $$-0.538053\pi$$
−0.119263 + 0.992863i $$0.538053\pi$$
$$194$$ −14.0711 −1.01024
$$195$$ 0 0
$$196$$ −26.1421 −1.86730
$$197$$ 4.75736 0.338948 0.169474 0.985535i $$-0.445793\pi$$
0.169474 + 0.985535i $$0.445793\pi$$
$$198$$ 0 0
$$199$$ 10.8284 0.767607 0.383803 0.923415i $$-0.374614\pi$$
0.383803 + 0.923415i $$0.374614\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −35.9706 −2.53088
$$203$$ 0.485281 0.0340601
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 32.9706 2.29717
$$207$$ 0 0
$$208$$ 8.48528 0.588348
$$209$$ 6.41421 0.443680
$$210$$ 0 0
$$211$$ −13.3137 −0.916553 −0.458277 0.888810i $$-0.651533\pi$$
−0.458277 + 0.888810i $$0.651533\pi$$
$$212$$ 29.3137 2.01327
$$213$$ 0 0
$$214$$ −12.8284 −0.876933
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.51472 0.238595
$$218$$ 12.8284 0.868851
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.82843 0.459330
$$222$$ 0 0
$$223$$ 21.1716 1.41775 0.708877 0.705332i $$-0.249202\pi$$
0.708877 + 0.705332i $$0.249202\pi$$
$$224$$ 0.656854 0.0438879
$$225$$ 0 0
$$226$$ −24.1421 −1.60591
$$227$$ −18.4853 −1.22691 −0.613456 0.789729i $$-0.710221\pi$$
−0.613456 + 0.789729i $$0.710221\pi$$
$$228$$ 0 0
$$229$$ −2.51472 −0.166177 −0.0830886 0.996542i $$-0.526478\pi$$
−0.0830886 + 0.996542i $$0.526478\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −5.17157 −0.339530
$$233$$ 16.5563 1.08464 0.542321 0.840171i $$-0.317546\pi$$
0.542321 + 0.840171i $$0.317546\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −42.1127 −2.74130
$$237$$ 0 0
$$238$$ −2.41421 −0.156490
$$239$$ −23.6569 −1.53023 −0.765117 0.643891i $$-0.777319\pi$$
−0.765117 + 0.643891i $$0.777319\pi$$
$$240$$ 0 0
$$241$$ 14.1421 0.910975 0.455488 0.890242i $$-0.349465\pi$$
0.455488 + 0.890242i $$0.349465\pi$$
$$242$$ 2.41421 0.155192
$$243$$ 0 0
$$244$$ 33.7990 2.16376
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 18.1421 1.15436
$$248$$ −37.4558 −2.37845
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −8.97056 −0.566217 −0.283108 0.959088i $$-0.591366\pi$$
−0.283108 + 0.959088i $$0.591366\pi$$
$$252$$ 0 0
$$253$$ 1.00000 0.0628695
$$254$$ −17.4853 −1.09712
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ 9.31371 0.580973 0.290487 0.956879i $$-0.406183\pi$$
0.290487 + 0.956879i $$0.406183\pi$$
$$258$$ 0 0
$$259$$ −0.0710678 −0.00441594
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −16.4853 −1.01846
$$263$$ 22.9706 1.41643 0.708213 0.705999i $$-0.249502\pi$$
0.708213 + 0.705999i $$0.249502\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.41421 −0.393281
$$267$$ 0 0
$$268$$ 1.31371 0.0802475
$$269$$ 15.7990 0.963281 0.481641 0.876369i $$-0.340041\pi$$
0.481641 + 0.876369i $$0.340041\pi$$
$$270$$ 0 0
$$271$$ 8.89949 0.540606 0.270303 0.962775i $$-0.412876\pi$$
0.270303 + 0.962775i $$0.412876\pi$$
$$272$$ 7.24264 0.439150
$$273$$ 0 0
$$274$$ −29.3137 −1.77091
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.82843 −0.290112 −0.145056 0.989423i $$-0.546336\pi$$
−0.145056 + 0.989423i $$0.546336\pi$$
$$278$$ −45.7990 −2.74684
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −32.0711 −1.91320 −0.956600 0.291405i $$-0.905877\pi$$
−0.956600 + 0.291405i $$0.905877\pi$$
$$282$$ 0 0
$$283$$ 0.899495 0.0534694 0.0267347 0.999643i $$-0.491489\pi$$
0.0267347 + 0.999643i $$0.491489\pi$$
$$284$$ −29.9706 −1.77843
$$285$$ 0 0
$$286$$ 6.82843 0.403773
$$287$$ −4.51472 −0.266495
$$288$$ 0 0
$$289$$ −11.1716 −0.657151
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 33.7990 1.97794
$$293$$ 17.5858 1.02737 0.513686 0.857978i $$-0.328280\pi$$
0.513686 + 0.857978i $$0.328280\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0.757359 0.0440206
$$297$$ 0 0
$$298$$ −18.6569 −1.08076
$$299$$ 2.82843 0.163572
$$300$$ 0 0
$$301$$ −4.82843 −0.278306
$$302$$ 33.7990 1.94491
$$303$$ 0 0
$$304$$ 19.2426 1.10364
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −6.68629 −0.381607 −0.190803 0.981628i $$-0.561109\pi$$
−0.190803 + 0.981628i $$0.561109\pi$$
$$308$$ −1.58579 −0.0903586
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −13.6569 −0.774409 −0.387205 0.921994i $$-0.626559\pi$$
−0.387205 + 0.921994i $$0.626559\pi$$
$$312$$ 0 0
$$313$$ 27.1421 1.53416 0.767082 0.641549i $$-0.221708\pi$$
0.767082 + 0.641549i $$0.221708\pi$$
$$314$$ −14.4853 −0.817452
$$315$$ 0 0
$$316$$ 50.6985 2.85201
$$317$$ 30.8284 1.73150 0.865748 0.500479i $$-0.166843\pi$$
0.865748 + 0.500479i $$0.166843\pi$$
$$318$$ 0 0
$$319$$ −1.17157 −0.0655955
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1.00000 −0.0557278
$$323$$ 15.4853 0.861624
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −38.1421 −2.11250
$$327$$ 0 0
$$328$$ 48.1127 2.65658
$$329$$ 3.10051 0.170936
$$330$$ 0 0
$$331$$ −32.1421 −1.76669 −0.883346 0.468722i $$-0.844715\pi$$
−0.883346 + 0.468722i $$0.844715\pi$$
$$332$$ −17.1716 −0.942412
$$333$$ 0 0
$$334$$ 52.6274 2.87964
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4.14214 −0.225637 −0.112818 0.993616i $$-0.535988\pi$$
−0.112818 + 0.993616i $$0.535988\pi$$
$$338$$ −12.0711 −0.656580
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −8.48528 −0.459504
$$342$$ 0 0
$$343$$ 5.72792 0.309279
$$344$$ 51.4558 2.77431
$$345$$ 0 0
$$346$$ −30.3137 −1.62968
$$347$$ −21.1716 −1.13655 −0.568275 0.822839i $$-0.692389\pi$$
−0.568275 + 0.822839i $$0.692389\pi$$
$$348$$ 0 0
$$349$$ −2.48528 −0.133034 −0.0665170 0.997785i $$-0.521189\pi$$
−0.0665170 + 0.997785i $$0.521189\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.58579 −0.0845227
$$353$$ −4.48528 −0.238727 −0.119364 0.992851i $$-0.538085\pi$$
−0.119364 + 0.992851i $$0.538085\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ −40.5563 −2.14347
$$359$$ −15.5147 −0.818836 −0.409418 0.912347i $$-0.634268\pi$$
−0.409418 + 0.912347i $$0.634268\pi$$
$$360$$ 0 0
$$361$$ 22.1421 1.16538
$$362$$ −53.0416 −2.78781
$$363$$ 0 0
$$364$$ −4.48528 −0.235093
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1.31371 −0.0685750 −0.0342875 0.999412i $$-0.510916\pi$$
−0.0342875 + 0.999412i $$0.510916\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3.17157 −0.164660
$$372$$ 0 0
$$373$$ 23.6569 1.22491 0.612453 0.790507i $$-0.290183\pi$$
0.612453 + 0.790507i $$0.290183\pi$$
$$374$$ 5.82843 0.301381
$$375$$ 0 0
$$376$$ −33.0416 −1.70399
$$377$$ −3.31371 −0.170665
$$378$$ 0 0
$$379$$ 9.17157 0.471112 0.235556 0.971861i $$-0.424309\pi$$
0.235556 + 0.971861i $$0.424309\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −14.8995 −0.762324
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 0 0
$$388$$ −22.3137 −1.13281
$$389$$ 17.6569 0.895238 0.447619 0.894224i $$-0.352272\pi$$
0.447619 + 0.894224i $$0.352272\pi$$
$$390$$ 0 0
$$391$$ 2.41421 0.122092
$$392$$ −30.1421 −1.52241
$$393$$ 0 0
$$394$$ 11.4853 0.578620
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 35.9411 1.80383 0.901917 0.431910i $$-0.142160\pi$$
0.901917 + 0.431910i $$0.142160\pi$$
$$398$$ 26.1421 1.31039
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −31.7990 −1.58797 −0.793983 0.607940i $$-0.791996\pi$$
−0.793983 + 0.607940i $$0.791996\pi$$
$$402$$ 0 0
$$403$$ −24.0000 −1.19553
$$404$$ −57.0416 −2.83793
$$405$$ 0 0
$$406$$ 1.17157 0.0581442
$$407$$ 0.171573 0.00850455
$$408$$ 0 0
$$409$$ −4.14214 −0.204815 −0.102408 0.994743i $$-0.532655\pi$$
−0.102408 + 0.994743i $$0.532655\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 52.2843 2.57586
$$413$$ 4.55635 0.224203
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.48528 −0.219909
$$417$$ 0 0
$$418$$ 15.4853 0.757410
$$419$$ 25.4853 1.24504 0.622519 0.782605i $$-0.286109\pi$$
0.622519 + 0.782605i $$0.286109\pi$$
$$420$$ 0 0
$$421$$ −27.0000 −1.31590 −0.657950 0.753062i $$-0.728576\pi$$
−0.657950 + 0.753062i $$0.728576\pi$$
$$422$$ −32.1421 −1.56465
$$423$$ 0 0
$$424$$ 33.7990 1.64142
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3.65685 −0.176968
$$428$$ −20.3431 −0.983323
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −1.17157 −0.0564327 −0.0282163 0.999602i $$-0.508983\pi$$
−0.0282163 + 0.999602i $$0.508983\pi$$
$$432$$ 0 0
$$433$$ 13.3137 0.639816 0.319908 0.947449i $$-0.396348\pi$$
0.319908 + 0.947449i $$0.396348\pi$$
$$434$$ 8.48528 0.407307
$$435$$ 0 0
$$436$$ 20.3431 0.974260
$$437$$ 6.41421 0.306833
$$438$$ 0 0
$$439$$ 2.27208 0.108440 0.0542202 0.998529i $$-0.482733\pi$$
0.0542202 + 0.998529i $$0.482733\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 16.4853 0.784125
$$443$$ −7.97056 −0.378693 −0.189346 0.981910i $$-0.560637\pi$$
−0.189346 + 0.981910i $$0.560637\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 51.1127 2.42026
$$447$$ 0 0
$$448$$ 4.07107 0.192340
$$449$$ 6.48528 0.306059 0.153030 0.988222i $$-0.451097\pi$$
0.153030 + 0.988222i $$0.451097\pi$$
$$450$$ 0 0
$$451$$ 10.8995 0.513237
$$452$$ −38.2843 −1.80074
$$453$$ 0 0
$$454$$ −44.6274 −2.09447
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.85786 −0.180463 −0.0902316 0.995921i $$-0.528761\pi$$
−0.0902316 + 0.995921i $$0.528761\pi$$
$$458$$ −6.07107 −0.283682
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 32.7696 1.52623 0.763115 0.646263i $$-0.223669\pi$$
0.763115 + 0.646263i $$0.223669\pi$$
$$462$$ 0 0
$$463$$ −34.9706 −1.62522 −0.812610 0.582808i $$-0.801954\pi$$
−0.812610 + 0.582808i $$0.801954\pi$$
$$464$$ −3.51472 −0.163167
$$465$$ 0 0
$$466$$ 39.9706 1.85160
$$467$$ 10.6274 0.491778 0.245889 0.969298i $$-0.420920\pi$$
0.245889 + 0.969298i $$0.420920\pi$$
$$468$$ 0 0
$$469$$ −0.142136 −0.00656321
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −48.5563 −2.23499
$$473$$ 11.6569 0.535983
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.82843 −0.175476
$$477$$ 0 0
$$478$$ −57.1127 −2.61227
$$479$$ 24.4853 1.11876 0.559381 0.828911i $$-0.311039\pi$$
0.559381 + 0.828911i $$0.311039\pi$$
$$480$$ 0 0
$$481$$ 0.485281 0.0221269
$$482$$ 34.1421 1.55513
$$483$$ 0 0
$$484$$ 3.82843 0.174019
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 6.48528 0.293876 0.146938 0.989146i $$-0.453058\pi$$
0.146938 + 0.989146i $$0.453058\pi$$
$$488$$ 38.9706 1.76411
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 24.1421 1.08952 0.544760 0.838592i $$-0.316621\pi$$
0.544760 + 0.838592i $$0.316621\pi$$
$$492$$ 0 0
$$493$$ −2.82843 −0.127386
$$494$$ 43.7990 1.97061
$$495$$ 0 0
$$496$$ −25.4558 −1.14300
$$497$$ 3.24264 0.145452
$$498$$ 0 0
$$499$$ 35.1716 1.57450 0.787248 0.616637i $$-0.211505\pi$$
0.787248 + 0.616637i $$0.211505\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −21.6569 −0.966593
$$503$$ −34.2843 −1.52866 −0.764330 0.644825i $$-0.776930\pi$$
−0.764330 + 0.644825i $$0.776930\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.41421 0.107325
$$507$$ 0 0
$$508$$ −27.7279 −1.23023
$$509$$ −4.62742 −0.205107 −0.102553 0.994728i $$-0.532701\pi$$
−0.102553 + 0.994728i $$0.532701\pi$$
$$510$$ 0 0
$$511$$ −3.65685 −0.161770
$$512$$ −31.2426 −1.38074
$$513$$ 0 0
$$514$$ 22.4853 0.991783
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −7.48528 −0.329202
$$518$$ −0.171573 −0.00753848
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 36.1421 1.58342 0.791708 0.610900i $$-0.209192\pi$$
0.791708 + 0.610900i $$0.209192\pi$$
$$522$$ 0 0
$$523$$ −42.2132 −1.84585 −0.922927 0.384974i $$-0.874210\pi$$
−0.922927 + 0.384974i $$0.874210\pi$$
$$524$$ −26.1421 −1.14202
$$525$$ 0 0
$$526$$ 55.4558 2.41799
$$527$$ −20.4853 −0.892353
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −10.1716 −0.440994
$$533$$ 30.8284 1.33533
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 1.51472 0.0654259
$$537$$ 0 0
$$538$$ 38.1421 1.64442
$$539$$ −6.82843 −0.294121
$$540$$ 0 0
$$541$$ 11.3137 0.486414 0.243207 0.969974i $$-0.421801\pi$$
0.243207 + 0.969974i $$0.421801\pi$$
$$542$$ 21.4853 0.922872
$$543$$ 0 0
$$544$$ −3.82843 −0.164142
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −35.8701 −1.53369 −0.766846 0.641831i $$-0.778175\pi$$
−0.766846 + 0.641831i $$0.778175\pi$$
$$548$$ −46.4853 −1.98575
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7.51472 −0.320138
$$552$$ 0 0
$$553$$ −5.48528 −0.233258
$$554$$ −11.6569 −0.495252
$$555$$ 0 0
$$556$$ −72.6274 −3.08009
$$557$$ −5.17157 −0.219127 −0.109563 0.993980i $$-0.534945\pi$$
−0.109563 + 0.993980i $$0.534945\pi$$
$$558$$ 0 0
$$559$$ 32.9706 1.39451
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −77.4264 −3.26604
$$563$$ 15.3137 0.645396 0.322698 0.946502i $$-0.395410\pi$$
0.322698 + 0.946502i $$0.395410\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 2.17157 0.0912780
$$567$$ 0 0
$$568$$ −34.5563 −1.44995
$$569$$ −15.2426 −0.639005 −0.319502 0.947585i $$-0.603516\pi$$
−0.319502 + 0.947585i $$0.603516\pi$$
$$570$$ 0 0
$$571$$ −9.02944 −0.377870 −0.188935 0.981990i $$-0.560504\pi$$
−0.188935 + 0.981990i $$0.560504\pi$$
$$572$$ 10.8284 0.452759
$$573$$ 0 0
$$574$$ −10.8995 −0.454936
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −23.9706 −0.997908 −0.498954 0.866629i $$-0.666282\pi$$
−0.498954 + 0.866629i $$0.666282\pi$$
$$578$$ −26.9706 −1.12183
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1.85786 0.0770772
$$582$$ 0 0
$$583$$ 7.65685 0.317115
$$584$$ 38.9706 1.61261
$$585$$ 0 0
$$586$$ 42.4558 1.75383
$$587$$ −36.6569 −1.51299 −0.756495 0.653999i $$-0.773090\pi$$
−0.756495 + 0.653999i $$0.773090\pi$$
$$588$$ 0 0
$$589$$ −54.4264 −2.24260
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0.514719 0.0211548
$$593$$ 3.79899 0.156006 0.0780029 0.996953i $$-0.475146\pi$$
0.0780029 + 0.996953i $$0.475146\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −29.5858 −1.21188
$$597$$ 0 0
$$598$$ 6.82843 0.279235
$$599$$ −36.3137 −1.48374 −0.741869 0.670545i $$-0.766060\pi$$
−0.741869 + 0.670545i $$0.766060\pi$$
$$600$$ 0 0
$$601$$ −14.8284 −0.604864 −0.302432 0.953171i $$-0.597799\pi$$
−0.302432 + 0.953171i $$0.597799\pi$$
$$602$$ −11.6569 −0.475098
$$603$$ 0 0
$$604$$ 53.5980 2.18087
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2.97056 0.120571 0.0602857 0.998181i $$-0.480799\pi$$
0.0602857 + 0.998181i $$0.480799\pi$$
$$608$$ −10.1716 −0.412512
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −21.1716 −0.856510
$$612$$ 0 0
$$613$$ 42.0000 1.69636 0.848182 0.529705i $$-0.177697\pi$$
0.848182 + 0.529705i $$0.177697\pi$$
$$614$$ −16.1421 −0.651444
$$615$$ 0 0
$$616$$ −1.82843 −0.0736694
$$617$$ −9.85786 −0.396863 −0.198431 0.980115i $$-0.563585\pi$$
−0.198431 + 0.980115i $$0.563585\pi$$
$$618$$ 0 0
$$619$$ 38.6274 1.55257 0.776283 0.630384i $$-0.217103\pi$$
0.776283 + 0.630384i $$0.217103\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −32.9706 −1.32200
$$623$$ 1.51472 0.0606859
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 65.5269 2.61898
$$627$$ 0 0
$$628$$ −22.9706 −0.916625
$$629$$ 0.414214 0.0165158
$$630$$ 0 0
$$631$$ 26.6274 1.06002 0.530010 0.847991i $$-0.322188\pi$$
0.530010 + 0.847991i $$0.322188\pi$$
$$632$$ 58.4558 2.32525
$$633$$ 0 0
$$634$$ 74.4264 2.95585
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −19.3137 −0.765237
$$638$$ −2.82843 −0.111979
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.4853 1.67807 0.839034 0.544079i $$-0.183121\pi$$
0.839034 + 0.544079i $$0.183121\pi$$
$$642$$ 0 0
$$643$$ 32.9706 1.30023 0.650116 0.759835i $$-0.274720\pi$$
0.650116 + 0.759835i $$0.274720\pi$$
$$644$$ −1.58579 −0.0624887
$$645$$ 0 0
$$646$$ 37.3848 1.47088
$$647$$ 17.3431 0.681829 0.340915 0.940094i $$-0.389263\pi$$
0.340915 + 0.940094i $$0.389263\pi$$
$$648$$ 0 0
$$649$$ −11.0000 −0.431788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −60.4853 −2.36879
$$653$$ −40.4853 −1.58431 −0.792156 0.610319i $$-0.791041\pi$$
−0.792156 + 0.610319i $$0.791041\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 32.6985 1.27666
$$657$$ 0 0
$$658$$ 7.48528 0.291807
$$659$$ −15.1127 −0.588707 −0.294354 0.955697i $$-0.595104\pi$$
−0.294354 + 0.955697i $$0.595104\pi$$
$$660$$ 0 0
$$661$$ −34.6569 −1.34800 −0.673998 0.738733i $$-0.735424\pi$$
−0.673998 + 0.738733i $$0.735424\pi$$
$$662$$ −77.5980 −3.01593
$$663$$ 0 0
$$664$$ −19.7990 −0.768350
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.17157 −0.0453635
$$668$$ 83.4558 3.22900
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.82843 0.340818
$$672$$ 0 0
$$673$$ −11.6569 −0.449339 −0.224669 0.974435i $$-0.572130\pi$$
−0.224669 + 0.974435i $$0.572130\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ 0 0
$$676$$ −19.1421 −0.736236
$$677$$ 40.4853 1.55598 0.777988 0.628279i $$-0.216240\pi$$
0.777988 + 0.628279i $$0.216240\pi$$
$$678$$ 0 0
$$679$$ 2.41421 0.0926490
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −20.4853 −0.784422
$$683$$ 29.4853 1.12822 0.564111 0.825699i $$-0.309219\pi$$
0.564111 + 0.825699i $$0.309219\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 13.8284 0.527972
$$687$$ 0 0
$$688$$ 34.9706 1.33324
$$689$$ 21.6569 0.825060
$$690$$ 0 0
$$691$$ 15.4558 0.587968 0.293984 0.955810i $$-0.405019\pi$$
0.293984 + 0.955810i $$0.405019\pi$$
$$692$$ −48.0711 −1.82739
$$693$$ 0 0
$$694$$ −51.1127 −1.94021
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 26.3137 0.996703
$$698$$ −6.00000 −0.227103
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 4.61522 0.174315 0.0871573 0.996195i $$-0.472222\pi$$
0.0871573 + 0.996195i $$0.472222\pi$$
$$702$$ 0 0
$$703$$ 1.10051 0.0415063
$$704$$ −9.82843 −0.370423
$$705$$ 0 0
$$706$$ −10.8284 −0.407533
$$707$$ 6.17157 0.232106
$$708$$ 0 0
$$709$$ −0.857864 −0.0322178 −0.0161089 0.999870i $$-0.505128\pi$$
−0.0161089 + 0.999870i $$0.505128\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −16.1421 −0.604952
$$713$$ −8.48528 −0.317776
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −64.3137 −2.40352
$$717$$ 0 0
$$718$$ −37.4558 −1.39784
$$719$$ 1.65685 0.0617902 0.0308951 0.999523i $$-0.490164\pi$$
0.0308951 + 0.999523i $$0.490164\pi$$
$$720$$ 0 0
$$721$$ −5.65685 −0.210672
$$722$$ 53.4558 1.98942
$$723$$ 0 0
$$724$$ −84.1127 −3.12602
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 16.9706 0.629403 0.314702 0.949191i $$-0.398096\pi$$
0.314702 + 0.949191i $$0.398096\pi$$
$$728$$ −5.17157 −0.191671
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 28.1421 1.04087
$$732$$ 0 0
$$733$$ −3.85786 −0.142493 −0.0712467 0.997459i $$-0.522698\pi$$
−0.0712467 + 0.997459i $$0.522698\pi$$
$$734$$ −3.17157 −0.117065
$$735$$ 0 0
$$736$$ −1.58579 −0.0584529
$$737$$ 0.343146 0.0126399
$$738$$ 0 0
$$739$$ 17.5858 0.646904 0.323452 0.946245i $$-0.395157\pi$$
0.323452 + 0.946245i $$0.395157\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −7.65685 −0.281092
$$743$$ 31.1127 1.14141 0.570707 0.821154i $$-0.306669\pi$$
0.570707 + 0.821154i $$0.306669\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 57.1127 2.09104
$$747$$ 0 0
$$748$$ 9.24264 0.337944
$$749$$ 2.20101 0.0804232
$$750$$ 0 0
$$751$$ −47.5980 −1.73687 −0.868437 0.495799i $$-0.834875\pi$$
−0.868437 + 0.495799i $$0.834875\pi$$
$$752$$ −22.4558 −0.818880
$$753$$ 0 0
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 33.3137 1.21081 0.605404 0.795919i $$-0.293012\pi$$
0.605404 + 0.795919i $$0.293012\pi$$
$$758$$ 22.1421 0.804239
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.8284 0.392530 0.196265 0.980551i $$-0.437119\pi$$
0.196265 + 0.980551i $$0.437119\pi$$
$$762$$ 0 0
$$763$$ −2.20101 −0.0796819
$$764$$ −23.6274 −0.854810
$$765$$ 0 0
$$766$$ 48.2843 1.74458
$$767$$ −31.1127 −1.12341
$$768$$ 0 0
$$769$$ 3.65685 0.131870 0.0659348 0.997824i $$-0.478997\pi$$
0.0659348 + 0.997824i $$0.478997\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −12.6863 −0.456590
$$773$$ 4.82843 0.173666 0.0868332 0.996223i $$-0.472325\pi$$
0.0868332 + 0.996223i $$0.472325\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −25.7279 −0.923579
$$777$$ 0 0
$$778$$ 42.6274 1.52827
$$779$$ 69.9117 2.50485
$$780$$ 0 0
$$781$$ −7.82843 −0.280123
$$782$$ 5.82843 0.208424
$$783$$ 0 0
$$784$$ −20.4853 −0.731617
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22.0711 0.786749 0.393374 0.919378i $$-0.371308\pi$$
0.393374 + 0.919378i $$0.371308\pi$$
$$788$$ 18.2132 0.648819
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 4.14214 0.147277
$$792$$ 0 0
$$793$$ 24.9706 0.886731
$$794$$ 86.7696 3.07934
$$795$$ 0 0
$$796$$ 41.4558 1.46936
$$797$$ 7.02944 0.248995 0.124498 0.992220i $$-0.460268\pi$$
0.124498 + 0.992220i $$0.460268\pi$$
$$798$$ 0 0
$$799$$ −18.0711 −0.639308
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −76.7696 −2.71083
$$803$$ 8.82843 0.311548
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −57.9411 −2.04089
$$807$$ 0 0
$$808$$ −65.7696 −2.31376
$$809$$ 17.7279 0.623281 0.311640 0.950200i $$-0.399122\pi$$
0.311640 + 0.950200i $$0.399122\pi$$
$$810$$ 0 0
$$811$$ −32.2132 −1.13116 −0.565579 0.824694i $$-0.691347\pi$$
−0.565579 + 0.824694i $$0.691347\pi$$
$$812$$ 1.85786 0.0651983
$$813$$ 0 0
$$814$$ 0.414214 0.0145182
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 74.7696 2.61586
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 23.7990 0.830590 0.415295 0.909687i $$-0.363678\pi$$
0.415295 + 0.909687i $$0.363678\pi$$
$$822$$ 0 0
$$823$$ −18.9706 −0.661272 −0.330636 0.943758i $$-0.607263\pi$$
−0.330636 + 0.943758i $$0.607263\pi$$
$$824$$ 60.2843 2.10010
$$825$$ 0 0
$$826$$ 11.0000 0.382739
$$827$$ −35.3137 −1.22798 −0.613989 0.789315i $$-0.710436\pi$$
−0.613989 + 0.789315i $$0.710436\pi$$
$$828$$ 0 0
$$829$$ 19.9411 0.692584 0.346292 0.938127i $$-0.387441\pi$$
0.346292 + 0.938127i $$0.387441\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −27.7990 −0.963757
$$833$$ −16.4853 −0.571181
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 24.5563 0.849299
$$837$$ 0 0
$$838$$ 61.5269 2.12541
$$839$$ 24.6863 0.852265 0.426133 0.904661i $$-0.359876\pi$$
0.426133 + 0.904661i $$0.359876\pi$$
$$840$$ 0 0
$$841$$ −27.6274 −0.952670
$$842$$ −65.1838 −2.24638
$$843$$ 0 0
$$844$$ −50.9706 −1.75448
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −0.414214 −0.0142325
$$848$$ 22.9706 0.788812
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0.171573 0.00588144
$$852$$ 0 0
$$853$$ 24.8284 0.850109 0.425055 0.905168i $$-0.360255\pi$$
0.425055 + 0.905168i $$0.360255\pi$$
$$854$$ −8.82843 −0.302103
$$855$$ 0 0
$$856$$ −23.4558 −0.801704
$$857$$ 22.6985 0.775365 0.387683 0.921793i $$-0.373276\pi$$
0.387683 + 0.921793i $$0.373276\pi$$
$$858$$ 0 0
$$859$$ −3.51472 −0.119921 −0.0599603 0.998201i $$-0.519097\pi$$
−0.0599603 + 0.998201i $$0.519097\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −2.82843 −0.0963366
$$863$$ −51.3137 −1.74674 −0.873369 0.487058i $$-0.838070\pi$$
−0.873369 + 0.487058i $$0.838070\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 32.1421 1.09223
$$867$$ 0 0
$$868$$ 13.4558 0.456721
$$869$$ 13.2426 0.449226
$$870$$ 0 0
$$871$$ 0.970563 0.0328863
$$872$$ 23.4558 0.794315
$$873$$ 0 0
$$874$$ 15.4853 0.523797
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 47.1127 1.59088 0.795441 0.606031i $$-0.207239\pi$$
0.795441 + 0.606031i $$0.207239\pi$$
$$878$$ 5.48528 0.185119
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 24.0000 0.808581 0.404290 0.914631i $$-0.367519\pi$$
0.404290 + 0.914631i $$0.367519\pi$$
$$882$$ 0 0
$$883$$ 6.62742 0.223030 0.111515 0.993763i $$-0.464430\pi$$
0.111515 + 0.993763i $$0.464430\pi$$
$$884$$ 26.1421 0.879255
$$885$$ 0 0
$$886$$ −19.2426 −0.646469
$$887$$ 6.14214 0.206233 0.103116 0.994669i $$-0.467119\pi$$
0.103116 + 0.994669i $$0.467119\pi$$
$$888$$ 0 0
$$889$$ 3.00000 0.100617
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 81.0538 2.71388
$$893$$ −48.0122 −1.60667
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8.51472 0.284457
$$897$$ 0 0
$$898$$ 15.6569 0.522476
$$899$$ 9.94113 0.331555
$$900$$ 0 0
$$901$$ 18.4853 0.615834
$$902$$ 26.3137 0.876151
$$903$$ 0 0
$$904$$ −44.1421 −1.46815
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −14.4853 −0.480976 −0.240488 0.970652i $$-0.577307\pi$$
−0.240488 + 0.970652i $$0.577307\pi$$
$$908$$ −70.7696 −2.34857
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −45.4853 −1.50699 −0.753497 0.657451i $$-0.771635\pi$$
−0.753497 + 0.657451i $$0.771635\pi$$
$$912$$ 0 0
$$913$$ −4.48528 −0.148441
$$914$$ −9.31371 −0.308070
$$915$$ 0 0
$$916$$ −9.62742 −0.318099
$$917$$ 2.82843 0.0934029
$$918$$ 0 0
$$919$$ 44.2132 1.45846 0.729230 0.684269i $$-0.239879\pi$$
0.729230 + 0.684269i $$0.239879\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 79.1127 2.60544
$$923$$ −22.1421 −0.728817
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −84.4264 −2.77442
$$927$$ 0 0
$$928$$ 1.85786 0.0609874
$$929$$ −25.7990 −0.846437 −0.423219 0.906028i $$-0.639100\pi$$
−0.423219 + 0.906028i $$0.639100\pi$$
$$930$$ 0 0
$$931$$ −43.7990 −1.43545
$$932$$ 63.3848 2.07624
$$933$$ 0 0
$$934$$ 25.6569 0.839518
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ −0.343146 −0.0112041
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 3.10051 0.101074 0.0505368 0.998722i $$-0.483907\pi$$
0.0505368 + 0.998722i $$0.483907\pi$$
$$942$$ 0 0
$$943$$ 10.8995 0.354936
$$944$$ −33.0000 −1.07406
$$945$$ 0 0
$$946$$ 28.1421 0.914980
$$947$$ 2.79899 0.0909549 0.0454775 0.998965i $$-0.485519\pi$$
0.0454775 + 0.998965i $$0.485519\pi$$
$$948$$ 0 0
$$949$$ 24.9706 0.810579
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4.41421 −0.143065
$$953$$ 19.0416 0.616819 0.308409 0.951254i $$-0.400203\pi$$
0.308409 + 0.951254i $$0.400203\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −90.5685 −2.92920
$$957$$ 0 0
$$958$$ 59.1127 1.90984
$$959$$ 5.02944 0.162409
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 1.17157 0.0377730
$$963$$ 0 0
$$964$$ 54.1421 1.74380
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −38.0000 −1.22200 −0.610999 0.791632i $$-0.709232\pi$$
−0.610999 + 0.791632i $$0.709232\pi$$
$$968$$ 4.41421 0.141878
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 27.6863 0.888495 0.444248 0.895904i $$-0.353471\pi$$
0.444248 + 0.895904i $$0.353471\pi$$
$$972$$ 0 0
$$973$$ 7.85786 0.251912
$$974$$ 15.6569 0.501678
$$975$$ 0 0
$$976$$ 26.4853 0.847773
$$977$$ −52.5685 −1.68182 −0.840908 0.541178i $$-0.817979\pi$$
−0.840908 + 0.541178i $$0.817979\pi$$
$$978$$ 0 0
$$979$$ −3.65685 −0.116874
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 58.2843 1.85993
$$983$$ −5.28427 −0.168542 −0.0842710 0.996443i $$-0.526856\pi$$
−0.0842710 + 0.996443i $$0.526856\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −6.82843 −0.217461
$$987$$ 0 0
$$988$$ 69.4558 2.20968
$$989$$ 11.6569 0.370666
$$990$$ 0 0
$$991$$ 14.2843 0.453755 0.226877 0.973923i $$-0.427148\pi$$
0.226877 + 0.973923i $$0.427148\pi$$
$$992$$ 13.4558 0.427223
$$993$$ 0 0
$$994$$ 7.82843 0.248303
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −43.2548 −1.36989 −0.684947 0.728593i $$-0.740175\pi$$
−0.684947 + 0.728593i $$0.740175\pi$$
$$998$$ 84.9117 2.68783
$$999$$ 0 0
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 2475.2.a.w.1.2 2
3.2 odd 2 825.2.a.d.1.1 2
5.2 odd 4 2475.2.c.o.199.4 4
5.3 odd 4 2475.2.c.o.199.1 4
5.4 even 2 2475.2.a.l.1.1 2
15.2 even 4 825.2.c.d.199.1 4
15.8 even 4 825.2.c.d.199.4 4
15.14 odd 2 825.2.a.f.1.2 yes 2
33.32 even 2 9075.2.a.ca.1.2 2
165.164 even 2 9075.2.a.w.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.1 2 3.2 odd 2
825.2.a.f.1.2 yes 2 15.14 odd 2
825.2.c.d.199.1 4 15.2 even 4
825.2.c.d.199.4 4 15.8 even 4
2475.2.a.l.1.1 2 5.4 even 2
2475.2.a.w.1.2 2 1.1 even 1 trivial
2475.2.c.o.199.1 4 5.3 odd 4
2475.2.c.o.199.4 4 5.2 odd 4
9075.2.a.w.1.1 2 165.164 even 2
9075.2.a.ca.1.2 2 33.32 even 2