# Properties

 Label 1815.2.a.k.1.1 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ 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 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.414214 q^{10} +1.82843 q^{12} -5.65685 q^{13} -2.00000 q^{14} +1.00000 q^{15} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} +1.17157 q^{19} +1.82843 q^{20} -4.82843 q^{21} -4.00000 q^{23} -1.58579 q^{24} +1.00000 q^{25} +2.34315 q^{26} -1.00000 q^{27} -8.82843 q^{28} -0.828427 q^{29} -0.414214 q^{30} -4.41421 q^{32} -2.82843 q^{34} -4.82843 q^{35} -1.82843 q^{36} +0.343146 q^{37} -0.485281 q^{38} +5.65685 q^{39} -1.58579 q^{40} +0.828427 q^{41} +2.00000 q^{42} +3.17157 q^{43} -1.00000 q^{45} +1.65685 q^{46} -4.00000 q^{47} -3.00000 q^{48} +16.3137 q^{49} -0.414214 q^{50} -6.82843 q^{51} +10.3431 q^{52} -13.3137 q^{53} +0.414214 q^{54} +7.65685 q^{56} -1.17157 q^{57} +0.343146 q^{58} -4.00000 q^{59} -1.82843 q^{60} +0.343146 q^{61} +4.82843 q^{63} -4.17157 q^{64} +5.65685 q^{65} +5.65685 q^{67} -12.4853 q^{68} +4.00000 q^{69} +2.00000 q^{70} +13.6569 q^{71} +1.58579 q^{72} +11.3137 q^{73} -0.142136 q^{74} -1.00000 q^{75} -2.14214 q^{76} -2.34315 q^{78} +8.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} -0.343146 q^{82} +10.0000 q^{83} +8.82843 q^{84} -6.82843 q^{85} -1.31371 q^{86} +0.828427 q^{87} -7.65685 q^{89} +0.414214 q^{90} -27.3137 q^{91} +7.31371 q^{92} +1.65685 q^{94} -1.17157 q^{95} +4.41421 q^{96} +0.343146 q^{97} -6.75736 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} - 2q^{10} - 2q^{12} - 4q^{14} + 2q^{15} + 6q^{16} + 8q^{17} + 2q^{18} + 8q^{19} - 2q^{20} - 4q^{21} - 8q^{23} - 6q^{24} + 2q^{25} + 16q^{26} - 2q^{27} - 12q^{28} + 4q^{29} + 2q^{30} - 6q^{32} - 4q^{35} + 2q^{36} + 12q^{37} + 16q^{38} - 6q^{40} - 4q^{41} + 4q^{42} + 12q^{43} - 2q^{45} - 8q^{46} - 8q^{47} - 6q^{48} + 10q^{49} + 2q^{50} - 8q^{51} + 32q^{52} - 4q^{53} - 2q^{54} + 4q^{56} - 8q^{57} + 12q^{58} - 8q^{59} + 2q^{60} + 12q^{61} + 4q^{63} - 14q^{64} - 8q^{68} + 8q^{69} + 4q^{70} + 16q^{71} + 6q^{72} + 28q^{74} - 2q^{75} + 24q^{76} - 16q^{78} - 6q^{80} + 2q^{81} - 12q^{82} + 20q^{83} + 12q^{84} - 8q^{85} + 20q^{86} - 4q^{87} - 4q^{89} - 2q^{90} - 32q^{91} - 8q^{92} - 8q^{94} - 8q^{95} + 6q^{96} + 12q^{97} - 22q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.82843 −0.914214
$$5$$ −1.00000 −0.447214
$$6$$ 0.414214 0.169102
$$7$$ 4.82843 1.82497 0.912487 0.409106i $$-0.134159\pi$$
0.912487 + 0.409106i $$0.134159\pi$$
$$8$$ 1.58579 0.560660
$$9$$ 1.00000 0.333333
$$10$$ 0.414214 0.130986
$$11$$ 0 0
$$12$$ 1.82843 0.527821
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 1.00000 0.258199
$$16$$ 3.00000 0.750000
$$17$$ 6.82843 1.65614 0.828068 0.560627i $$-0.189440\pi$$
0.828068 + 0.560627i $$0.189440\pi$$
$$18$$ −0.414214 −0.0976311
$$19$$ 1.17157 0.268777 0.134389 0.990929i $$-0.457093\pi$$
0.134389 + 0.990929i $$0.457093\pi$$
$$20$$ 1.82843 0.408849
$$21$$ −4.82843 −1.05365
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −1.58579 −0.323697
$$25$$ 1.00000 0.200000
$$26$$ 2.34315 0.459529
$$27$$ −1.00000 −0.192450
$$28$$ −8.82843 −1.66842
$$29$$ −0.828427 −0.153835 −0.0769175 0.997037i $$-0.524508\pi$$
−0.0769175 + 0.997037i $$0.524508\pi$$
$$30$$ −0.414214 −0.0756247
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ 0 0
$$34$$ −2.82843 −0.485071
$$35$$ −4.82843 −0.816153
$$36$$ −1.82843 −0.304738
$$37$$ 0.343146 0.0564128 0.0282064 0.999602i $$-0.491020\pi$$
0.0282064 + 0.999602i $$0.491020\pi$$
$$38$$ −0.485281 −0.0787230
$$39$$ 5.65685 0.905822
$$40$$ −1.58579 −0.250735
$$41$$ 0.828427 0.129379 0.0646893 0.997905i $$-0.479394\pi$$
0.0646893 + 0.997905i $$0.479394\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 3.17157 0.483660 0.241830 0.970319i $$-0.422252\pi$$
0.241830 + 0.970319i $$0.422252\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 1.65685 0.244290
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ 16.3137 2.33053
$$50$$ −0.414214 −0.0585786
$$51$$ −6.82843 −0.956171
$$52$$ 10.3431 1.43434
$$53$$ −13.3137 −1.82878 −0.914389 0.404836i $$-0.867329\pi$$
−0.914389 + 0.404836i $$0.867329\pi$$
$$54$$ 0.414214 0.0563673
$$55$$ 0 0
$$56$$ 7.65685 1.02319
$$57$$ −1.17157 −0.155179
$$58$$ 0.343146 0.0450572
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ −1.82843 −0.236049
$$61$$ 0.343146 0.0439353 0.0219677 0.999759i $$-0.493007\pi$$
0.0219677 + 0.999759i $$0.493007\pi$$
$$62$$ 0 0
$$63$$ 4.82843 0.608325
$$64$$ −4.17157 −0.521447
$$65$$ 5.65685 0.701646
$$66$$ 0 0
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ −12.4853 −1.51406
$$69$$ 4.00000 0.481543
$$70$$ 2.00000 0.239046
$$71$$ 13.6569 1.62077 0.810385 0.585897i $$-0.199258\pi$$
0.810385 + 0.585897i $$0.199258\pi$$
$$72$$ 1.58579 0.186887
$$73$$ 11.3137 1.32417 0.662085 0.749429i $$-0.269672\pi$$
0.662085 + 0.749429i $$0.269672\pi$$
$$74$$ −0.142136 −0.0165229
$$75$$ −1.00000 −0.115470
$$76$$ −2.14214 −0.245720
$$77$$ 0 0
$$78$$ −2.34315 −0.265309
$$79$$ 8.48528 0.954669 0.477334 0.878722i $$-0.341603\pi$$
0.477334 + 0.878722i $$0.341603\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ −0.343146 −0.0378941
$$83$$ 10.0000 1.09764 0.548821 0.835940i $$-0.315077\pi$$
0.548821 + 0.835940i $$0.315077\pi$$
$$84$$ 8.82843 0.963260
$$85$$ −6.82843 −0.740647
$$86$$ −1.31371 −0.141661
$$87$$ 0.828427 0.0888167
$$88$$ 0 0
$$89$$ −7.65685 −0.811625 −0.405812 0.913956i $$-0.633011\pi$$
−0.405812 + 0.913956i $$0.633011\pi$$
$$90$$ 0.414214 0.0436619
$$91$$ −27.3137 −2.86325
$$92$$ 7.31371 0.762507
$$93$$ 0 0
$$94$$ 1.65685 0.170891
$$95$$ −1.17157 −0.120201
$$96$$ 4.41421 0.450524
$$97$$ 0.343146 0.0348412 0.0174206 0.999848i $$-0.494455\pi$$
0.0174206 + 0.999848i $$0.494455\pi$$
$$98$$ −6.75736 −0.682596
$$99$$ 0 0
$$100$$ −1.82843 −0.182843
$$101$$ −4.82843 −0.480446 −0.240223 0.970718i $$-0.577221\pi$$
−0.240223 + 0.970718i $$0.577221\pi$$
$$102$$ 2.82843 0.280056
$$103$$ 19.3137 1.90304 0.951518 0.307593i $$-0.0995234\pi$$
0.951518 + 0.307593i $$0.0995234\pi$$
$$104$$ −8.97056 −0.879636
$$105$$ 4.82843 0.471206
$$106$$ 5.51472 0.535637
$$107$$ 5.31371 0.513696 0.256848 0.966452i $$-0.417316\pi$$
0.256848 + 0.966452i $$0.417316\pi$$
$$108$$ 1.82843 0.175940
$$109$$ 5.31371 0.508961 0.254480 0.967078i $$-0.418096\pi$$
0.254480 + 0.967078i $$0.418096\pi$$
$$110$$ 0 0
$$111$$ −0.343146 −0.0325700
$$112$$ 14.4853 1.36873
$$113$$ 14.9706 1.40831 0.704156 0.710045i $$-0.251326\pi$$
0.704156 + 0.710045i $$0.251326\pi$$
$$114$$ 0.485281 0.0454508
$$115$$ 4.00000 0.373002
$$116$$ 1.51472 0.140638
$$117$$ −5.65685 −0.522976
$$118$$ 1.65685 0.152526
$$119$$ 32.9706 3.02241
$$120$$ 1.58579 0.144762
$$121$$ 0 0
$$122$$ −0.142136 −0.0128684
$$123$$ −0.828427 −0.0746968
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −2.48528 −0.220533 −0.110267 0.993902i $$-0.535170\pi$$
−0.110267 + 0.993902i $$0.535170\pi$$
$$128$$ 10.5563 0.933058
$$129$$ −3.17157 −0.279241
$$130$$ −2.34315 −0.205507
$$131$$ 19.3137 1.68745 0.843723 0.536778i $$-0.180359\pi$$
0.843723 + 0.536778i $$0.180359\pi$$
$$132$$ 0 0
$$133$$ 5.65685 0.490511
$$134$$ −2.34315 −0.202417
$$135$$ 1.00000 0.0860663
$$136$$ 10.8284 0.928530
$$137$$ 9.31371 0.795724 0.397862 0.917445i $$-0.369752\pi$$
0.397862 + 0.917445i $$0.369752\pi$$
$$138$$ −1.65685 −0.141041
$$139$$ 16.4853 1.39826 0.699132 0.714993i $$-0.253570\pi$$
0.699132 + 0.714993i $$0.253570\pi$$
$$140$$ 8.82843 0.746138
$$141$$ 4.00000 0.336861
$$142$$ −5.65685 −0.474713
$$143$$ 0 0
$$144$$ 3.00000 0.250000
$$145$$ 0.828427 0.0687971
$$146$$ −4.68629 −0.387840
$$147$$ −16.3137 −1.34553
$$148$$ −0.627417 −0.0515734
$$149$$ −18.4853 −1.51437 −0.757187 0.653199i $$-0.773427\pi$$
−0.757187 + 0.653199i $$0.773427\pi$$
$$150$$ 0.414214 0.0338204
$$151$$ 0.485281 0.0394916 0.0197458 0.999805i $$-0.493714\pi$$
0.0197458 + 0.999805i $$0.493714\pi$$
$$152$$ 1.85786 0.150693
$$153$$ 6.82843 0.552046
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −10.3431 −0.828114
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −3.51472 −0.279616
$$159$$ 13.3137 1.05585
$$160$$ 4.41421 0.348974
$$161$$ −19.3137 −1.52213
$$162$$ −0.414214 −0.0325437
$$163$$ 15.3137 1.19946 0.599731 0.800202i $$-0.295274\pi$$
0.599731 + 0.800202i $$0.295274\pi$$
$$164$$ −1.51472 −0.118280
$$165$$ 0 0
$$166$$ −4.14214 −0.321492
$$167$$ −9.31371 −0.720716 −0.360358 0.932814i $$-0.617346\pi$$
−0.360358 + 0.932814i $$0.617346\pi$$
$$168$$ −7.65685 −0.590739
$$169$$ 19.0000 1.46154
$$170$$ 2.82843 0.216930
$$171$$ 1.17157 0.0895924
$$172$$ −5.79899 −0.442169
$$173$$ 2.82843 0.215041 0.107521 0.994203i $$-0.465709\pi$$
0.107521 + 0.994203i $$0.465709\pi$$
$$174$$ −0.343146 −0.0260138
$$175$$ 4.82843 0.364995
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 3.17157 0.237719
$$179$$ −6.34315 −0.474109 −0.237054 0.971496i $$-0.576182\pi$$
−0.237054 + 0.971496i $$0.576182\pi$$
$$180$$ 1.82843 0.136283
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 11.3137 0.838628
$$183$$ −0.343146 −0.0253661
$$184$$ −6.34315 −0.467623
$$185$$ −0.343146 −0.0252286
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 7.31371 0.533407
$$189$$ −4.82843 −0.351216
$$190$$ 0.485281 0.0352060
$$191$$ −5.65685 −0.409316 −0.204658 0.978834i $$-0.565608\pi$$
−0.204658 + 0.978834i $$0.565608\pi$$
$$192$$ 4.17157 0.301057
$$193$$ −2.34315 −0.168663 −0.0843317 0.996438i $$-0.526876\pi$$
−0.0843317 + 0.996438i $$0.526876\pi$$
$$194$$ −0.142136 −0.0102047
$$195$$ −5.65685 −0.405096
$$196$$ −29.8284 −2.13060
$$197$$ −8.48528 −0.604551 −0.302276 0.953221i $$-0.597746\pi$$
−0.302276 + 0.953221i $$0.597746\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 1.58579 0.112132
$$201$$ −5.65685 −0.399004
$$202$$ 2.00000 0.140720
$$203$$ −4.00000 −0.280745
$$204$$ 12.4853 0.874145
$$205$$ −0.828427 −0.0578599
$$206$$ −8.00000 −0.557386
$$207$$ −4.00000 −0.278019
$$208$$ −16.9706 −1.17670
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ −6.82843 −0.470088 −0.235044 0.971985i $$-0.575523\pi$$
−0.235044 + 0.971985i $$0.575523\pi$$
$$212$$ 24.3431 1.67189
$$213$$ −13.6569 −0.935752
$$214$$ −2.20101 −0.150458
$$215$$ −3.17157 −0.216299
$$216$$ −1.58579 −0.107899
$$217$$ 0 0
$$218$$ −2.20101 −0.149071
$$219$$ −11.3137 −0.764510
$$220$$ 0 0
$$221$$ −38.6274 −2.59836
$$222$$ 0.142136 0.00953952
$$223$$ −17.6569 −1.18239 −0.591195 0.806529i $$-0.701344\pi$$
−0.591195 + 0.806529i $$0.701344\pi$$
$$224$$ −21.3137 −1.42408
$$225$$ 1.00000 0.0666667
$$226$$ −6.20101 −0.412485
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 2.14214 0.141866
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ −1.65685 −0.109250
$$231$$ 0 0
$$232$$ −1.31371 −0.0862492
$$233$$ 13.1716 0.862898 0.431449 0.902137i $$-0.358002\pi$$
0.431449 + 0.902137i $$0.358002\pi$$
$$234$$ 2.34315 0.153176
$$235$$ 4.00000 0.260931
$$236$$ 7.31371 0.476082
$$237$$ −8.48528 −0.551178
$$238$$ −13.6569 −0.885242
$$239$$ −6.34315 −0.410304 −0.205152 0.978730i $$-0.565769\pi$$
−0.205152 + 0.978730i $$0.565769\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 23.6569 1.52387 0.761936 0.647652i $$-0.224249\pi$$
0.761936 + 0.647652i $$0.224249\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ −0.627417 −0.0401663
$$245$$ −16.3137 −1.04224
$$246$$ 0.343146 0.0218782
$$247$$ −6.62742 −0.421692
$$248$$ 0 0
$$249$$ −10.0000 −0.633724
$$250$$ 0.414214 0.0261972
$$251$$ −12.9706 −0.818695 −0.409347 0.912379i $$-0.634244\pi$$
−0.409347 + 0.912379i $$0.634244\pi$$
$$252$$ −8.82843 −0.556139
$$253$$ 0 0
$$254$$ 1.02944 0.0645926
$$255$$ 6.82843 0.427613
$$256$$ 3.97056 0.248160
$$257$$ −27.6569 −1.72519 −0.862594 0.505898i $$-0.831161\pi$$
−0.862594 + 0.505898i $$0.831161\pi$$
$$258$$ 1.31371 0.0817879
$$259$$ 1.65685 0.102952
$$260$$ −10.3431 −0.641455
$$261$$ −0.828427 −0.0512784
$$262$$ −8.00000 −0.494242
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 13.3137 0.817855
$$266$$ −2.34315 −0.143667
$$267$$ 7.65685 0.468592
$$268$$ −10.3431 −0.631808
$$269$$ −24.6274 −1.50156 −0.750780 0.660552i $$-0.770322\pi$$
−0.750780 + 0.660552i $$0.770322\pi$$
$$270$$ −0.414214 −0.0252082
$$271$$ −27.7990 −1.68867 −0.844334 0.535817i $$-0.820004\pi$$
−0.844334 + 0.535817i $$0.820004\pi$$
$$272$$ 20.4853 1.24210
$$273$$ 27.3137 1.65310
$$274$$ −3.85786 −0.233062
$$275$$ 0 0
$$276$$ −7.31371 −0.440234
$$277$$ 13.6569 0.820561 0.410280 0.911959i $$-0.365431\pi$$
0.410280 + 0.911959i $$0.365431\pi$$
$$278$$ −6.82843 −0.409542
$$279$$ 0 0
$$280$$ −7.65685 −0.457585
$$281$$ 16.8284 1.00390 0.501950 0.864897i $$-0.332616\pi$$
0.501950 + 0.864897i $$0.332616\pi$$
$$282$$ −1.65685 −0.0986642
$$283$$ 3.17157 0.188530 0.0942652 0.995547i $$-0.469950\pi$$
0.0942652 + 0.995547i $$0.469950\pi$$
$$284$$ −24.9706 −1.48173
$$285$$ 1.17157 0.0693980
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ −4.41421 −0.260110
$$289$$ 29.6274 1.74279
$$290$$ −0.343146 −0.0201502
$$291$$ −0.343146 −0.0201156
$$292$$ −20.6863 −1.21057
$$293$$ 1.17157 0.0684440 0.0342220 0.999414i $$-0.489105\pi$$
0.0342220 + 0.999414i $$0.489105\pi$$
$$294$$ 6.75736 0.394097
$$295$$ 4.00000 0.232889
$$296$$ 0.544156 0.0316284
$$297$$ 0 0
$$298$$ 7.65685 0.443550
$$299$$ 22.6274 1.30858
$$300$$ 1.82843 0.105564
$$301$$ 15.3137 0.882667
$$302$$ −0.201010 −0.0115668
$$303$$ 4.82843 0.277386
$$304$$ 3.51472 0.201583
$$305$$ −0.343146 −0.0196485
$$306$$ −2.82843 −0.161690
$$307$$ 8.82843 0.503865 0.251932 0.967745i $$-0.418934\pi$$
0.251932 + 0.967745i $$0.418934\pi$$
$$308$$ 0 0
$$309$$ −19.3137 −1.09872
$$310$$ 0 0
$$311$$ 19.3137 1.09518 0.547590 0.836747i $$-0.315545\pi$$
0.547590 + 0.836747i $$0.315545\pi$$
$$312$$ 8.97056 0.507858
$$313$$ 4.34315 0.245489 0.122745 0.992438i $$-0.460830\pi$$
0.122745 + 0.992438i $$0.460830\pi$$
$$314$$ −7.45584 −0.420758
$$315$$ −4.82843 −0.272051
$$316$$ −15.5147 −0.872771
$$317$$ 30.2843 1.70093 0.850467 0.526028i $$-0.176319\pi$$
0.850467 + 0.526028i $$0.176319\pi$$
$$318$$ −5.51472 −0.309250
$$319$$ 0 0
$$320$$ 4.17157 0.233198
$$321$$ −5.31371 −0.296582
$$322$$ 8.00000 0.445823
$$323$$ 8.00000 0.445132
$$324$$ −1.82843 −0.101579
$$325$$ −5.65685 −0.313786
$$326$$ −6.34315 −0.351314
$$327$$ −5.31371 −0.293849
$$328$$ 1.31371 0.0725374
$$329$$ −19.3137 −1.06480
$$330$$ 0 0
$$331$$ 17.6569 0.970508 0.485254 0.874373i $$-0.338727\pi$$
0.485254 + 0.874373i $$0.338727\pi$$
$$332$$ −18.2843 −1.00348
$$333$$ 0.343146 0.0188043
$$334$$ 3.85786 0.211093
$$335$$ −5.65685 −0.309067
$$336$$ −14.4853 −0.790237
$$337$$ 19.3137 1.05208 0.526042 0.850458i $$-0.323675\pi$$
0.526042 + 0.850458i $$0.323675\pi$$
$$338$$ −7.87006 −0.428075
$$339$$ −14.9706 −0.813089
$$340$$ 12.4853 0.677109
$$341$$ 0 0
$$342$$ −0.485281 −0.0262410
$$343$$ 44.9706 2.42818
$$344$$ 5.02944 0.271169
$$345$$ −4.00000 −0.215353
$$346$$ −1.17157 −0.0629841
$$347$$ 6.68629 0.358939 0.179469 0.983764i $$-0.442562\pi$$
0.179469 + 0.983764i $$0.442562\pi$$
$$348$$ −1.51472 −0.0811974
$$349$$ 22.9706 1.22959 0.614793 0.788688i $$-0.289240\pi$$
0.614793 + 0.788688i $$0.289240\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 5.65685 0.301941
$$352$$ 0 0
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ −1.65685 −0.0880608
$$355$$ −13.6569 −0.724831
$$356$$ 14.0000 0.741999
$$357$$ −32.9706 −1.74499
$$358$$ 2.62742 0.138863
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ −1.58579 −0.0835783
$$361$$ −17.6274 −0.927759
$$362$$ 5.79899 0.304788
$$363$$ 0 0
$$364$$ 49.9411 2.61763
$$365$$ −11.3137 −0.592187
$$366$$ 0.142136 0.00742955
$$367$$ −1.65685 −0.0864871 −0.0432435 0.999065i $$-0.513769\pi$$
−0.0432435 + 0.999065i $$0.513769\pi$$
$$368$$ −12.0000 −0.625543
$$369$$ 0.828427 0.0431262
$$370$$ 0.142136 0.00738928
$$371$$ −64.2843 −3.33747
$$372$$ 0 0
$$373$$ 34.6274 1.79294 0.896470 0.443105i $$-0.146123\pi$$
0.896470 + 0.443105i $$0.146123\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ −6.34315 −0.327123
$$377$$ 4.68629 0.241356
$$378$$ 2.00000 0.102869
$$379$$ −0.686292 −0.0352524 −0.0176262 0.999845i $$-0.505611\pi$$
−0.0176262 + 0.999845i $$0.505611\pi$$
$$380$$ 2.14214 0.109889
$$381$$ 2.48528 0.127325
$$382$$ 2.34315 0.119886
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ −10.5563 −0.538701
$$385$$ 0 0
$$386$$ 0.970563 0.0494003
$$387$$ 3.17157 0.161220
$$388$$ −0.627417 −0.0318523
$$389$$ −12.3431 −0.625822 −0.312911 0.949782i $$-0.601304\pi$$
−0.312911 + 0.949782i $$0.601304\pi$$
$$390$$ 2.34315 0.118650
$$391$$ −27.3137 −1.38131
$$392$$ 25.8701 1.30664
$$393$$ −19.3137 −0.974248
$$394$$ 3.51472 0.177069
$$395$$ −8.48528 −0.426941
$$396$$ 0 0
$$397$$ 18.9706 0.952105 0.476053 0.879417i $$-0.342067\pi$$
0.476053 + 0.879417i $$0.342067\pi$$
$$398$$ 4.28427 0.214751
$$399$$ −5.65685 −0.283197
$$400$$ 3.00000 0.150000
$$401$$ −29.3137 −1.46386 −0.731928 0.681382i $$-0.761379\pi$$
−0.731928 + 0.681382i $$0.761379\pi$$
$$402$$ 2.34315 0.116865
$$403$$ 0 0
$$404$$ 8.82843 0.439231
$$405$$ −1.00000 −0.0496904
$$406$$ 1.65685 0.0822283
$$407$$ 0 0
$$408$$ −10.8284 −0.536087
$$409$$ 8.34315 0.412542 0.206271 0.978495i $$-0.433867\pi$$
0.206271 + 0.978495i $$0.433867\pi$$
$$410$$ 0.343146 0.0169468
$$411$$ −9.31371 −0.459411
$$412$$ −35.3137 −1.73978
$$413$$ −19.3137 −0.950365
$$414$$ 1.65685 0.0814299
$$415$$ −10.0000 −0.490881
$$416$$ 24.9706 1.22428
$$417$$ −16.4853 −0.807288
$$418$$ 0 0
$$419$$ −3.02944 −0.147998 −0.0739988 0.997258i $$-0.523576\pi$$
−0.0739988 + 0.997258i $$0.523576\pi$$
$$420$$ −8.82843 −0.430783
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 2.82843 0.137686
$$423$$ −4.00000 −0.194487
$$424$$ −21.1127 −1.02532
$$425$$ 6.82843 0.331227
$$426$$ 5.65685 0.274075
$$427$$ 1.65685 0.0801808
$$428$$ −9.71573 −0.469627
$$429$$ 0 0
$$430$$ 1.31371 0.0633526
$$431$$ −10.3431 −0.498212 −0.249106 0.968476i $$-0.580137\pi$$
−0.249106 + 0.968476i $$0.580137\pi$$
$$432$$ −3.00000 −0.144338
$$433$$ −4.34315 −0.208718 −0.104359 0.994540i $$-0.533279\pi$$
−0.104359 + 0.994540i $$0.533279\pi$$
$$434$$ 0 0
$$435$$ −0.828427 −0.0397200
$$436$$ −9.71573 −0.465299
$$437$$ −4.68629 −0.224176
$$438$$ 4.68629 0.223920
$$439$$ 3.51472 0.167748 0.0838742 0.996476i $$-0.473271\pi$$
0.0838742 + 0.996476i $$0.473271\pi$$
$$440$$ 0 0
$$441$$ 16.3137 0.776843
$$442$$ 16.0000 0.761042
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0.627417 0.0297759
$$445$$ 7.65685 0.362970
$$446$$ 7.31371 0.346314
$$447$$ 18.4853 0.874324
$$448$$ −20.1421 −0.951626
$$449$$ −2.97056 −0.140190 −0.0700948 0.997540i $$-0.522330\pi$$
−0.0700948 + 0.997540i $$0.522330\pi$$
$$450$$ −0.414214 −0.0195262
$$451$$ 0 0
$$452$$ −27.3726 −1.28750
$$453$$ −0.485281 −0.0228005
$$454$$ −5.79899 −0.272160
$$455$$ 27.3137 1.28049
$$456$$ −1.85786 −0.0870025
$$457$$ 0.686292 0.0321034 0.0160517 0.999871i $$-0.494890\pi$$
0.0160517 + 0.999871i $$0.494890\pi$$
$$458$$ 0.828427 0.0387099
$$459$$ −6.82843 −0.318724
$$460$$ −7.31371 −0.341003
$$461$$ 28.1421 1.31071 0.655355 0.755321i $$-0.272519\pi$$
0.655355 + 0.755321i $$0.272519\pi$$
$$462$$ 0 0
$$463$$ −28.9706 −1.34638 −0.673188 0.739471i $$-0.735076\pi$$
−0.673188 + 0.739471i $$0.735076\pi$$
$$464$$ −2.48528 −0.115376
$$465$$ 0 0
$$466$$ −5.45584 −0.252737
$$467$$ 22.6274 1.04707 0.523536 0.852004i $$-0.324613\pi$$
0.523536 + 0.852004i $$0.324613\pi$$
$$468$$ 10.3431 0.478112
$$469$$ 27.3137 1.26123
$$470$$ −1.65685 −0.0764250
$$471$$ −18.0000 −0.829396
$$472$$ −6.34315 −0.291967
$$473$$ 0 0
$$474$$ 3.51472 0.161436
$$475$$ 1.17157 0.0537555
$$476$$ −60.2843 −2.76313
$$477$$ −13.3137 −0.609593
$$478$$ 2.62742 0.120175
$$479$$ −3.02944 −0.138419 −0.0692093 0.997602i $$-0.522048\pi$$
−0.0692093 + 0.997602i $$0.522048\pi$$
$$480$$ −4.41421 −0.201480
$$481$$ −1.94113 −0.0885077
$$482$$ −9.79899 −0.446332
$$483$$ 19.3137 0.878804
$$484$$ 0 0
$$485$$ −0.343146 −0.0155814
$$486$$ 0.414214 0.0187891
$$487$$ 20.9706 0.950267 0.475133 0.879914i $$-0.342400\pi$$
0.475133 + 0.879914i $$0.342400\pi$$
$$488$$ 0.544156 0.0246328
$$489$$ −15.3137 −0.692510
$$490$$ 6.75736 0.305266
$$491$$ −25.6569 −1.15788 −0.578939 0.815371i $$-0.696533\pi$$
−0.578939 + 0.815371i $$0.696533\pi$$
$$492$$ 1.51472 0.0682888
$$493$$ −5.65685 −0.254772
$$494$$ 2.74517 0.123511
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 65.9411 2.95786
$$498$$ 4.14214 0.185614
$$499$$ −33.6569 −1.50669 −0.753344 0.657627i $$-0.771560\pi$$
−0.753344 + 0.657627i $$0.771560\pi$$
$$500$$ 1.82843 0.0817697
$$501$$ 9.31371 0.416106
$$502$$ 5.37258 0.239790
$$503$$ 5.31371 0.236927 0.118463 0.992958i $$-0.462203\pi$$
0.118463 + 0.992958i $$0.462203\pi$$
$$504$$ 7.65685 0.341063
$$505$$ 4.82843 0.214862
$$506$$ 0 0
$$507$$ −19.0000 −0.843820
$$508$$ 4.54416 0.201614
$$509$$ 41.3137 1.83120 0.915599 0.402093i $$-0.131717\pi$$
0.915599 + 0.402093i $$0.131717\pi$$
$$510$$ −2.82843 −0.125245
$$511$$ 54.6274 2.41657
$$512$$ −22.7574 −1.00574
$$513$$ −1.17157 −0.0517262
$$514$$ 11.4558 0.505296
$$515$$ −19.3137 −0.851064
$$516$$ 5.79899 0.255286
$$517$$ 0 0
$$518$$ −0.686292 −0.0301539
$$519$$ −2.82843 −0.124154
$$520$$ 8.97056 0.393385
$$521$$ 12.6274 0.553217 0.276609 0.960983i $$-0.410789\pi$$
0.276609 + 0.960983i $$0.410789\pi$$
$$522$$ 0.343146 0.0150191
$$523$$ 26.4853 1.15812 0.579060 0.815285i $$-0.303420\pi$$
0.579060 + 0.815285i $$0.303420\pi$$
$$524$$ −35.3137 −1.54269
$$525$$ −4.82843 −0.210730
$$526$$ 7.45584 0.325090
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −5.51472 −0.239544
$$531$$ −4.00000 −0.173585
$$532$$ −10.3431 −0.448432
$$533$$ −4.68629 −0.202986
$$534$$ −3.17157 −0.137247
$$535$$ −5.31371 −0.229732
$$536$$ 8.97056 0.387469
$$537$$ 6.34315 0.273727
$$538$$ 10.2010 0.439797
$$539$$ 0 0
$$540$$ −1.82843 −0.0786830
$$541$$ 5.31371 0.228454 0.114227 0.993455i $$-0.463561\pi$$
0.114227 + 0.993455i $$0.463561\pi$$
$$542$$ 11.5147 0.494600
$$543$$ 14.0000 0.600798
$$544$$ −30.1421 −1.29233
$$545$$ −5.31371 −0.227614
$$546$$ −11.3137 −0.484182
$$547$$ −20.1421 −0.861216 −0.430608 0.902539i $$-0.641701\pi$$
−0.430608 + 0.902539i $$0.641701\pi$$
$$548$$ −17.0294 −0.727462
$$549$$ 0.343146 0.0146451
$$550$$ 0 0
$$551$$ −0.970563 −0.0413474
$$552$$ 6.34315 0.269982
$$553$$ 40.9706 1.74225
$$554$$ −5.65685 −0.240337
$$555$$ 0.343146 0.0145657
$$556$$ −30.1421 −1.27831
$$557$$ −10.8284 −0.458815 −0.229408 0.973330i $$-0.573679\pi$$
−0.229408 + 0.973330i $$0.573679\pi$$
$$558$$ 0 0
$$559$$ −17.9411 −0.758829
$$560$$ −14.4853 −0.612115
$$561$$ 0 0
$$562$$ −6.97056 −0.294035
$$563$$ −20.3431 −0.857361 −0.428681 0.903456i $$-0.641021\pi$$
−0.428681 + 0.903456i $$0.641021\pi$$
$$564$$ −7.31371 −0.307963
$$565$$ −14.9706 −0.629816
$$566$$ −1.31371 −0.0552193
$$567$$ 4.82843 0.202775
$$568$$ 21.6569 0.908701
$$569$$ −15.4558 −0.647943 −0.323971 0.946067i $$-0.605018\pi$$
−0.323971 + 0.946067i $$0.605018\pi$$
$$570$$ −0.485281 −0.0203262
$$571$$ −0.485281 −0.0203084 −0.0101542 0.999948i $$-0.503232\pi$$
−0.0101542 + 0.999948i $$0.503232\pi$$
$$572$$ 0 0
$$573$$ 5.65685 0.236318
$$574$$ −1.65685 −0.0691558
$$575$$ −4.00000 −0.166812
$$576$$ −4.17157 −0.173816
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ −12.2721 −0.510451
$$579$$ 2.34315 0.0973778
$$580$$ −1.51472 −0.0628953
$$581$$ 48.2843 2.00317
$$582$$ 0.142136 0.00589171
$$583$$ 0 0
$$584$$ 17.9411 0.742409
$$585$$ 5.65685 0.233882
$$586$$ −0.485281 −0.0200468
$$587$$ −30.6274 −1.26413 −0.632064 0.774916i $$-0.717792\pi$$
−0.632064 + 0.774916i $$0.717792\pi$$
$$588$$ 29.8284 1.23010
$$589$$ 0 0
$$590$$ −1.65685 −0.0682116
$$591$$ 8.48528 0.349038
$$592$$ 1.02944 0.0423096
$$593$$ 17.1716 0.705152 0.352576 0.935783i $$-0.385306\pi$$
0.352576 + 0.935783i $$0.385306\pi$$
$$594$$ 0 0
$$595$$ −32.9706 −1.35166
$$596$$ 33.7990 1.38446
$$597$$ 10.3431 0.423317
$$598$$ −9.37258 −0.383273
$$599$$ 4.68629 0.191477 0.0957383 0.995407i $$-0.469479\pi$$
0.0957383 + 0.995407i $$0.469479\pi$$
$$600$$ −1.58579 −0.0647395
$$601$$ −17.3137 −0.706241 −0.353120 0.935578i $$-0.614879\pi$$
−0.353120 + 0.935578i $$0.614879\pi$$
$$602$$ −6.34315 −0.258527
$$603$$ 5.65685 0.230365
$$604$$ −0.887302 −0.0361038
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ −18.4853 −0.750294 −0.375147 0.926965i $$-0.622408\pi$$
−0.375147 + 0.926965i $$0.622408\pi$$
$$608$$ −5.17157 −0.209735
$$609$$ 4.00000 0.162088
$$610$$ 0.142136 0.00575490
$$611$$ 22.6274 0.915407
$$612$$ −12.4853 −0.504688
$$613$$ −21.9411 −0.886194 −0.443097 0.896474i $$-0.646120\pi$$
−0.443097 + 0.896474i $$0.646120\pi$$
$$614$$ −3.65685 −0.147579
$$615$$ 0.828427 0.0334054
$$616$$ 0 0
$$617$$ 11.6569 0.469287 0.234644 0.972081i $$-0.424608\pi$$
0.234644 + 0.972081i $$0.424608\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −25.6569 −1.03124 −0.515618 0.856819i $$-0.672438\pi$$
−0.515618 + 0.856819i $$0.672438\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −8.00000 −0.320771
$$623$$ −36.9706 −1.48119
$$624$$ 16.9706 0.679366
$$625$$ 1.00000 0.0400000
$$626$$ −1.79899 −0.0719021
$$627$$ 0 0
$$628$$ −32.9117 −1.31332
$$629$$ 2.34315 0.0934273
$$630$$ 2.00000 0.0796819
$$631$$ 34.3431 1.36718 0.683590 0.729867i $$-0.260418\pi$$
0.683590 + 0.729867i $$0.260418\pi$$
$$632$$ 13.4558 0.535245
$$633$$ 6.82843 0.271406
$$634$$ −12.5442 −0.498192
$$635$$ 2.48528 0.0986254
$$636$$ −24.3431 −0.965269
$$637$$ −92.2843 −3.65644
$$638$$ 0 0
$$639$$ 13.6569 0.540257
$$640$$ −10.5563 −0.417276
$$641$$ −26.9706 −1.06527 −0.532637 0.846344i $$-0.678799\pi$$
−0.532637 + 0.846344i $$0.678799\pi$$
$$642$$ 2.20101 0.0868669
$$643$$ −29.9411 −1.18076 −0.590381 0.807124i $$-0.701023\pi$$
−0.590381 + 0.807124i $$0.701023\pi$$
$$644$$ 35.3137 1.39156
$$645$$ 3.17157 0.124881
$$646$$ −3.31371 −0.130376
$$647$$ 27.3137 1.07381 0.536906 0.843642i $$-0.319593\pi$$
0.536906 + 0.843642i $$0.319593\pi$$
$$648$$ 1.58579 0.0622956
$$649$$ 0 0
$$650$$ 2.34315 0.0919057
$$651$$ 0 0
$$652$$ −28.0000 −1.09656
$$653$$ −26.9706 −1.05544 −0.527720 0.849418i $$-0.676953\pi$$
−0.527720 + 0.849418i $$0.676953\pi$$
$$654$$ 2.20101 0.0860663
$$655$$ −19.3137 −0.754649
$$656$$ 2.48528 0.0970339
$$657$$ 11.3137 0.441390
$$658$$ 8.00000 0.311872
$$659$$ −7.31371 −0.284902 −0.142451 0.989802i $$-0.545498\pi$$
−0.142451 + 0.989802i $$0.545498\pi$$
$$660$$ 0 0
$$661$$ −13.3137 −0.517843 −0.258922 0.965898i $$-0.583367\pi$$
−0.258922 + 0.965898i $$0.583367\pi$$
$$662$$ −7.31371 −0.284255
$$663$$ 38.6274 1.50016
$$664$$ 15.8579 0.615404
$$665$$ −5.65685 −0.219363
$$666$$ −0.142136 −0.00550764
$$667$$ 3.31371 0.128307
$$668$$ 17.0294 0.658889
$$669$$ 17.6569 0.682653
$$670$$ 2.34315 0.0905236
$$671$$ 0 0
$$672$$ 21.3137 0.822194
$$673$$ −29.6569 −1.14319 −0.571594 0.820537i $$-0.693675\pi$$
−0.571594 + 0.820537i $$0.693675\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ −1.00000 −0.0384900
$$676$$ −34.7401 −1.33616
$$677$$ 21.4558 0.824615 0.412308 0.911045i $$-0.364723\pi$$
0.412308 + 0.911045i $$0.364723\pi$$
$$678$$ 6.20101 0.238148
$$679$$ 1.65685 0.0635842
$$680$$ −10.8284 −0.415251
$$681$$ −14.0000 −0.536481
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ −2.14214 −0.0819066
$$685$$ −9.31371 −0.355859
$$686$$ −18.6274 −0.711198
$$687$$ 2.00000 0.0763048
$$688$$ 9.51472 0.362745
$$689$$ 75.3137 2.86922
$$690$$ 1.65685 0.0630754
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −5.17157 −0.196594
$$693$$ 0 0
$$694$$ −2.76955 −0.105131
$$695$$ −16.4853 −0.625322
$$696$$ 1.31371 0.0497960
$$697$$ 5.65685 0.214269
$$698$$ −9.51472 −0.360137
$$699$$ −13.1716 −0.498195
$$700$$ −8.82843 −0.333683
$$701$$ −7.85786 −0.296787 −0.148394 0.988928i $$-0.547410\pi$$
−0.148394 + 0.988928i $$0.547410\pi$$
$$702$$ −2.34315 −0.0884363
$$703$$ 0.402020 0.0151625
$$704$$ 0 0
$$705$$ −4.00000 −0.150649
$$706$$ −10.7696 −0.405317
$$707$$ −23.3137 −0.876802
$$708$$ −7.31371 −0.274866
$$709$$ 29.3137 1.10090 0.550450 0.834868i $$-0.314456\pi$$
0.550450 + 0.834868i $$0.314456\pi$$
$$710$$ 5.65685 0.212298
$$711$$ 8.48528 0.318223
$$712$$ −12.1421 −0.455046
$$713$$ 0 0
$$714$$ 13.6569 0.511095
$$715$$ 0 0
$$716$$ 11.5980 0.433437
$$717$$ 6.34315 0.236889
$$718$$ 4.97056 0.185500
$$719$$ 31.5980 1.17841 0.589203 0.807985i $$-0.299442\pi$$
0.589203 + 0.807985i $$0.299442\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 93.2548 3.47299
$$722$$ 7.30152 0.271734
$$723$$ −23.6569 −0.879808
$$724$$ 25.5980 0.951341
$$725$$ −0.828427 −0.0307670
$$726$$ 0 0
$$727$$ −33.9411 −1.25881 −0.629403 0.777079i $$-0.716701\pi$$
−0.629403 + 0.777079i $$0.716701\pi$$
$$728$$ −43.3137 −1.60531
$$729$$ 1.00000 0.0370370
$$730$$ 4.68629 0.173447
$$731$$ 21.6569 0.801008
$$732$$ 0.627417 0.0231900
$$733$$ 17.6569 0.652171 0.326085 0.945340i $$-0.394270\pi$$
0.326085 + 0.945340i $$0.394270\pi$$
$$734$$ 0.686292 0.0253315
$$735$$ 16.3137 0.601740
$$736$$ 17.6569 0.650840
$$737$$ 0 0
$$738$$ −0.343146 −0.0126314
$$739$$ −47.1127 −1.73307 −0.866534 0.499118i $$-0.833658\pi$$
−0.866534 + 0.499118i $$0.833658\pi$$
$$740$$ 0.627417 0.0230643
$$741$$ 6.62742 0.243464
$$742$$ 26.6274 0.977523
$$743$$ −47.6569 −1.74836 −0.874180 0.485602i $$-0.838601\pi$$
−0.874180 + 0.485602i $$0.838601\pi$$
$$744$$ 0 0
$$745$$ 18.4853 0.677248
$$746$$ −14.3431 −0.525140
$$747$$ 10.0000 0.365881
$$748$$ 0 0
$$749$$ 25.6569 0.937481
$$750$$ −0.414214 −0.0151249
$$751$$ −36.2843 −1.32403 −0.662016 0.749490i $$-0.730299\pi$$
−0.662016 + 0.749490i $$0.730299\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 12.9706 0.472674
$$754$$ −1.94113 −0.0706916
$$755$$ −0.485281 −0.0176612
$$756$$ 8.82843 0.321087
$$757$$ 8.62742 0.313569 0.156784 0.987633i $$-0.449887\pi$$
0.156784 + 0.987633i $$0.449887\pi$$
$$758$$ 0.284271 0.0103252
$$759$$ 0 0
$$760$$ −1.85786 −0.0673918
$$761$$ 23.1716 0.839969 0.419984 0.907531i $$-0.362036\pi$$
0.419984 + 0.907531i $$0.362036\pi$$
$$762$$ −1.02944 −0.0372926
$$763$$ 25.6569 0.928840
$$764$$ 10.3431 0.374202
$$765$$ −6.82843 −0.246882
$$766$$ 3.31371 0.119729
$$767$$ 22.6274 0.817029
$$768$$ −3.97056 −0.143275
$$769$$ −33.3137 −1.20132 −0.600662 0.799503i $$-0.705096\pi$$
−0.600662 + 0.799503i $$0.705096\pi$$
$$770$$ 0 0
$$771$$ 27.6569 0.996037
$$772$$ 4.28427 0.154194
$$773$$ −7.65685 −0.275398 −0.137699 0.990474i $$-0.543971\pi$$
−0.137699 + 0.990474i $$0.543971\pi$$
$$774$$ −1.31371 −0.0472203
$$775$$ 0 0
$$776$$ 0.544156 0.0195341
$$777$$ −1.65685 −0.0594393
$$778$$ 5.11270 0.183299
$$779$$ 0.970563 0.0347740
$$780$$ 10.3431 0.370344
$$781$$ 0 0
$$782$$ 11.3137 0.404577
$$783$$ 0.828427 0.0296056
$$784$$ 48.9411 1.74790
$$785$$ −18.0000 −0.642448
$$786$$ 8.00000 0.285351
$$787$$ −8.14214 −0.290236 −0.145118 0.989414i $$-0.546356\pi$$
−0.145118 + 0.989414i $$0.546356\pi$$
$$788$$ 15.5147 0.552689
$$789$$ 18.0000 0.640817
$$790$$ 3.51472 0.125048
$$791$$ 72.2843 2.57013
$$792$$ 0 0
$$793$$ −1.94113 −0.0689314
$$794$$ −7.85786 −0.278865
$$795$$ −13.3137 −0.472189
$$796$$ 18.9117 0.670307
$$797$$ 1.02944 0.0364645 0.0182323 0.999834i $$-0.494196\pi$$
0.0182323 + 0.999834i $$0.494196\pi$$
$$798$$ 2.34315 0.0829465
$$799$$ −27.3137 −0.966290
$$800$$ −4.41421 −0.156066
$$801$$ −7.65685 −0.270542
$$802$$ 12.1421 0.428754
$$803$$ 0 0
$$804$$ 10.3431 0.364775
$$805$$ 19.3137 0.680719
$$806$$ 0 0
$$807$$ 24.6274 0.866926
$$808$$ −7.65685 −0.269367
$$809$$ 56.4264 1.98385 0.991923 0.126838i $$-0.0404829\pi$$
0.991923 + 0.126838i $$0.0404829\pi$$
$$810$$ 0.414214 0.0145540
$$811$$ 16.4853 0.578877 0.289438 0.957197i $$-0.406532\pi$$
0.289438 + 0.957197i $$0.406532\pi$$
$$812$$ 7.31371 0.256661
$$813$$ 27.7990 0.974953
$$814$$ 0 0
$$815$$ −15.3137 −0.536416
$$816$$ −20.4853 −0.717128
$$817$$ 3.71573 0.129997
$$818$$ −3.45584 −0.120831
$$819$$ −27.3137 −0.954418
$$820$$ 1.51472 0.0528963
$$821$$ 7.17157 0.250290 0.125145 0.992138i $$-0.460060\pi$$
0.125145 + 0.992138i $$0.460060\pi$$
$$822$$ 3.85786 0.134558
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 30.6274 1.06696
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ −18.6863 −0.649786 −0.324893 0.945751i $$-0.605328\pi$$
−0.324893 + 0.945751i $$0.605328\pi$$
$$828$$ 7.31371 0.254169
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 4.14214 0.143776
$$831$$ −13.6569 −0.473751
$$832$$ 23.5980 0.818113
$$833$$ 111.397 3.85968
$$834$$ 6.82843 0.236449
$$835$$ 9.31371 0.322314
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 1.25483 0.0433475
$$839$$ 22.6274 0.781185 0.390593 0.920564i $$-0.372270\pi$$
0.390593 + 0.920564i $$0.372270\pi$$
$$840$$ 7.65685 0.264187
$$841$$ −28.3137 −0.976335
$$842$$ 2.48528 0.0856485
$$843$$ −16.8284 −0.579602
$$844$$ 12.4853 0.429761
$$845$$ −19.0000 −0.653620
$$846$$ 1.65685 0.0569638
$$847$$ 0 0
$$848$$ −39.9411 −1.37158
$$849$$ −3.17157 −0.108848
$$850$$ −2.82843 −0.0970143
$$851$$ −1.37258 −0.0470515
$$852$$ 24.9706 0.855477
$$853$$ 31.3137 1.07216 0.536080 0.844167i $$-0.319904\pi$$
0.536080 + 0.844167i $$0.319904\pi$$
$$854$$ −0.686292 −0.0234844
$$855$$ −1.17157 −0.0400669
$$856$$ 8.42641 0.288009
$$857$$ 11.5147 0.393335 0.196668 0.980470i $$-0.436988\pi$$
0.196668 + 0.980470i $$0.436988\pi$$
$$858$$ 0 0
$$859$$ −19.0294 −0.649276 −0.324638 0.945838i $$-0.605242\pi$$
−0.324638 + 0.945838i $$0.605242\pi$$
$$860$$ 5.79899 0.197744
$$861$$ −4.00000 −0.136320
$$862$$ 4.28427 0.145923
$$863$$ −43.3137 −1.47442 −0.737208 0.675666i $$-0.763856\pi$$
−0.737208 + 0.675666i $$0.763856\pi$$
$$864$$ 4.41421 0.150175
$$865$$ −2.82843 −0.0961694
$$866$$ 1.79899 0.0611322
$$867$$ −29.6274 −1.00620
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0.343146 0.0116337
$$871$$ −32.0000 −1.08428
$$872$$ 8.42641 0.285354
$$873$$ 0.343146 0.0116137
$$874$$ 1.94113 0.0656595
$$875$$ −4.82843 −0.163231
$$876$$ 20.6863 0.698925
$$877$$ 42.6274 1.43943 0.719713 0.694272i $$-0.244273\pi$$
0.719713 + 0.694272i $$0.244273\pi$$
$$878$$ −1.45584 −0.0491324
$$879$$ −1.17157 −0.0395162
$$880$$ 0 0
$$881$$ −13.0294 −0.438973 −0.219486 0.975616i $$-0.570438\pi$$
−0.219486 + 0.975616i $$0.570438\pi$$
$$882$$ −6.75736 −0.227532
$$883$$ −50.6274 −1.70375 −0.851874 0.523747i $$-0.824534\pi$$
−0.851874 + 0.523747i $$0.824534\pi$$
$$884$$ 70.6274 2.37546
$$885$$ −4.00000 −0.134459
$$886$$ −4.97056 −0.166989
$$887$$ 4.34315 0.145829 0.0729143 0.997338i $$-0.476770\pi$$
0.0729143 + 0.997338i $$0.476770\pi$$
$$888$$ −0.544156 −0.0182607
$$889$$ −12.0000 −0.402467
$$890$$ −3.17157 −0.106311
$$891$$ 0 0
$$892$$ 32.2843 1.08096
$$893$$ −4.68629 −0.156821
$$894$$ −7.65685 −0.256084
$$895$$ 6.34315 0.212028
$$896$$ 50.9706 1.70281
$$897$$ −22.6274 −0.755507
$$898$$ 1.23045 0.0410606
$$899$$ 0 0
$$900$$ −1.82843 −0.0609476
$$901$$ −90.9117 −3.02871
$$902$$ 0 0
$$903$$ −15.3137 −0.509608
$$904$$ 23.7401 0.789584
$$905$$ 14.0000 0.465376
$$906$$ 0.201010 0.00667811
$$907$$ 7.02944 0.233409 0.116704 0.993167i $$-0.462767\pi$$
0.116704 + 0.993167i $$0.462767\pi$$
$$908$$ −25.5980 −0.849499
$$909$$ −4.82843 −0.160149
$$910$$ −11.3137 −0.375046
$$911$$ −15.0294 −0.497947 −0.248974 0.968510i $$-0.580093\pi$$
−0.248974 + 0.968510i $$0.580093\pi$$
$$912$$ −3.51472 −0.116384
$$913$$ 0 0
$$914$$ −0.284271 −0.00940286
$$915$$ 0.343146 0.0113440
$$916$$ 3.65685 0.120826
$$917$$ 93.2548 3.07955
$$918$$ 2.82843 0.0933520
$$919$$ −28.4853 −0.939643 −0.469821 0.882762i $$-0.655682\pi$$
−0.469821 + 0.882762i $$0.655682\pi$$
$$920$$ 6.34315 0.209127
$$921$$ −8.82843 −0.290907
$$922$$ −11.6569 −0.383898
$$923$$ −77.2548 −2.54287
$$924$$ 0 0
$$925$$ 0.343146 0.0112826
$$926$$ 12.0000 0.394344
$$927$$ 19.3137 0.634345
$$928$$ 3.65685 0.120042
$$929$$ 33.5980 1.10231 0.551157 0.834402i $$-0.314187\pi$$
0.551157 + 0.834402i $$0.314187\pi$$
$$930$$ 0 0
$$931$$ 19.1127 0.626393
$$932$$ −24.0833 −0.788873
$$933$$ −19.3137 −0.632302
$$934$$ −9.37258 −0.306680
$$935$$ 0 0
$$936$$ −8.97056 −0.293212
$$937$$ −44.9706 −1.46912 −0.734562 0.678541i $$-0.762612\pi$$
−0.734562 + 0.678541i $$0.762612\pi$$
$$938$$ −11.3137 −0.369406
$$939$$ −4.34315 −0.141733
$$940$$ −7.31371 −0.238547
$$941$$ −38.7696 −1.26385 −0.631926 0.775029i $$-0.717735\pi$$
−0.631926 + 0.775029i $$0.717735\pi$$
$$942$$ 7.45584 0.242925
$$943$$ −3.31371 −0.107909
$$944$$ −12.0000 −0.390567
$$945$$ 4.82843 0.157069
$$946$$ 0 0
$$947$$ −38.6274 −1.25522 −0.627611 0.778527i $$-0.715967\pi$$
−0.627611 + 0.778527i $$0.715967\pi$$
$$948$$ 15.5147 0.503895
$$949$$ −64.0000 −2.07753
$$950$$ −0.485281 −0.0157446
$$951$$ −30.2843 −0.982035
$$952$$ 52.2843 1.69454
$$953$$ −27.7990 −0.900498 −0.450249 0.892903i $$-0.648665\pi$$
−0.450249 + 0.892903i $$0.648665\pi$$
$$954$$ 5.51472 0.178546
$$955$$ 5.65685 0.183052
$$956$$ 11.5980 0.375105
$$957$$ 0 0
$$958$$ 1.25483 0.0405418
$$959$$ 44.9706 1.45218
$$960$$ −4.17157 −0.134637
$$961$$ −31.0000 −1.00000
$$962$$ 0.804041 0.0259233
$$963$$ 5.31371 0.171232
$$964$$ −43.2548 −1.39314
$$965$$ 2.34315 0.0754285
$$966$$ −8.00000 −0.257396
$$967$$ 39.4558 1.26881 0.634407 0.772999i $$-0.281244\pi$$
0.634407 + 0.772999i $$0.281244\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0.142136 0.00456370
$$971$$ 10.6274 0.341050 0.170525 0.985353i $$-0.445454\pi$$
0.170525 + 0.985353i $$0.445454\pi$$
$$972$$ 1.82843 0.0586468
$$973$$ 79.5980 2.55179
$$974$$ −8.68629 −0.278327
$$975$$ 5.65685 0.181164
$$976$$ 1.02944 0.0329515
$$977$$ 25.3137 0.809857 0.404929 0.914348i $$-0.367296\pi$$
0.404929 + 0.914348i $$0.367296\pi$$
$$978$$ 6.34315 0.202831
$$979$$ 0 0
$$980$$ 29.8284 0.952834
$$981$$ 5.31371 0.169654
$$982$$ 10.6274 0.339135
$$983$$ 14.6274 0.466542 0.233271 0.972412i $$-0.425057\pi$$
0.233271 + 0.972412i $$0.425057\pi$$
$$984$$ −1.31371 −0.0418795
$$985$$ 8.48528 0.270364
$$986$$ 2.34315 0.0746210
$$987$$ 19.3137 0.614762
$$988$$ 12.1177 0.385517
$$989$$ −12.6863 −0.403401
$$990$$ 0 0
$$991$$ 14.6274 0.464655 0.232328 0.972638i $$-0.425366\pi$$
0.232328 + 0.972638i $$0.425366\pi$$
$$992$$ 0 0
$$993$$ −17.6569 −0.560323
$$994$$ −27.3137 −0.866338
$$995$$ 10.3431 0.327900
$$996$$ 18.2843 0.579359
$$997$$ 16.6863 0.528460 0.264230 0.964460i $$-0.414882\pi$$
0.264230 + 0.964460i $$0.414882\pi$$
$$998$$ 13.9411 0.441299
$$999$$ −0.343146 −0.0108567
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 1815.2.a.k.1.1 2
3.2 odd 2 5445.2.a.m.1.2 2
5.4 even 2 9075.2.a.v.1.2 2
11.10 odd 2 165.2.a.a.1.2 2
33.32 even 2 495.2.a.d.1.1 2
44.43 even 2 2640.2.a.bb.1.2 2
55.32 even 4 825.2.c.e.199.3 4
55.43 even 4 825.2.c.e.199.2 4
55.54 odd 2 825.2.a.g.1.1 2
77.76 even 2 8085.2.a.ba.1.2 2
132.131 odd 2 7920.2.a.cg.1.2 2
165.32 odd 4 2475.2.c.m.199.2 4
165.98 odd 4 2475.2.c.m.199.3 4
165.164 even 2 2475.2.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 11.10 odd 2
495.2.a.d.1.1 2 33.32 even 2
825.2.a.g.1.1 2 55.54 odd 2
825.2.c.e.199.2 4 55.43 even 4
825.2.c.e.199.3 4 55.32 even 4
1815.2.a.k.1.1 2 1.1 even 1 trivial
2475.2.a.m.1.2 2 165.164 even 2
2475.2.c.m.199.2 4 165.32 odd 4
2475.2.c.m.199.3 4 165.98 odd 4
2640.2.a.bb.1.2 2 44.43 even 2
5445.2.a.m.1.2 2 3.2 odd 2
7920.2.a.cg.1.2 2 132.131 odd 2
8085.2.a.ba.1.2 2 77.76 even 2
9075.2.a.v.1.2 2 5.4 even 2