# Properties

 Label 1110.2.a.q.1.1 Level $1110$ Weight $2$ Character 1110.1 Self dual yes Analytic conductor $8.863$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{113})$$ Defining polynomial: $$x^{2} - x - 28$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.81507$$ of defining polynomial Character $$\chi$$ $$=$$ 1110.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -3.81507 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +7.81507 q^{19} +1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} +5.81507 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.81507 q^{26} +1.00000 q^{27} +3.00000 q^{28} -6.81507 q^{29} -1.00000 q^{30} +6.81507 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -7.81507 q^{38} -3.81507 q^{39} -1.00000 q^{40} -4.81507 q^{41} -3.00000 q^{42} -4.81507 q^{43} -1.00000 q^{44} +1.00000 q^{45} -5.81507 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -3.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +7.81507 q^{57} +6.81507 q^{58} +2.00000 q^{59} +1.00000 q^{60} +6.81507 q^{61} -6.81507 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.81507 q^{65} +1.00000 q^{66} +7.63015 q^{67} +1.00000 q^{68} +5.81507 q^{69} -3.00000 q^{70} -11.6301 q^{71} -1.00000 q^{72} -5.81507 q^{73} -1.00000 q^{74} +1.00000 q^{75} +7.81507 q^{76} -3.00000 q^{77} +3.81507 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.81507 q^{82} -3.81507 q^{83} +3.00000 q^{84} +1.00000 q^{85} +4.81507 q^{86} -6.81507 q^{87} +1.00000 q^{88} +11.8151 q^{89} -1.00000 q^{90} -11.4452 q^{91} +5.81507 q^{92} +6.81507 q^{93} +8.00000 q^{94} +7.81507 q^{95} -1.00000 q^{96} -4.81507 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 3q^{13} - 6q^{14} + 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 5q^{19} + 2q^{20} + 6q^{21} + 2q^{22} + q^{23} - 2q^{24} + 2q^{25} - 3q^{26} + 2q^{27} + 6q^{28} - 3q^{29} - 2q^{30} + 3q^{31} - 2q^{32} - 2q^{33} - 2q^{34} + 6q^{35} + 2q^{36} + 2q^{37} - 5q^{38} + 3q^{39} - 2q^{40} + q^{41} - 6q^{42} + q^{43} - 2q^{44} + 2q^{45} - q^{46} - 16q^{47} + 2q^{48} + 4q^{49} - 2q^{50} + 2q^{51} + 3q^{52} - 2q^{54} - 2q^{55} - 6q^{56} + 5q^{57} + 3q^{58} + 4q^{59} + 2q^{60} + 3q^{61} - 3q^{62} + 6q^{63} + 2q^{64} + 3q^{65} + 2q^{66} - 6q^{67} + 2q^{68} + q^{69} - 6q^{70} - 2q^{71} - 2q^{72} - q^{73} - 2q^{74} + 2q^{75} + 5q^{76} - 6q^{77} - 3q^{78} + 24q^{79} + 2q^{80} + 2q^{81} - q^{82} + 3q^{83} + 6q^{84} + 2q^{85} - q^{86} - 3q^{87} + 2q^{88} + 13q^{89} - 2q^{90} + 9q^{91} + q^{92} + 3q^{93} + 16q^{94} + 5q^{95} - 2q^{96} + q^{97} - 4q^{98} - 2q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −3.81507 −1.05811 −0.529055 0.848587i $$-0.677454\pi$$
−0.529055 + 0.848587i $$0.677454\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 7.81507 1.79290 0.896450 0.443144i $$-0.146137\pi$$
0.896450 + 0.443144i $$0.146137\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 3.00000 0.654654
$$22$$ 1.00000 0.213201
$$23$$ 5.81507 1.21253 0.606263 0.795264i $$-0.292668\pi$$
0.606263 + 0.795264i $$0.292668\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 3.81507 0.748197
$$27$$ 1.00000 0.192450
$$28$$ 3.00000 0.566947
$$29$$ −6.81507 −1.26553 −0.632764 0.774345i $$-0.718080\pi$$
−0.632764 + 0.774345i $$0.718080\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 6.81507 1.22402 0.612012 0.790849i $$-0.290361\pi$$
0.612012 + 0.790849i $$0.290361\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −1.00000 −0.171499
$$35$$ 3.00000 0.507093
$$36$$ 1.00000 0.166667
$$37$$ 1.00000 0.164399
$$38$$ −7.81507 −1.26777
$$39$$ −3.81507 −0.610901
$$40$$ −1.00000 −0.158114
$$41$$ −4.81507 −0.751988 −0.375994 0.926622i $$-0.622699\pi$$
−0.375994 + 0.926622i $$0.622699\pi$$
$$42$$ −3.00000 −0.462910
$$43$$ −4.81507 −0.734292 −0.367146 0.930163i $$-0.619665\pi$$
−0.367146 + 0.930163i $$0.619665\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 1.00000 0.149071
$$46$$ −5.81507 −0.857386
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 2.00000 0.285714
$$50$$ −1.00000 −0.141421
$$51$$ 1.00000 0.140028
$$52$$ −3.81507 −0.529055
$$53$$ 10.6301 1.46016 0.730081 0.683360i $$-0.239482\pi$$
0.730081 + 0.683360i $$0.239482\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.00000 −0.134840
$$56$$ −3.00000 −0.400892
$$57$$ 7.81507 1.03513
$$58$$ 6.81507 0.894863
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 6.81507 0.872581 0.436290 0.899806i $$-0.356292\pi$$
0.436290 + 0.899806i $$0.356292\pi$$
$$62$$ −6.81507 −0.865515
$$63$$ 3.00000 0.377964
$$64$$ 1.00000 0.125000
$$65$$ −3.81507 −0.473202
$$66$$ 1.00000 0.123091
$$67$$ 7.63015 0.932171 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 5.81507 0.700053
$$70$$ −3.00000 −0.358569
$$71$$ −11.6301 −1.38024 −0.690122 0.723693i $$-0.742443\pi$$
−0.690122 + 0.723693i $$0.742443\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −5.81507 −0.680603 −0.340301 0.940316i $$-0.610529\pi$$
−0.340301 + 0.940316i $$0.610529\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 1.00000 0.115470
$$76$$ 7.81507 0.896450
$$77$$ −3.00000 −0.341882
$$78$$ 3.81507 0.431972
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 4.81507 0.531736
$$83$$ −3.81507 −0.418759 −0.209379 0.977834i $$-0.567144\pi$$
−0.209379 + 0.977834i $$0.567144\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 1.00000 0.108465
$$86$$ 4.81507 0.519223
$$87$$ −6.81507 −0.730653
$$88$$ 1.00000 0.106600
$$89$$ 11.8151 1.25240 0.626198 0.779664i $$-0.284610\pi$$
0.626198 + 0.779664i $$0.284610\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ −11.4452 −1.19978
$$92$$ 5.81507 0.606263
$$93$$ 6.81507 0.706690
$$94$$ 8.00000 0.825137
$$95$$ 7.81507 0.801810
$$96$$ −1.00000 −0.102062
$$97$$ −4.81507 −0.488897 −0.244448 0.969662i $$-0.578607\pi$$
−0.244448 + 0.969662i $$0.578607\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ −1.00000 −0.100504
$$100$$ 1.00000 0.100000
$$101$$ 11.6301 1.15724 0.578621 0.815596i $$-0.303591\pi$$
0.578621 + 0.815596i $$0.303591\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 3.81507 0.374099
$$105$$ 3.00000 0.292770
$$106$$ −10.6301 −1.03249
$$107$$ −8.18493 −0.791267 −0.395633 0.918409i $$-0.629475\pi$$
−0.395633 + 0.918409i $$0.629475\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 13.0000 1.24517 0.622587 0.782551i $$-0.286082\pi$$
0.622587 + 0.782551i $$0.286082\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 1.00000 0.0949158
$$112$$ 3.00000 0.283473
$$113$$ −6.81507 −0.641108 −0.320554 0.947230i $$-0.603869\pi$$
−0.320554 + 0.947230i $$0.603869\pi$$
$$114$$ −7.81507 −0.731949
$$115$$ 5.81507 0.542258
$$116$$ −6.81507 −0.632764
$$117$$ −3.81507 −0.352704
$$118$$ −2.00000 −0.184115
$$119$$ 3.00000 0.275010
$$120$$ −1.00000 −0.0912871
$$121$$ −10.0000 −0.909091
$$122$$ −6.81507 −0.617008
$$123$$ −4.81507 −0.434161
$$124$$ 6.81507 0.612012
$$125$$ 1.00000 0.0894427
$$126$$ −3.00000 −0.267261
$$127$$ −21.4452 −1.90296 −0.951478 0.307718i $$-0.900435\pi$$
−0.951478 + 0.307718i $$0.900435\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.81507 −0.423944
$$130$$ 3.81507 0.334604
$$131$$ −9.63015 −0.841390 −0.420695 0.907202i $$-0.638214\pi$$
−0.420695 + 0.907202i $$0.638214\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 23.4452 2.03296
$$134$$ −7.63015 −0.659144
$$135$$ 1.00000 0.0860663
$$136$$ −1.00000 −0.0857493
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −5.81507 −0.495012
$$139$$ 22.8151 1.93515 0.967575 0.252585i $$-0.0812809\pi$$
0.967575 + 0.252585i $$0.0812809\pi$$
$$140$$ 3.00000 0.253546
$$141$$ −8.00000 −0.673722
$$142$$ 11.6301 0.975980
$$143$$ 3.81507 0.319032
$$144$$ 1.00000 0.0833333
$$145$$ −6.81507 −0.565961
$$146$$ 5.81507 0.481259
$$147$$ 2.00000 0.164957
$$148$$ 1.00000 0.0821995
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 3.81507 0.310466 0.155233 0.987878i $$-0.450387\pi$$
0.155233 + 0.987878i $$0.450387\pi$$
$$152$$ −7.81507 −0.633886
$$153$$ 1.00000 0.0808452
$$154$$ 3.00000 0.241747
$$155$$ 6.81507 0.547400
$$156$$ −3.81507 −0.305450
$$157$$ −12.8151 −1.02275 −0.511377 0.859356i $$-0.670864\pi$$
−0.511377 + 0.859356i $$0.670864\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ 10.6301 0.843025
$$160$$ −1.00000 −0.0790569
$$161$$ 17.4452 1.37488
$$162$$ −1.00000 −0.0785674
$$163$$ 16.6301 1.30257 0.651287 0.758832i $$-0.274230\pi$$
0.651287 + 0.758832i $$0.274230\pi$$
$$164$$ −4.81507 −0.375994
$$165$$ −1.00000 −0.0778499
$$166$$ 3.81507 0.296107
$$167$$ −21.8151 −1.68810 −0.844051 0.536264i $$-0.819835\pi$$
−0.844051 + 0.536264i $$0.819835\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ 1.55478 0.119599
$$170$$ −1.00000 −0.0766965
$$171$$ 7.81507 0.597634
$$172$$ −4.81507 −0.367146
$$173$$ −1.00000 −0.0760286 −0.0380143 0.999277i $$-0.512103\pi$$
−0.0380143 + 0.999277i $$0.512103\pi$$
$$174$$ 6.81507 0.516649
$$175$$ 3.00000 0.226779
$$176$$ −1.00000 −0.0753778
$$177$$ 2.00000 0.150329
$$178$$ −11.8151 −0.885577
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 11.4452 0.848376
$$183$$ 6.81507 0.503785
$$184$$ −5.81507 −0.428693
$$185$$ 1.00000 0.0735215
$$186$$ −6.81507 −0.499705
$$187$$ −1.00000 −0.0731272
$$188$$ −8.00000 −0.583460
$$189$$ 3.00000 0.218218
$$190$$ −7.81507 −0.566965
$$191$$ 4.63015 0.335026 0.167513 0.985870i $$-0.446426\pi$$
0.167513 + 0.985870i $$0.446426\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 4.81507 0.345702
$$195$$ −3.81507 −0.273203
$$196$$ 2.00000 0.142857
$$197$$ −4.18493 −0.298164 −0.149082 0.988825i $$-0.547632\pi$$
−0.149082 + 0.988825i $$0.547632\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 7.63015 0.538189
$$202$$ −11.6301 −0.818294
$$203$$ −20.4452 −1.43497
$$204$$ 1.00000 0.0700140
$$205$$ −4.81507 −0.336299
$$206$$ 0 0
$$207$$ 5.81507 0.404176
$$208$$ −3.81507 −0.264528
$$209$$ −7.81507 −0.540580
$$210$$ −3.00000 −0.207020
$$211$$ 8.44522 0.581393 0.290696 0.956815i $$-0.406113\pi$$
0.290696 + 0.956815i $$0.406113\pi$$
$$212$$ 10.6301 0.730081
$$213$$ −11.6301 −0.796884
$$214$$ 8.18493 0.559510
$$215$$ −4.81507 −0.328385
$$216$$ −1.00000 −0.0680414
$$217$$ 20.4452 1.38791
$$218$$ −13.0000 −0.880471
$$219$$ −5.81507 −0.392946
$$220$$ −1.00000 −0.0674200
$$221$$ −3.81507 −0.256630
$$222$$ −1.00000 −0.0671156
$$223$$ −16.8151 −1.12602 −0.563010 0.826450i $$-0.690357\pi$$
−0.563010 + 0.826450i $$0.690357\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 1.00000 0.0666667
$$226$$ 6.81507 0.453332
$$227$$ −24.8151 −1.64703 −0.823517 0.567291i $$-0.807991\pi$$
−0.823517 + 0.567291i $$0.807991\pi$$
$$228$$ 7.81507 0.517566
$$229$$ 0.369854 0.0244407 0.0122203 0.999925i $$-0.496110\pi$$
0.0122203 + 0.999925i $$0.496110\pi$$
$$230$$ −5.81507 −0.383435
$$231$$ −3.00000 −0.197386
$$232$$ 6.81507 0.447431
$$233$$ −7.63015 −0.499867 −0.249934 0.968263i $$-0.580409\pi$$
−0.249934 + 0.968263i $$0.580409\pi$$
$$234$$ 3.81507 0.249399
$$235$$ −8.00000 −0.521862
$$236$$ 2.00000 0.130189
$$237$$ 12.0000 0.779484
$$238$$ −3.00000 −0.194461
$$239$$ 6.44522 0.416907 0.208453 0.978032i $$-0.433157\pi$$
0.208453 + 0.978032i $$0.433157\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 10.0000 0.642824
$$243$$ 1.00000 0.0641500
$$244$$ 6.81507 0.436290
$$245$$ 2.00000 0.127775
$$246$$ 4.81507 0.306998
$$247$$ −29.8151 −1.89709
$$248$$ −6.81507 −0.432758
$$249$$ −3.81507 −0.241770
$$250$$ −1.00000 −0.0632456
$$251$$ −11.6301 −0.734088 −0.367044 0.930204i $$-0.619630\pi$$
−0.367044 + 0.930204i $$0.619630\pi$$
$$252$$ 3.00000 0.188982
$$253$$ −5.81507 −0.365591
$$254$$ 21.4452 1.34559
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −3.81507 −0.237978 −0.118989 0.992896i $$-0.537965\pi$$
−0.118989 + 0.992896i $$0.537965\pi$$
$$258$$ 4.81507 0.299773
$$259$$ 3.00000 0.186411
$$260$$ −3.81507 −0.236601
$$261$$ −6.81507 −0.421842
$$262$$ 9.63015 0.594952
$$263$$ −30.4452 −1.87733 −0.938666 0.344827i $$-0.887938\pi$$
−0.938666 + 0.344827i $$0.887938\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 10.6301 0.653005
$$266$$ −23.4452 −1.43752
$$267$$ 11.8151 0.723071
$$268$$ 7.63015 0.466085
$$269$$ 9.81507 0.598436 0.299218 0.954185i $$-0.403274\pi$$
0.299218 + 0.954185i $$0.403274\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ −11.4452 −0.692696
$$274$$ −2.00000 −0.120824
$$275$$ −1.00000 −0.0603023
$$276$$ 5.81507 0.350026
$$277$$ 21.4452 1.28852 0.644259 0.764807i $$-0.277166\pi$$
0.644259 + 0.764807i $$0.277166\pi$$
$$278$$ −22.8151 −1.36836
$$279$$ 6.81507 0.408008
$$280$$ −3.00000 −0.179284
$$281$$ −19.4452 −1.16000 −0.580002 0.814615i $$-0.696948\pi$$
−0.580002 + 0.814615i $$0.696948\pi$$
$$282$$ 8.00000 0.476393
$$283$$ −21.8151 −1.29677 −0.648386 0.761312i $$-0.724556\pi$$
−0.648386 + 0.761312i $$0.724556\pi$$
$$284$$ −11.6301 −0.690122
$$285$$ 7.81507 0.462925
$$286$$ −3.81507 −0.225590
$$287$$ −14.4452 −0.852674
$$288$$ −1.00000 −0.0589256
$$289$$ −16.0000 −0.941176
$$290$$ 6.81507 0.400195
$$291$$ −4.81507 −0.282265
$$292$$ −5.81507 −0.340301
$$293$$ 28.2603 1.65098 0.825492 0.564414i $$-0.190898\pi$$
0.825492 + 0.564414i $$0.190898\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 2.00000 0.116445
$$296$$ −1.00000 −0.0581238
$$297$$ −1.00000 −0.0580259
$$298$$ −2.00000 −0.115857
$$299$$ −22.1849 −1.28299
$$300$$ 1.00000 0.0577350
$$301$$ −14.4452 −0.832609
$$302$$ −3.81507 −0.219533
$$303$$ 11.6301 0.668134
$$304$$ 7.81507 0.448225
$$305$$ 6.81507 0.390230
$$306$$ −1.00000 −0.0571662
$$307$$ −18.0000 −1.02731 −0.513657 0.857996i $$-0.671710\pi$$
−0.513657 + 0.857996i $$0.671710\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ −6.81507 −0.387070
$$311$$ 10.4452 0.592294 0.296147 0.955142i $$-0.404298\pi$$
0.296147 + 0.955142i $$0.404298\pi$$
$$312$$ 3.81507 0.215986
$$313$$ 32.8904 1.85908 0.929539 0.368725i $$-0.120205\pi$$
0.929539 + 0.368725i $$0.120205\pi$$
$$314$$ 12.8151 0.723196
$$315$$ 3.00000 0.169031
$$316$$ 12.0000 0.675053
$$317$$ −2.81507 −0.158110 −0.0790551 0.996870i $$-0.525190\pi$$
−0.0790551 + 0.996870i $$0.525190\pi$$
$$318$$ −10.6301 −0.596109
$$319$$ 6.81507 0.381571
$$320$$ 1.00000 0.0559017
$$321$$ −8.18493 −0.456838
$$322$$ −17.4452 −0.972184
$$323$$ 7.81507 0.434842
$$324$$ 1.00000 0.0555556
$$325$$ −3.81507 −0.211622
$$326$$ −16.6301 −0.921059
$$327$$ 13.0000 0.718902
$$328$$ 4.81507 0.265868
$$329$$ −24.0000 −1.32316
$$330$$ 1.00000 0.0550482
$$331$$ −23.2603 −1.27850 −0.639251 0.768998i $$-0.720755\pi$$
−0.639251 + 0.768998i $$0.720755\pi$$
$$332$$ −3.81507 −0.209379
$$333$$ 1.00000 0.0547997
$$334$$ 21.8151 1.19367
$$335$$ 7.63015 0.416879
$$336$$ 3.00000 0.163663
$$337$$ −4.18493 −0.227968 −0.113984 0.993483i $$-0.536361\pi$$
−0.113984 + 0.993483i $$0.536361\pi$$
$$338$$ −1.55478 −0.0845690
$$339$$ −6.81507 −0.370144
$$340$$ 1.00000 0.0542326
$$341$$ −6.81507 −0.369057
$$342$$ −7.81507 −0.422591
$$343$$ −15.0000 −0.809924
$$344$$ 4.81507 0.259611
$$345$$ 5.81507 0.313073
$$346$$ 1.00000 0.0537603
$$347$$ 11.2603 0.604484 0.302242 0.953231i $$-0.402265\pi$$
0.302242 + 0.953231i $$0.402265\pi$$
$$348$$ −6.81507 −0.365326
$$349$$ 7.26029 0.388635 0.194317 0.980939i $$-0.437751\pi$$
0.194317 + 0.980939i $$0.437751\pi$$
$$350$$ −3.00000 −0.160357
$$351$$ −3.81507 −0.203634
$$352$$ 1.00000 0.0533002
$$353$$ −13.1849 −0.701763 −0.350881 0.936420i $$-0.614118\pi$$
−0.350881 + 0.936420i $$0.614118\pi$$
$$354$$ −2.00000 −0.106299
$$355$$ −11.6301 −0.617264
$$356$$ 11.8151 0.626198
$$357$$ 3.00000 0.158777
$$358$$ −16.0000 −0.845626
$$359$$ 6.36985 0.336188 0.168094 0.985771i $$-0.446239\pi$$
0.168094 + 0.985771i $$0.446239\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 42.0754 2.21449
$$362$$ 10.0000 0.525588
$$363$$ −10.0000 −0.524864
$$364$$ −11.4452 −0.599892
$$365$$ −5.81507 −0.304375
$$366$$ −6.81507 −0.356230
$$367$$ −21.0000 −1.09619 −0.548096 0.836416i $$-0.684647\pi$$
−0.548096 + 0.836416i $$0.684647\pi$$
$$368$$ 5.81507 0.303132
$$369$$ −4.81507 −0.250663
$$370$$ −1.00000 −0.0519875
$$371$$ 31.8904 1.65567
$$372$$ 6.81507 0.353345
$$373$$ −17.2603 −0.893704 −0.446852 0.894608i $$-0.647455\pi$$
−0.446852 + 0.894608i $$0.647455\pi$$
$$374$$ 1.00000 0.0517088
$$375$$ 1.00000 0.0516398
$$376$$ 8.00000 0.412568
$$377$$ 26.0000 1.33907
$$378$$ −3.00000 −0.154303
$$379$$ −19.6301 −1.00833 −0.504166 0.863607i $$-0.668200\pi$$
−0.504166 + 0.863607i $$0.668200\pi$$
$$380$$ 7.81507 0.400905
$$381$$ −21.4452 −1.09867
$$382$$ −4.63015 −0.236899
$$383$$ 33.4452 1.70897 0.854485 0.519475i $$-0.173873\pi$$
0.854485 + 0.519475i $$0.173873\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −3.00000 −0.152894
$$386$$ 2.00000 0.101797
$$387$$ −4.81507 −0.244764
$$388$$ −4.81507 −0.244448
$$389$$ −35.7055 −1.81034 −0.905171 0.425048i $$-0.860257\pi$$
−0.905171 + 0.425048i $$0.860257\pi$$
$$390$$ 3.81507 0.193184
$$391$$ 5.81507 0.294081
$$392$$ −2.00000 −0.101015
$$393$$ −9.63015 −0.485777
$$394$$ 4.18493 0.210834
$$395$$ 12.0000 0.603786
$$396$$ −1.00000 −0.0502519
$$397$$ −5.63015 −0.282569 −0.141284 0.989969i $$-0.545123\pi$$
−0.141284 + 0.989969i $$0.545123\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 23.4452 1.17373
$$400$$ 1.00000 0.0500000
$$401$$ 0.554781 0.0277045 0.0138522 0.999904i $$-0.495591\pi$$
0.0138522 + 0.999904i $$0.495591\pi$$
$$402$$ −7.63015 −0.380557
$$403$$ −26.0000 −1.29515
$$404$$ 11.6301 0.578621
$$405$$ 1.00000 0.0496904
$$406$$ 20.4452 1.01468
$$407$$ −1.00000 −0.0495682
$$408$$ −1.00000 −0.0495074
$$409$$ 6.36985 0.314969 0.157485 0.987521i $$-0.449662\pi$$
0.157485 + 0.987521i $$0.449662\pi$$
$$410$$ 4.81507 0.237800
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ 6.00000 0.295241
$$414$$ −5.81507 −0.285795
$$415$$ −3.81507 −0.187275
$$416$$ 3.81507 0.187049
$$417$$ 22.8151 1.11726
$$418$$ 7.81507 0.382248
$$419$$ −25.4452 −1.24308 −0.621540 0.783382i $$-0.713493\pi$$
−0.621540 + 0.783382i $$0.713493\pi$$
$$420$$ 3.00000 0.146385
$$421$$ −29.2603 −1.42606 −0.713030 0.701134i $$-0.752678\pi$$
−0.713030 + 0.701134i $$0.752678\pi$$
$$422$$ −8.44522 −0.411107
$$423$$ −8.00000 −0.388973
$$424$$ −10.6301 −0.516246
$$425$$ 1.00000 0.0485071
$$426$$ 11.6301 0.563482
$$427$$ 20.4452 0.989413
$$428$$ −8.18493 −0.395633
$$429$$ 3.81507 0.184193
$$430$$ 4.81507 0.232203
$$431$$ 21.0000 1.01153 0.505767 0.862670i $$-0.331209\pi$$
0.505767 + 0.862670i $$0.331209\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 1.81507 0.0872268 0.0436134 0.999048i $$-0.486113\pi$$
0.0436134 + 0.999048i $$0.486113\pi$$
$$434$$ −20.4452 −0.981402
$$435$$ −6.81507 −0.326758
$$436$$ 13.0000 0.622587
$$437$$ 45.4452 2.17394
$$438$$ 5.81507 0.277855
$$439$$ −18.4452 −0.880342 −0.440171 0.897914i $$-0.645082\pi$$
−0.440171 + 0.897914i $$0.645082\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 2.00000 0.0952381
$$442$$ 3.81507 0.181465
$$443$$ 13.6301 0.647588 0.323794 0.946128i $$-0.395042\pi$$
0.323794 + 0.946128i $$0.395042\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 11.8151 0.560088
$$446$$ 16.8151 0.796217
$$447$$ 2.00000 0.0945968
$$448$$ 3.00000 0.141737
$$449$$ 30.8904 1.45781 0.728905 0.684615i $$-0.240030\pi$$
0.728905 + 0.684615i $$0.240030\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 4.81507 0.226733
$$452$$ −6.81507 −0.320554
$$453$$ 3.81507 0.179248
$$454$$ 24.8151 1.16463
$$455$$ −11.4452 −0.536560
$$456$$ −7.81507 −0.365974
$$457$$ −28.0754 −1.31331 −0.656655 0.754191i $$-0.728029\pi$$
−0.656655 + 0.754191i $$0.728029\pi$$
$$458$$ −0.369854 −0.0172822
$$459$$ 1.00000 0.0466760
$$460$$ 5.81507 0.271129
$$461$$ 34.8151 1.62150 0.810750 0.585393i $$-0.199060\pi$$
0.810750 + 0.585393i $$0.199060\pi$$
$$462$$ 3.00000 0.139573
$$463$$ −17.6301 −0.819342 −0.409671 0.912233i $$-0.634357\pi$$
−0.409671 + 0.912233i $$0.634357\pi$$
$$464$$ −6.81507 −0.316382
$$465$$ 6.81507 0.316041
$$466$$ 7.63015 0.353460
$$467$$ 28.8151 1.33340 0.666701 0.745325i $$-0.267706\pi$$
0.666701 + 0.745325i $$0.267706\pi$$
$$468$$ −3.81507 −0.176352
$$469$$ 22.8904 1.05698
$$470$$ 8.00000 0.369012
$$471$$ −12.8151 −0.590487
$$472$$ −2.00000 −0.0920575
$$473$$ 4.81507 0.221397
$$474$$ −12.0000 −0.551178
$$475$$ 7.81507 0.358580
$$476$$ 3.00000 0.137505
$$477$$ 10.6301 0.486721
$$478$$ −6.44522 −0.294797
$$479$$ 9.81507 0.448462 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ −3.81507 −0.173952
$$482$$ 26.0000 1.18427
$$483$$ 17.4452 0.793785
$$484$$ −10.0000 −0.454545
$$485$$ −4.81507 −0.218641
$$486$$ −1.00000 −0.0453609
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ −6.81507 −0.308504
$$489$$ 16.6301 0.752041
$$490$$ −2.00000 −0.0903508
$$491$$ 2.18493 0.0986044 0.0493022 0.998784i $$-0.484300\pi$$
0.0493022 + 0.998784i $$0.484300\pi$$
$$492$$ −4.81507 −0.217080
$$493$$ −6.81507 −0.306935
$$494$$ 29.8151 1.34144
$$495$$ −1.00000 −0.0449467
$$496$$ 6.81507 0.306006
$$497$$ −34.8904 −1.56505
$$498$$ 3.81507 0.170958
$$499$$ −17.4452 −0.780955 −0.390478 0.920612i $$-0.627690\pi$$
−0.390478 + 0.920612i $$0.627690\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −21.8151 −0.974626
$$502$$ 11.6301 0.519079
$$503$$ 0.739708 0.0329820 0.0164910 0.999864i $$-0.494751\pi$$
0.0164910 + 0.999864i $$0.494751\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 11.6301 0.517535
$$506$$ 5.81507 0.258512
$$507$$ 1.55478 0.0690503
$$508$$ −21.4452 −0.951478
$$509$$ 17.4452 0.773246 0.386623 0.922238i $$-0.373642\pi$$
0.386623 + 0.922238i $$0.373642\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ −17.4452 −0.771731
$$512$$ −1.00000 −0.0441942
$$513$$ 7.81507 0.345044
$$514$$ 3.81507 0.168276
$$515$$ 0 0
$$516$$ −4.81507 −0.211972
$$517$$ 8.00000 0.351840
$$518$$ −3.00000 −0.131812
$$519$$ −1.00000 −0.0438951
$$520$$ 3.81507 0.167302
$$521$$ 0.815073 0.0357090 0.0178545 0.999841i $$-0.494316\pi$$
0.0178545 + 0.999841i $$0.494316\pi$$
$$522$$ 6.81507 0.298288
$$523$$ −0.739708 −0.0323452 −0.0161726 0.999869i $$-0.505148\pi$$
−0.0161726 + 0.999869i $$0.505148\pi$$
$$524$$ −9.63015 −0.420695
$$525$$ 3.00000 0.130931
$$526$$ 30.4452 1.32747
$$527$$ 6.81507 0.296869
$$528$$ −1.00000 −0.0435194
$$529$$ 10.8151 0.470221
$$530$$ −10.6301 −0.461744
$$531$$ 2.00000 0.0867926
$$532$$ 23.4452 1.01648
$$533$$ 18.3699 0.795687
$$534$$ −11.8151 −0.511288
$$535$$ −8.18493 −0.353865
$$536$$ −7.63015 −0.329572
$$537$$ 16.0000 0.690451
$$538$$ −9.81507 −0.423158
$$539$$ −2.00000 −0.0861461
$$540$$ 1.00000 0.0430331
$$541$$ 27.8151 1.19586 0.597932 0.801547i $$-0.295989\pi$$
0.597932 + 0.801547i $$0.295989\pi$$
$$542$$ 20.0000 0.859074
$$543$$ −10.0000 −0.429141
$$544$$ −1.00000 −0.0428746
$$545$$ 13.0000 0.556859
$$546$$ 11.4452 0.489810
$$547$$ 23.8904 1.02148 0.510741 0.859735i $$-0.329371\pi$$
0.510741 + 0.859735i $$0.329371\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 6.81507 0.290860
$$550$$ 1.00000 0.0426401
$$551$$ −53.2603 −2.26896
$$552$$ −5.81507 −0.247506
$$553$$ 36.0000 1.53088
$$554$$ −21.4452 −0.911120
$$555$$ 1.00000 0.0424476
$$556$$ 22.8151 0.967575
$$557$$ −21.6301 −0.916499 −0.458249 0.888824i $$-0.651523\pi$$
−0.458249 + 0.888824i $$0.651523\pi$$
$$558$$ −6.81507 −0.288505
$$559$$ 18.3699 0.776962
$$560$$ 3.00000 0.126773
$$561$$ −1.00000 −0.0422200
$$562$$ 19.4452 0.820247
$$563$$ −8.44522 −0.355924 −0.177962 0.984037i $$-0.556950\pi$$
−0.177962 + 0.984037i $$0.556950\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ −6.81507 −0.286712
$$566$$ 21.8151 0.916956
$$567$$ 3.00000 0.125988
$$568$$ 11.6301 0.487990
$$569$$ −35.8151 −1.50145 −0.750723 0.660617i $$-0.770295\pi$$
−0.750723 + 0.660617i $$0.770295\pi$$
$$570$$ −7.81507 −0.327337
$$571$$ −14.8151 −0.619992 −0.309996 0.950738i $$-0.600328\pi$$
−0.309996 + 0.950738i $$0.600328\pi$$
$$572$$ 3.81507 0.159516
$$573$$ 4.63015 0.193427
$$574$$ 14.4452 0.602932
$$575$$ 5.81507 0.242505
$$576$$ 1.00000 0.0416667
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 16.0000 0.665512
$$579$$ −2.00000 −0.0831172
$$580$$ −6.81507 −0.282980
$$581$$ −11.4452 −0.474828
$$582$$ 4.81507 0.199591
$$583$$ −10.6301 −0.440256
$$584$$ 5.81507 0.240629
$$585$$ −3.81507 −0.157734
$$586$$ −28.2603 −1.16742
$$587$$ 20.8151 0.859130 0.429565 0.903036i $$-0.358667\pi$$
0.429565 + 0.903036i $$0.358667\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ 53.2603 2.19455
$$590$$ −2.00000 −0.0823387
$$591$$ −4.18493 −0.172145
$$592$$ 1.00000 0.0410997
$$593$$ −13.6301 −0.559723 −0.279862 0.960040i $$-0.590289\pi$$
−0.279862 + 0.960040i $$0.590289\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 3.00000 0.122988
$$596$$ 2.00000 0.0819232
$$597$$ −4.00000 −0.163709
$$598$$ 22.1849 0.907209
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −19.3699 −0.790113 −0.395056 0.918657i $$-0.629275\pi$$
−0.395056 + 0.918657i $$0.629275\pi$$
$$602$$ 14.4452 0.588743
$$603$$ 7.63015 0.310724
$$604$$ 3.81507 0.155233
$$605$$ −10.0000 −0.406558
$$606$$ −11.6301 −0.472442
$$607$$ −37.2603 −1.51235 −0.756174 0.654370i $$-0.772934\pi$$
−0.756174 + 0.654370i $$0.772934\pi$$
$$608$$ −7.81507 −0.316943
$$609$$ −20.4452 −0.828482
$$610$$ −6.81507 −0.275934
$$611$$ 30.5206 1.23473
$$612$$ 1.00000 0.0404226
$$613$$ −6.07536 −0.245382 −0.122691 0.992445i $$-0.539152\pi$$
−0.122691 + 0.992445i $$0.539152\pi$$
$$614$$ 18.0000 0.726421
$$615$$ −4.81507 −0.194162
$$616$$ 3.00000 0.120873
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ −13.5548 −0.544813 −0.272406 0.962182i $$-0.587819\pi$$
−0.272406 + 0.962182i $$0.587819\pi$$
$$620$$ 6.81507 0.273700
$$621$$ 5.81507 0.233351
$$622$$ −10.4452 −0.418815
$$623$$ 35.4452 1.42008
$$624$$ −3.81507 −0.152725
$$625$$ 1.00000 0.0400000
$$626$$ −32.8904 −1.31457
$$627$$ −7.81507 −0.312104
$$628$$ −12.8151 −0.511377
$$629$$ 1.00000 0.0398726
$$630$$ −3.00000 −0.119523
$$631$$ 33.7055 1.34180 0.670898 0.741550i $$-0.265909\pi$$
0.670898 + 0.741550i $$0.265909\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ 8.44522 0.335667
$$634$$ 2.81507 0.111801
$$635$$ −21.4452 −0.851028
$$636$$ 10.6301 0.421513
$$637$$ −7.63015 −0.302317
$$638$$ −6.81507 −0.269811
$$639$$ −11.6301 −0.460081
$$640$$ −1.00000 −0.0395285
$$641$$ −13.5548 −0.535382 −0.267691 0.963505i $$-0.586261\pi$$
−0.267691 + 0.963505i $$0.586261\pi$$
$$642$$ 8.18493 0.323033
$$643$$ 17.0000 0.670415 0.335207 0.942144i $$-0.391194\pi$$
0.335207 + 0.942144i $$0.391194\pi$$
$$644$$ 17.4452 0.687438
$$645$$ −4.81507 −0.189593
$$646$$ −7.81507 −0.307480
$$647$$ 25.0754 0.985814 0.492907 0.870082i $$-0.335934\pi$$
0.492907 + 0.870082i $$0.335934\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −2.00000 −0.0785069
$$650$$ 3.81507 0.149639
$$651$$ 20.4452 0.801311
$$652$$ 16.6301 0.651287
$$653$$ 4.00000 0.156532 0.0782660 0.996933i $$-0.475062\pi$$
0.0782660 + 0.996933i $$0.475062\pi$$
$$654$$ −13.0000 −0.508340
$$655$$ −9.63015 −0.376281
$$656$$ −4.81507 −0.187997
$$657$$ −5.81507 −0.226868
$$658$$ 24.0000 0.935617
$$659$$ −31.2603 −1.21773 −0.608864 0.793275i $$-0.708375\pi$$
−0.608864 + 0.793275i $$0.708375\pi$$
$$660$$ −1.00000 −0.0389249
$$661$$ 3.73971 0.145458 0.0727289 0.997352i $$-0.476829\pi$$
0.0727289 + 0.997352i $$0.476829\pi$$
$$662$$ 23.2603 0.904037
$$663$$ −3.81507 −0.148165
$$664$$ 3.81507 0.148054
$$665$$ 23.4452 0.909167
$$666$$ −1.00000 −0.0387492
$$667$$ −39.6301 −1.53449
$$668$$ −21.8151 −0.844051
$$669$$ −16.8151 −0.650108
$$670$$ −7.63015 −0.294778
$$671$$ −6.81507 −0.263093
$$672$$ −3.00000 −0.115728
$$673$$ 6.18493 0.238411 0.119206 0.992870i $$-0.461965\pi$$
0.119206 + 0.992870i $$0.461965\pi$$
$$674$$ 4.18493 0.161197
$$675$$ 1.00000 0.0384900
$$676$$ 1.55478 0.0597993
$$677$$ 47.0754 1.80925 0.904627 0.426205i $$-0.140150\pi$$
0.904627 + 0.426205i $$0.140150\pi$$
$$678$$ 6.81507 0.261731
$$679$$ −14.4452 −0.554357
$$680$$ −1.00000 −0.0383482
$$681$$ −24.8151 −0.950916
$$682$$ 6.81507 0.260963
$$683$$ −47.3357 −1.81125 −0.905624 0.424081i $$-0.860597\pi$$
−0.905624 + 0.424081i $$0.860597\pi$$
$$684$$ 7.81507 0.298817
$$685$$ 2.00000 0.0764161
$$686$$ 15.0000 0.572703
$$687$$ 0.369854 0.0141108
$$688$$ −4.81507 −0.183573
$$689$$ −40.5548 −1.54501
$$690$$ −5.81507 −0.221376
$$691$$ −34.0754 −1.29629 −0.648144 0.761518i $$-0.724455\pi$$
−0.648144 + 0.761518i $$0.724455\pi$$
$$692$$ −1.00000 −0.0380143
$$693$$ −3.00000 −0.113961
$$694$$ −11.2603 −0.427435
$$695$$ 22.8151 0.865425
$$696$$ 6.81507 0.258325
$$697$$ −4.81507 −0.182384
$$698$$ −7.26029 −0.274806
$$699$$ −7.63015 −0.288599
$$700$$ 3.00000 0.113389
$$701$$ 33.6301 1.27019 0.635097 0.772433i $$-0.280960\pi$$
0.635097 + 0.772433i $$0.280960\pi$$
$$702$$ 3.81507 0.143991
$$703$$ 7.81507 0.294751
$$704$$ −1.00000 −0.0376889
$$705$$ −8.00000 −0.301297
$$706$$ 13.1849 0.496221
$$707$$ 34.8904 1.31219
$$708$$ 2.00000 0.0751646
$$709$$ 20.6301 0.774781 0.387391 0.921916i $$-0.373376\pi$$
0.387391 + 0.921916i $$0.373376\pi$$
$$710$$ 11.6301 0.436472
$$711$$ 12.0000 0.450035
$$712$$ −11.8151 −0.442789
$$713$$ 39.6301 1.48416
$$714$$ −3.00000 −0.112272
$$715$$ 3.81507 0.142676
$$716$$ 16.0000 0.597948
$$717$$ 6.44522 0.240701
$$718$$ −6.36985 −0.237721
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ −42.0754 −1.56588
$$723$$ −26.0000 −0.966950
$$724$$ −10.0000 −0.371647
$$725$$ −6.81507 −0.253105
$$726$$ 10.0000 0.371135
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ 11.4452 0.424188
$$729$$ 1.00000 0.0370370
$$730$$ 5.81507 0.215226
$$731$$ −4.81507 −0.178092
$$732$$ 6.81507 0.251892
$$733$$ 40.4452 1.49388 0.746939 0.664892i $$-0.231523\pi$$
0.746939 + 0.664892i $$0.231523\pi$$
$$734$$ 21.0000 0.775124
$$735$$ 2.00000 0.0737711
$$736$$ −5.81507 −0.214346
$$737$$ −7.63015 −0.281060
$$738$$ 4.81507 0.177245
$$739$$ 34.0754 1.25348 0.626741 0.779227i $$-0.284388\pi$$
0.626741 + 0.779227i $$0.284388\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ −29.8151 −1.09528
$$742$$ −31.8904 −1.17073
$$743$$ 4.44522 0.163079 0.0815396 0.996670i $$-0.474016\pi$$
0.0815396 + 0.996670i $$0.474016\pi$$
$$744$$ −6.81507 −0.249853
$$745$$ 2.00000 0.0732743
$$746$$ 17.2603 0.631944
$$747$$ −3.81507 −0.139586
$$748$$ −1.00000 −0.0365636
$$749$$ −24.5548 −0.897212
$$750$$ −1.00000 −0.0365148
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −11.6301 −0.423826
$$754$$ −26.0000 −0.946864
$$755$$ 3.81507 0.138845
$$756$$ 3.00000 0.109109
$$757$$ 23.8151 0.865574 0.432787 0.901496i $$-0.357530\pi$$
0.432787 + 0.901496i $$0.357530\pi$$
$$758$$ 19.6301 0.712999
$$759$$ −5.81507 −0.211074
$$760$$ −7.81507 −0.283482
$$761$$ −9.18493 −0.332953 −0.166477 0.986045i $$-0.553239\pi$$
−0.166477 + 0.986045i $$0.553239\pi$$
$$762$$ 21.4452 0.776878
$$763$$ 39.0000 1.41189
$$764$$ 4.63015 0.167513
$$765$$ 1.00000 0.0361551
$$766$$ −33.4452 −1.20842
$$767$$ −7.63015 −0.275509
$$768$$ 1.00000 0.0360844
$$769$$ 49.2603 1.77637 0.888186 0.459485i $$-0.151966\pi$$
0.888186 + 0.459485i $$0.151966\pi$$
$$770$$ 3.00000 0.108112
$$771$$ −3.81507 −0.137396
$$772$$ −2.00000 −0.0719816
$$773$$ −7.36985 −0.265075 −0.132538 0.991178i $$-0.542313\pi$$
−0.132538 + 0.991178i $$0.542313\pi$$
$$774$$ 4.81507 0.173074
$$775$$ 6.81507 0.244805
$$776$$ 4.81507 0.172851
$$777$$ 3.00000 0.107624
$$778$$ 35.7055 1.28010
$$779$$ −37.6301 −1.34824
$$780$$ −3.81507 −0.136602
$$781$$ 11.6301 0.416159
$$782$$ −5.81507 −0.207947
$$783$$ −6.81507 −0.243551
$$784$$ 2.00000 0.0714286
$$785$$ −12.8151 −0.457390
$$786$$ 9.63015 0.343496
$$787$$ −39.6301 −1.41266 −0.706331 0.707882i $$-0.749651\pi$$
−0.706331 + 0.707882i $$0.749651\pi$$
$$788$$ −4.18493 −0.149082
$$789$$ −30.4452 −1.08388
$$790$$ −12.0000 −0.426941
$$791$$ −20.4452 −0.726948
$$792$$ 1.00000 0.0355335
$$793$$ −26.0000 −0.923287
$$794$$ 5.63015 0.199806
$$795$$ 10.6301 0.377012
$$796$$ −4.00000 −0.141776
$$797$$ −34.8904 −1.23588 −0.617941 0.786224i $$-0.712033\pi$$
−0.617941 + 0.786224i $$0.712033\pi$$
$$798$$ −23.4452 −0.829952
$$799$$ −8.00000 −0.283020
$$800$$ −1.00000 −0.0353553
$$801$$ 11.8151 0.417465
$$802$$ −0.554781 −0.0195900
$$803$$ 5.81507 0.205209
$$804$$ 7.63015 0.269094
$$805$$ 17.4452 0.614863
$$806$$ 26.0000 0.915811
$$807$$ 9.81507 0.345507
$$808$$ −11.6301 −0.409147
$$809$$ −13.0754 −0.459705 −0.229853 0.973225i $$-0.573824\pi$$
−0.229853 + 0.973225i $$0.573824\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −20.4452 −0.717487
$$813$$ −20.0000 −0.701431
$$814$$ 1.00000 0.0350500
$$815$$ 16.6301 0.582529
$$816$$ 1.00000 0.0350070
$$817$$ −37.6301 −1.31651
$$818$$ −6.36985 −0.222717
$$819$$ −11.4452 −0.399928
$$820$$ −4.81507 −0.168150
$$821$$ 11.0754 0.386533 0.193266 0.981146i $$-0.438092\pi$$
0.193266 + 0.981146i $$0.438092\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 41.0754 1.43180 0.715899 0.698204i $$-0.246017\pi$$
0.715899 + 0.698204i $$0.246017\pi$$
$$824$$ 0 0
$$825$$ −1.00000 −0.0348155
$$826$$ −6.00000 −0.208767
$$827$$ −28.0754 −0.976276 −0.488138 0.872766i $$-0.662324\pi$$
−0.488138 + 0.872766i $$0.662324\pi$$
$$828$$ 5.81507 0.202088
$$829$$ −53.5206 −1.85885 −0.929423 0.369015i $$-0.879695\pi$$
−0.929423 + 0.369015i $$0.879695\pi$$
$$830$$ 3.81507 0.132423
$$831$$ 21.4452 0.743926
$$832$$ −3.81507 −0.132264
$$833$$ 2.00000 0.0692959
$$834$$ −22.8151 −0.790021
$$835$$ −21.8151 −0.754942
$$836$$ −7.81507 −0.270290
$$837$$ 6.81507 0.235563
$$838$$ 25.4452 0.878990
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ 17.4452 0.601559
$$842$$ 29.2603 1.00838
$$843$$ −19.4452 −0.669729
$$844$$ 8.44522 0.290696
$$845$$ 1.55478 0.0534861
$$846$$ 8.00000 0.275046
$$847$$ −30.0000 −1.03081
$$848$$ 10.6301 0.365041
$$849$$ −21.8151 −0.748691
$$850$$ −1.00000 −0.0342997
$$851$$ 5.81507 0.199338
$$852$$ −11.6301 −0.398442
$$853$$ −11.0754 −0.379213 −0.189607 0.981860i $$-0.560721\pi$$
−0.189607 + 0.981860i $$0.560721\pi$$
$$854$$ −20.4452 −0.699621
$$855$$ 7.81507 0.267270
$$856$$ 8.18493 0.279755
$$857$$ 30.2603 1.03367 0.516836 0.856084i $$-0.327110\pi$$
0.516836 + 0.856084i $$0.327110\pi$$
$$858$$ −3.81507 −0.130244
$$859$$ 31.4452 1.07290 0.536449 0.843933i $$-0.319766\pi$$
0.536449 + 0.843933i $$0.319766\pi$$
$$860$$ −4.81507 −0.164193
$$861$$ −14.4452 −0.492292
$$862$$ −21.0000 −0.715263
$$863$$ −1.55478 −0.0529254 −0.0264627 0.999650i $$-0.508424\pi$$
−0.0264627 + 0.999650i $$0.508424\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −1.00000 −0.0340010
$$866$$ −1.81507 −0.0616787
$$867$$ −16.0000 −0.543388
$$868$$ 20.4452 0.693956
$$869$$ −12.0000 −0.407072
$$870$$ 6.81507 0.231053
$$871$$ −29.1096 −0.986340
$$872$$ −13.0000 −0.440236
$$873$$ −4.81507 −0.162966
$$874$$ −45.4452 −1.53721
$$875$$ 3.00000 0.101419
$$876$$ −5.81507 −0.196473
$$877$$ 49.7055 1.67844 0.839218 0.543795i $$-0.183013\pi$$
0.839218 + 0.543795i $$0.183013\pi$$
$$878$$ 18.4452 0.622496
$$879$$ 28.2603 0.953196
$$880$$ −1.00000 −0.0337100
$$881$$ 43.1849 1.45494 0.727469 0.686141i $$-0.240697\pi$$
0.727469 + 0.686141i $$0.240697\pi$$
$$882$$ −2.00000 −0.0673435
$$883$$ 27.0000 0.908622 0.454311 0.890843i $$-0.349885\pi$$
0.454311 + 0.890843i $$0.349885\pi$$
$$884$$ −3.81507 −0.128315
$$885$$ 2.00000 0.0672293
$$886$$ −13.6301 −0.457914
$$887$$ −14.0754 −0.472604 −0.236302 0.971680i $$-0.575936\pi$$
−0.236302 + 0.971680i $$0.575936\pi$$
$$888$$ −1.00000 −0.0335578
$$889$$ −64.3357 −2.15775
$$890$$ −11.8151 −0.396042
$$891$$ −1.00000 −0.0335013
$$892$$ −16.8151 −0.563010
$$893$$ −62.5206 −2.09217
$$894$$ −2.00000 −0.0668900
$$895$$ 16.0000 0.534821
$$896$$ −3.00000 −0.100223
$$897$$ −22.1849 −0.740733
$$898$$ −30.8904 −1.03083
$$899$$ −46.4452 −1.54903
$$900$$ 1.00000 0.0333333
$$901$$ 10.6301 0.354142
$$902$$ −4.81507 −0.160324
$$903$$ −14.4452 −0.480707
$$904$$ 6.81507 0.226666
$$905$$ −10.0000 −0.332411
$$906$$ −3.81507 −0.126747
$$907$$ −45.8151 −1.52126 −0.760632 0.649183i $$-0.775111\pi$$
−0.760632 + 0.649183i $$0.775111\pi$$
$$908$$ −24.8151 −0.823517
$$909$$ 11.6301 0.385748
$$910$$ 11.4452 0.379405
$$911$$ −54.5206 −1.80635 −0.903174 0.429275i $$-0.858769\pi$$
−0.903174 + 0.429275i $$0.858769\pi$$
$$912$$ 7.81507 0.258783
$$913$$ 3.81507 0.126260
$$914$$ 28.0754 0.928651
$$915$$ 6.81507 0.225299
$$916$$ 0.369854 0.0122203
$$917$$ −28.8904 −0.954046
$$918$$ −1.00000 −0.0330049
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ −5.81507 −0.191717
$$921$$ −18.0000 −0.593120
$$922$$ −34.8151 −1.14657
$$923$$ 44.3699 1.46045
$$924$$ −3.00000 −0.0986928
$$925$$ 1.00000 0.0328798
$$926$$ 17.6301 0.579363
$$927$$ 0 0
$$928$$ 6.81507 0.223716
$$929$$ 24.4452 0.802022 0.401011 0.916073i $$-0.368659\pi$$
0.401011 + 0.916073i $$0.368659\pi$$
$$930$$ −6.81507 −0.223475
$$931$$ 15.6301 0.512257
$$932$$ −7.63015 −0.249934
$$933$$ 10.4452 0.341961
$$934$$ −28.8151 −0.942858
$$935$$ −1.00000 −0.0327035
$$936$$ 3.81507 0.124700
$$937$$ −11.6301 −0.379940 −0.189970 0.981790i $$-0.560839\pi$$
−0.189970 + 0.981790i $$0.560839\pi$$
$$938$$ −22.8904 −0.747399
$$939$$ 32.8904 1.07334
$$940$$ −8.00000 −0.260931
$$941$$ −1.26029 −0.0410843 −0.0205422 0.999789i $$-0.506539\pi$$
−0.0205422 + 0.999789i $$0.506539\pi$$
$$942$$ 12.8151 0.417538
$$943$$ −28.0000 −0.911805
$$944$$ 2.00000 0.0650945
$$945$$ 3.00000 0.0975900
$$946$$ −4.81507 −0.156552
$$947$$ −25.5548 −0.830419 −0.415209 0.909726i $$-0.636292\pi$$
−0.415209 + 0.909726i $$0.636292\pi$$
$$948$$ 12.0000 0.389742
$$949$$ 22.1849 0.720153
$$950$$ −7.81507 −0.253554
$$951$$ −2.81507 −0.0912850
$$952$$ −3.00000 −0.0972306
$$953$$ 14.8904 0.482349 0.241174 0.970482i $$-0.422467\pi$$
0.241174 + 0.970482i $$0.422467\pi$$
$$954$$ −10.6301 −0.344164
$$955$$ 4.63015 0.149828
$$956$$ 6.44522 0.208453
$$957$$ 6.81507 0.220300
$$958$$ −9.81507 −0.317111
$$959$$ 6.00000 0.193750
$$960$$ 1.00000 0.0322749
$$961$$ 15.4452 0.498233
$$962$$ 3.81507 0.123003
$$963$$ −8.18493 −0.263756
$$964$$ −26.0000 −0.837404
$$965$$ −2.00000 −0.0643823
$$966$$ −17.4452 −0.561291
$$967$$ 7.63015 0.245369 0.122684 0.992446i $$-0.460850\pi$$
0.122684 + 0.992446i $$0.460850\pi$$
$$968$$ 10.0000 0.321412
$$969$$ 7.81507 0.251056
$$970$$ 4.81507 0.154603
$$971$$ −48.0754 −1.54281 −0.771406 0.636343i $$-0.780446\pi$$
−0.771406 + 0.636343i $$0.780446\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 68.4452 2.19425
$$974$$ −2.00000 −0.0640841
$$975$$ −3.81507 −0.122180
$$976$$ 6.81507 0.218145
$$977$$ 21.0000 0.671850 0.335925 0.941889i $$-0.390951\pi$$
0.335925 + 0.941889i $$0.390951\pi$$
$$978$$ −16.6301 −0.531773
$$979$$ −11.8151 −0.377611
$$980$$ 2.00000 0.0638877
$$981$$ 13.0000 0.415058
$$982$$ −2.18493 −0.0697238
$$983$$ 57.3357 1.82872 0.914362 0.404898i $$-0.132693\pi$$
0.914362 + 0.404898i $$0.132693\pi$$
$$984$$ 4.81507 0.153499
$$985$$ −4.18493 −0.133343
$$986$$ 6.81507 0.217036
$$987$$ −24.0000 −0.763928
$$988$$ −29.8151 −0.948544
$$989$$ −28.0000 −0.890348
$$990$$ 1.00000 0.0317821
$$991$$ −38.4452 −1.22125 −0.610626 0.791919i $$-0.709082\pi$$
−0.610626 + 0.791919i $$0.709082\pi$$
$$992$$ −6.81507 −0.216379
$$993$$ −23.2603 −0.738143
$$994$$ 34.8904 1.10666
$$995$$ −4.00000 −0.126809
$$996$$ −3.81507 −0.120885
$$997$$ 55.0754 1.74425 0.872127 0.489279i $$-0.162740\pi$$
0.872127 + 0.489279i $$0.162740\pi$$
$$998$$ 17.4452 0.552219
$$999$$ 1.00000 0.0316386
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 1110.2.a.q.1.1 2
3.2 odd 2 3330.2.a.bf.1.1 2
4.3 odd 2 8880.2.a.bh.1.1 2
5.4 even 2 5550.2.a.bx.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.1 2 1.1 even 1 trivial
3330.2.a.bf.1.1 2 3.2 odd 2
5550.2.a.bx.1.2 2 5.4 even 2
8880.2.a.bh.1.1 2 4.3 odd 2