Properties

 Label 5550.2.a.bx.1.1 Level $5550$ Weight $2$ Character 5550.1 Self dual yes Analytic conductor $44.317$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$44.3169731218$$ 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: no (minimal twist has level 1110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-4.81507$$ of defining polynomial Character $$\chi$$ $$=$$ 5550.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -6.81507 q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.81507 q^{19} +3.00000 q^{21} -1.00000 q^{22} +4.81507 q^{23} -1.00000 q^{24} -6.81507 q^{26} -1.00000 q^{27} -3.00000 q^{28} +3.81507 q^{29} -3.81507 q^{31} +1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.81507 q^{38} +6.81507 q^{39} +5.81507 q^{41} +3.00000 q^{42} -5.81507 q^{43} -1.00000 q^{44} +4.81507 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} -6.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -3.00000 q^{56} +2.81507 q^{57} +3.81507 q^{58} +2.00000 q^{59} -3.81507 q^{61} -3.81507 q^{62} -3.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +13.6301 q^{67} -1.00000 q^{68} -4.81507 q^{69} +9.63015 q^{71} +1.00000 q^{72} -4.81507 q^{73} -1.00000 q^{74} -2.81507 q^{76} +3.00000 q^{77} +6.81507 q^{78} +12.0000 q^{79} +1.00000 q^{81} +5.81507 q^{82} -6.81507 q^{83} +3.00000 q^{84} -5.81507 q^{86} -3.81507 q^{87} -1.00000 q^{88} +1.18493 q^{89} +20.4452 q^{91} +4.81507 q^{92} +3.81507 q^{93} +8.00000 q^{94} -1.00000 q^{96} -5.81507 q^{97} +2.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} + 2q^{8} + 2q^{9} - 2q^{11} - 2q^{12} - 3q^{13} - 6q^{14} + 2q^{16} - 2q^{17} + 2q^{18} + 5q^{19} + 6q^{21} - 2q^{22} - q^{23} - 2q^{24} - 3q^{26} - 2q^{27} - 6q^{28} - 3q^{29} + 3q^{31} + 2q^{32} + 2q^{33} - 2q^{34} + 2q^{36} - 2q^{37} + 5q^{38} + 3q^{39} + q^{41} + 6q^{42} - q^{43} - 2q^{44} - q^{46} + 16q^{47} - 2q^{48} + 4q^{49} + 2q^{51} - 3q^{52} - 2q^{54} - 6q^{56} - 5q^{57} - 3q^{58} + 4q^{59} + 3q^{61} + 3q^{62} - 6q^{63} + 2q^{64} + 2q^{66} + 6q^{67} - 2q^{68} + q^{69} - 2q^{71} + 2q^{72} + q^{73} - 2q^{74} + 5q^{76} + 6q^{77} + 3q^{78} + 24q^{79} + 2q^{81} + q^{82} - 3q^{83} + 6q^{84} - q^{86} + 3q^{87} - 2q^{88} + 13q^{89} + 9q^{91} - q^{92} - 3q^{93} + 16q^{94} - 2q^{96} - q^{97} + 4q^{98} - 2q^{99} + O(q^{100})$$

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
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −6.81507 −1.89016 −0.945081 0.326837i $$-0.894017\pi$$
−0.945081 + 0.326837i $$0.894017\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.81507 −0.645822 −0.322911 0.946429i $$-0.604661\pi$$
−0.322911 + 0.946429i $$0.604661\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ −1.00000 −0.213201
$$23$$ 4.81507 1.00401 0.502006 0.864864i $$-0.332596\pi$$
0.502006 + 0.864864i $$0.332596\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −6.81507 −1.33655
$$27$$ −1.00000 −0.192450
$$28$$ −3.00000 −0.566947
$$29$$ 3.81507 0.708441 0.354221 0.935162i $$-0.384746\pi$$
0.354221 + 0.935162i $$0.384746\pi$$
$$30$$ 0 0
$$31$$ −3.81507 −0.685207 −0.342604 0.939480i $$-0.611309\pi$$
−0.342604 + 0.939480i $$0.611309\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ −2.81507 −0.456665
$$39$$ 6.81507 1.09129
$$40$$ 0 0
$$41$$ 5.81507 0.908162 0.454081 0.890960i $$-0.349968\pi$$
0.454081 + 0.890960i $$0.349968\pi$$
$$42$$ 3.00000 0.462910
$$43$$ −5.81507 −0.886790 −0.443395 0.896326i $$-0.646226\pi$$
−0.443395 + 0.896326i $$0.646226\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.81507 0.709944
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ −6.81507 −0.945081
$$53$$ 10.6301 1.46016 0.730081 0.683360i $$-0.239482\pi$$
0.730081 + 0.683360i $$0.239482\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −3.00000 −0.400892
$$57$$ 2.81507 0.372866
$$58$$ 3.81507 0.500944
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ −3.81507 −0.488470 −0.244235 0.969716i $$-0.578537\pi$$
−0.244235 + 0.969716i $$0.578537\pi$$
$$62$$ −3.81507 −0.484515
$$63$$ −3.00000 −0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ 13.6301 1.66519 0.832594 0.553884i $$-0.186855\pi$$
0.832594 + 0.553884i $$0.186855\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −4.81507 −0.579667
$$70$$ 0 0
$$71$$ 9.63015 1.14289 0.571444 0.820641i $$-0.306383\pi$$
0.571444 + 0.820641i $$0.306383\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −4.81507 −0.563562 −0.281781 0.959479i $$-0.590925\pi$$
−0.281781 + 0.959479i $$0.590925\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −2.81507 −0.322911
$$77$$ 3.00000 0.341882
$$78$$ 6.81507 0.771655
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 5.81507 0.642167
$$83$$ −6.81507 −0.748051 −0.374026 0.927418i $$-0.622023\pi$$
−0.374026 + 0.927418i $$0.622023\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 0 0
$$86$$ −5.81507 −0.627055
$$87$$ −3.81507 −0.409019
$$88$$ −1.00000 −0.106600
$$89$$ 1.18493 0.125602 0.0628010 0.998026i $$-0.479997\pi$$
0.0628010 + 0.998026i $$0.479997\pi$$
$$90$$ 0 0
$$91$$ 20.4452 2.14324
$$92$$ 4.81507 0.502006
$$93$$ 3.81507 0.395605
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −5.81507 −0.590431 −0.295216 0.955431i $$-0.595391\pi$$
−0.295216 + 0.955431i $$0.595391\pi$$
$$98$$ 2.00000 0.202031
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −9.63015 −0.958235 −0.479118 0.877751i $$-0.659043\pi$$
−0.479118 + 0.877751i $$0.659043\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −6.81507 −0.668273
$$105$$ 0 0
$$106$$ 10.6301 1.03249
$$107$$ 18.8151 1.81892 0.909461 0.415790i $$-0.136495\pi$$
0.909461 + 0.415790i $$0.136495\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 13.0000 1.24517 0.622587 0.782551i $$-0.286082\pi$$
0.622587 + 0.782551i $$0.286082\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ −3.00000 −0.283473
$$113$$ −3.81507 −0.358892 −0.179446 0.983768i $$-0.557430\pi$$
−0.179446 + 0.983768i $$0.557430\pi$$
$$114$$ 2.81507 0.263656
$$115$$ 0 0
$$116$$ 3.81507 0.354221
$$117$$ −6.81507 −0.630054
$$118$$ 2.00000 0.184115
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −3.81507 −0.345400
$$123$$ −5.81507 −0.524327
$$124$$ −3.81507 −0.342604
$$125$$ 0 0
$$126$$ −3.00000 −0.267261
$$127$$ −10.4452 −0.926863 −0.463432 0.886133i $$-0.653382\pi$$
−0.463432 + 0.886133i $$0.653382\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 5.81507 0.511989
$$130$$ 0 0
$$131$$ 11.6301 1.01613 0.508065 0.861319i $$-0.330361\pi$$
0.508065 + 0.861319i $$0.330361\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 8.44522 0.732293
$$134$$ 13.6301 1.17747
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ −4.81507 −0.409886
$$139$$ 12.1849 1.03351 0.516756 0.856133i $$-0.327139\pi$$
0.516756 + 0.856133i $$0.327139\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 9.63015 0.808144
$$143$$ 6.81507 0.569905
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −4.81507 −0.398498
$$147$$ −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 $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −6.81507 −0.554603 −0.277301 0.960783i $$-0.589440\pi$$
−0.277301 + 0.960783i $$0.589440\pi$$
$$152$$ −2.81507 −0.228333
$$153$$ −1.00000 −0.0808452
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 6.81507 0.545643
$$157$$ 2.18493 0.174376 0.0871881 0.996192i $$-0.472212\pi$$
0.0871881 + 0.996192i $$0.472212\pi$$
$$158$$ 12.0000 0.954669
$$159$$ −10.6301 −0.843025
$$160$$ 0 0
$$161$$ −14.4452 −1.13844
$$162$$ 1.00000 0.0785674
$$163$$ 4.63015 0.362661 0.181331 0.983422i $$-0.441960\pi$$
0.181331 + 0.983422i $$0.441960\pi$$
$$164$$ 5.81507 0.454081
$$165$$ 0 0
$$166$$ −6.81507 −0.528952
$$167$$ 11.1849 0.865516 0.432758 0.901510i $$-0.357541\pi$$
0.432758 + 0.901510i $$0.357541\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 33.4452 2.57271
$$170$$ 0 0
$$171$$ −2.81507 −0.215274
$$172$$ −5.81507 −0.443395
$$173$$ 1.00000 0.0760286 0.0380143 0.999277i $$-0.487897\pi$$
0.0380143 + 0.999277i $$0.487897\pi$$
$$174$$ −3.81507 −0.289220
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −2.00000 −0.150329
$$178$$ 1.18493 0.0888140
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 20.4452 1.51550
$$183$$ 3.81507 0.282018
$$184$$ 4.81507 0.354972
$$185$$ 0 0
$$186$$ 3.81507 0.279735
$$187$$ 1.00000 0.0731272
$$188$$ 8.00000 0.583460
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −16.6301 −1.20332 −0.601658 0.798754i $$-0.705493\pi$$
−0.601658 + 0.798754i $$0.705493\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −5.81507 −0.417498
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 14.8151 1.05553 0.527765 0.849390i $$-0.323030\pi$$
0.527765 + 0.849390i $$0.323030\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −13.6301 −0.961396
$$202$$ −9.63015 −0.677575
$$203$$ −11.4452 −0.803297
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.81507 0.334671
$$208$$ −6.81507 −0.472540
$$209$$ 2.81507 0.194723
$$210$$ 0 0
$$211$$ −23.4452 −1.61404 −0.807018 0.590527i $$-0.798920\pi$$
−0.807018 + 0.590527i $$0.798920\pi$$
$$212$$ 10.6301 0.730081
$$213$$ −9.63015 −0.659847
$$214$$ 18.8151 1.28617
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 11.4452 0.776952
$$218$$ 13.0000 0.880471
$$219$$ 4.81507 0.325372
$$220$$ 0 0
$$221$$ 6.81507 0.458431
$$222$$ 1.00000 0.0671156
$$223$$ 6.18493 0.414173 0.207087 0.978323i $$-0.433602\pi$$
0.207087 + 0.978323i $$0.433602\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ −3.81507 −0.253775
$$227$$ 14.1849 0.941487 0.470743 0.882270i $$-0.343986\pi$$
0.470743 + 0.882270i $$0.343986\pi$$
$$228$$ 2.81507 0.186433
$$229$$ 21.6301 1.42936 0.714680 0.699451i $$-0.246572\pi$$
0.714680 + 0.699451i $$0.246572\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 3.81507 0.250472
$$233$$ −13.6301 −0.892941 −0.446470 0.894798i $$-0.647319\pi$$
−0.446470 + 0.894798i $$0.647319\pi$$
$$234$$ −6.81507 −0.445515
$$235$$ 0 0
$$236$$ 2.00000 0.130189
$$237$$ −12.0000 −0.779484
$$238$$ 3.00000 0.194461
$$239$$ −25.4452 −1.64591 −0.822957 0.568103i $$-0.807677\pi$$
−0.822957 + 0.568103i $$0.807677\pi$$
$$240$$ 0 0
$$241$$ −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
$$244$$ −3.81507 −0.244235
$$245$$ 0 0
$$246$$ −5.81507 −0.370756
$$247$$ 19.1849 1.22071
$$248$$ −3.81507 −0.242257
$$249$$ 6.81507 0.431888
$$250$$ 0 0
$$251$$ 9.63015 0.607849 0.303925 0.952696i $$-0.401703\pi$$
0.303925 + 0.952696i $$0.401703\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ −4.81507 −0.302721
$$254$$ −10.4452 −0.655391
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.81507 −0.425113 −0.212556 0.977149i $$-0.568179\pi$$
−0.212556 + 0.977149i $$0.568179\pi$$
$$258$$ 5.81507 0.362031
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ 3.81507 0.236147
$$262$$ 11.6301 0.718513
$$263$$ −1.44522 −0.0891160 −0.0445580 0.999007i $$-0.514188\pi$$
−0.0445580 + 0.999007i $$0.514188\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 0 0
$$266$$ 8.44522 0.517810
$$267$$ −1.18493 −0.0725164
$$268$$ 13.6301 0.832594
$$269$$ −0.815073 −0.0496959 −0.0248479 0.999691i $$-0.507910\pi$$
−0.0248479 + 0.999691i $$0.507910\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ −20.4452 −1.23740
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ −4.81507 −0.289833
$$277$$ 10.4452 0.627592 0.313796 0.949490i $$-0.398399\pi$$
0.313796 + 0.949490i $$0.398399\pi$$
$$278$$ 12.1849 0.730803
$$279$$ −3.81507 −0.228402
$$280$$ 0 0
$$281$$ 12.4452 0.742420 0.371210 0.928549i $$-0.378943\pi$$
0.371210 + 0.928549i $$0.378943\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 11.1849 0.664875 0.332437 0.943125i $$-0.392129\pi$$
0.332437 + 0.943125i $$0.392129\pi$$
$$284$$ 9.63015 0.571444
$$285$$ 0 0
$$286$$ 6.81507 0.402984
$$287$$ −17.4452 −1.02976
$$288$$ 1.00000 0.0589256
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 5.81507 0.340886
$$292$$ −4.81507 −0.281781
$$293$$ 14.2603 0.833095 0.416548 0.909114i $$-0.363240\pi$$
0.416548 + 0.909114i $$0.363240\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 1.00000 0.0580259
$$298$$ 2.00000 0.115857
$$299$$ −32.8151 −1.89774
$$300$$ 0 0
$$301$$ 17.4452 1.00553
$$302$$ −6.81507 −0.392163
$$303$$ 9.63015 0.553237
$$304$$ −2.81507 −0.161456
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ 18.0000 1.02731 0.513657 0.857996i $$-0.328290\pi$$
0.513657 + 0.857996i $$0.328290\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −21.4452 −1.21605 −0.608023 0.793919i $$-0.708037\pi$$
−0.608023 + 0.793919i $$0.708037\pi$$
$$312$$ 6.81507 0.385828
$$313$$ 30.8904 1.74603 0.873015 0.487693i $$-0.162161\pi$$
0.873015 + 0.487693i $$0.162161\pi$$
$$314$$ 2.18493 0.123303
$$315$$ 0 0
$$316$$ 12.0000 0.675053
$$317$$ −7.81507 −0.438938 −0.219469 0.975619i $$-0.570433\pi$$
−0.219469 + 0.975619i $$0.570433\pi$$
$$318$$ −10.6301 −0.596109
$$319$$ −3.81507 −0.213603
$$320$$ 0 0
$$321$$ −18.8151 −1.05015
$$322$$ −14.4452 −0.805001
$$323$$ 2.81507 0.156635
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 4.63015 0.256440
$$327$$ −13.0000 −0.718902
$$328$$ 5.81507 0.321084
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 19.2603 1.05864 0.529321 0.848422i $$-0.322447\pi$$
0.529321 + 0.848422i $$0.322447\pi$$
$$332$$ −6.81507 −0.374026
$$333$$ −1.00000 −0.0547997
$$334$$ 11.1849 0.612012
$$335$$ 0 0
$$336$$ 3.00000 0.163663
$$337$$ 14.8151 0.807028 0.403514 0.914973i $$-0.367789\pi$$
0.403514 + 0.914973i $$0.367789\pi$$
$$338$$ 33.4452 1.81918
$$339$$ 3.81507 0.207206
$$340$$ 0 0
$$341$$ 3.81507 0.206598
$$342$$ −2.81507 −0.152222
$$343$$ 15.0000 0.809924
$$344$$ −5.81507 −0.313528
$$345$$ 0 0
$$346$$ 1.00000 0.0537603
$$347$$ 31.2603 1.67814 0.839070 0.544023i $$-0.183100\pi$$
0.839070 + 0.544023i $$0.183100\pi$$
$$348$$ −3.81507 −0.204509
$$349$$ −35.2603 −1.88744 −0.943720 0.330745i $$-0.892700\pi$$
−0.943720 + 0.330745i $$0.892700\pi$$
$$350$$ 0 0
$$351$$ 6.81507 0.363762
$$352$$ −1.00000 −0.0533002
$$353$$ 23.8151 1.26755 0.633774 0.773518i $$-0.281505\pi$$
0.633774 + 0.773518i $$0.281505\pi$$
$$354$$ −2.00000 −0.106299
$$355$$ 0 0
$$356$$ 1.18493 0.0628010
$$357$$ −3.00000 −0.158777
$$358$$ 16.0000 0.845626
$$359$$ 27.6301 1.45826 0.729132 0.684373i $$-0.239924\pi$$
0.729132 + 0.684373i $$0.239924\pi$$
$$360$$ 0 0
$$361$$ −11.0754 −0.582914
$$362$$ −10.0000 −0.525588
$$363$$ 10.0000 0.524864
$$364$$ 20.4452 1.07162
$$365$$ 0 0
$$366$$ 3.81507 0.199417
$$367$$ 21.0000 1.09619 0.548096 0.836416i $$-0.315353\pi$$
0.548096 + 0.836416i $$0.315353\pi$$
$$368$$ 4.81507 0.251003
$$369$$ 5.81507 0.302721
$$370$$ 0 0
$$371$$ −31.8904 −1.65567
$$372$$ 3.81507 0.197802
$$373$$ −25.2603 −1.30793 −0.653964 0.756526i $$-0.726895\pi$$
−0.653964 + 0.756526i $$0.726895\pi$$
$$374$$ 1.00000 0.0517088
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −26.0000 −1.33907
$$378$$ 3.00000 0.154303
$$379$$ 1.63015 0.0837350 0.0418675 0.999123i $$-0.486669\pi$$
0.0418675 + 0.999123i $$0.486669\pi$$
$$380$$ 0 0
$$381$$ 10.4452 0.535125
$$382$$ −16.6301 −0.850872
$$383$$ −1.55478 −0.0794456 −0.0397228 0.999211i $$-0.512647\pi$$
−0.0397228 + 0.999211i $$0.512647\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ −5.81507 −0.295597
$$388$$ −5.81507 −0.295216
$$389$$ 38.7055 1.96245 0.981224 0.192873i $$-0.0617807\pi$$
0.981224 + 0.192873i $$0.0617807\pi$$
$$390$$ 0 0
$$391$$ −4.81507 −0.243509
$$392$$ 2.00000 0.101015
$$393$$ −11.6301 −0.586663
$$394$$ 14.8151 0.746373
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ −15.6301 −0.784455 −0.392227 0.919868i $$-0.628295\pi$$
−0.392227 + 0.919868i $$0.628295\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ −8.44522 −0.422790
$$400$$ 0 0
$$401$$ 32.4452 1.62024 0.810118 0.586266i $$-0.199403\pi$$
0.810118 + 0.586266i $$0.199403\pi$$
$$402$$ −13.6301 −0.679810
$$403$$ 26.0000 1.29515
$$404$$ −9.63015 −0.479118
$$405$$ 0 0
$$406$$ −11.4452 −0.568017
$$407$$ 1.00000 0.0495682
$$408$$ 1.00000 0.0495074
$$409$$ 27.6301 1.36622 0.683111 0.730314i $$-0.260626\pi$$
0.683111 + 0.730314i $$0.260626\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ −6.00000 −0.295241
$$414$$ 4.81507 0.236648
$$415$$ 0 0
$$416$$ −6.81507 −0.334136
$$417$$ −12.1849 −0.596698
$$418$$ 2.81507 0.137690
$$419$$ 6.44522 0.314870 0.157435 0.987529i $$-0.449678\pi$$
0.157435 + 0.987529i $$0.449678\pi$$
$$420$$ 0 0
$$421$$ 13.2603 0.646267 0.323134 0.946353i $$-0.395264\pi$$
0.323134 + 0.946353i $$0.395264\pi$$
$$422$$ −23.4452 −1.14130
$$423$$ 8.00000 0.388973
$$424$$ 10.6301 0.516246
$$425$$ 0 0
$$426$$ −9.63015 −0.466582
$$427$$ 11.4452 0.553873
$$428$$ 18.8151 0.909461
$$429$$ −6.81507 −0.329035
$$430$$ 0 0
$$431$$ 21.0000 1.01153 0.505767 0.862670i $$-0.331209\pi$$
0.505767 + 0.862670i $$0.331209\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 8.81507 0.423625 0.211813 0.977310i $$-0.432063\pi$$
0.211813 + 0.977310i $$0.432063\pi$$
$$434$$ 11.4452 0.549388
$$435$$ 0 0
$$436$$ 13.0000 0.622587
$$437$$ −13.5548 −0.648413
$$438$$ 4.81507 0.230073
$$439$$ 13.4452 0.641705 0.320853 0.947129i $$-0.396031\pi$$
0.320853 + 0.947129i $$0.396031\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 6.81507 0.324160
$$443$$ 7.63015 0.362519 0.181260 0.983435i $$-0.441983\pi$$
0.181260 + 0.983435i $$0.441983\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 0 0
$$446$$ 6.18493 0.292865
$$447$$ −2.00000 −0.0945968
$$448$$ −3.00000 −0.141737
$$449$$ −32.8904 −1.55220 −0.776098 0.630612i $$-0.782804\pi$$
−0.776098 + 0.630612i $$0.782804\pi$$
$$450$$ 0 0
$$451$$ −5.81507 −0.273821
$$452$$ −3.81507 −0.179446
$$453$$ 6.81507 0.320200
$$454$$ 14.1849 0.665732
$$455$$ 0 0
$$456$$ 2.81507 0.131828
$$457$$ −25.0754 −1.17298 −0.586488 0.809958i $$-0.699490\pi$$
−0.586488 + 0.809958i $$0.699490\pi$$
$$458$$ 21.6301 1.01071
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 24.1849 1.12640 0.563202 0.826319i $$-0.309569\pi$$
0.563202 + 0.826319i $$0.309569\pi$$
$$462$$ −3.00000 −0.139573
$$463$$ −3.63015 −0.168707 −0.0843536 0.996436i $$-0.526883\pi$$
−0.0843536 + 0.996436i $$0.526883\pi$$
$$464$$ 3.81507 0.177110
$$465$$ 0 0
$$466$$ −13.6301 −0.631404
$$467$$ −18.1849 −0.841498 −0.420749 0.907177i $$-0.638233\pi$$
−0.420749 + 0.907177i $$0.638233\pi$$
$$468$$ −6.81507 −0.315027
$$469$$ −40.8904 −1.88814
$$470$$ 0 0
$$471$$ −2.18493 −0.100676
$$472$$ 2.00000 0.0920575
$$473$$ 5.81507 0.267377
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ 10.6301 0.486721
$$478$$ −25.4452 −1.16384
$$479$$ −0.815073 −0.0372416 −0.0186208 0.999827i $$-0.505928\pi$$
−0.0186208 + 0.999827i $$0.505928\pi$$
$$480$$ 0 0
$$481$$ 6.81507 0.310741
$$482$$ −26.0000 −1.18427
$$483$$ 14.4452 0.657280
$$484$$ −10.0000 −0.454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ −3.81507 −0.172700
$$489$$ −4.63015 −0.209382
$$490$$ 0 0
$$491$$ 12.8151 0.578336 0.289168 0.957278i $$-0.406621\pi$$
0.289168 + 0.957278i $$0.406621\pi$$
$$492$$ −5.81507 −0.262164
$$493$$ −3.81507 −0.171822
$$494$$ 19.1849 0.863171
$$495$$ 0 0
$$496$$ −3.81507 −0.171302
$$497$$ −28.8904 −1.29591
$$498$$ 6.81507 0.305391
$$499$$ 14.4452 0.646657 0.323328 0.946287i $$-0.395198\pi$$
0.323328 + 0.946287i $$0.395198\pi$$
$$500$$ 0 0
$$501$$ −11.1849 −0.499706
$$502$$ 9.63015 0.429814
$$503$$ −43.2603 −1.92888 −0.964441 0.264300i $$-0.914859\pi$$
−0.964441 + 0.264300i $$0.914859\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −4.81507 −0.214056
$$507$$ −33.4452 −1.48535
$$508$$ −10.4452 −0.463432
$$509$$ −14.4452 −0.640273 −0.320137 0.947371i $$-0.603729\pi$$
−0.320137 + 0.947371i $$0.603729\pi$$
$$510$$ 0 0
$$511$$ 14.4452 0.639019
$$512$$ 1.00000 0.0441942
$$513$$ 2.81507 0.124289
$$514$$ −6.81507 −0.300600
$$515$$ 0 0
$$516$$ 5.81507 0.255994
$$517$$ −8.00000 −0.351840
$$518$$ 3.00000 0.131812
$$519$$ −1.00000 −0.0438951
$$520$$ 0 0
$$521$$ −9.81507 −0.430006 −0.215003 0.976613i $$-0.568976\pi$$
−0.215003 + 0.976613i $$0.568976\pi$$
$$522$$ 3.81507 0.166981
$$523$$ 43.2603 1.89164 0.945820 0.324691i $$-0.105260\pi$$
0.945820 + 0.324691i $$0.105260\pi$$
$$524$$ 11.6301 0.508065
$$525$$ 0 0
$$526$$ −1.44522 −0.0630145
$$527$$ 3.81507 0.166187
$$528$$ 1.00000 0.0435194
$$529$$ 0.184927 0.00804031
$$530$$ 0 0
$$531$$ 2.00000 0.0867926
$$532$$ 8.44522 0.366147
$$533$$ −39.6301 −1.71657
$$534$$ −1.18493 −0.0512768
$$535$$ 0 0
$$536$$ 13.6301 0.588733
$$537$$ −16.0000 −0.690451
$$538$$ −0.815073 −0.0351403
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ 17.1849 0.738838 0.369419 0.929263i $$-0.379557\pi$$
0.369419 + 0.929263i $$0.379557\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 10.0000 0.429141
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ −20.4452 −0.874975
$$547$$ 39.8904 1.70559 0.852796 0.522244i $$-0.174905\pi$$
0.852796 + 0.522244i $$0.174905\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ −3.81507 −0.162823
$$550$$ 0 0
$$551$$ −10.7397 −0.457527
$$552$$ −4.81507 −0.204943
$$553$$ −36.0000 −1.53088
$$554$$ 10.4452 0.443775
$$555$$ 0 0
$$556$$ 12.1849 0.516756
$$557$$ 0.369854 0.0156712 0.00783561 0.999969i $$-0.497506\pi$$
0.00783561 + 0.999969i $$0.497506\pi$$
$$558$$ −3.81507 −0.161505
$$559$$ 39.6301 1.67618
$$560$$ 0 0
$$561$$ −1.00000 −0.0422200
$$562$$ 12.4452 0.524970
$$563$$ −23.4452 −0.988098 −0.494049 0.869434i $$-0.664484\pi$$
−0.494049 + 0.869434i $$0.664484\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 11.1849 0.470138
$$567$$ −3.00000 −0.125988
$$568$$ 9.63015 0.404072
$$569$$ −25.1849 −1.05581 −0.527904 0.849304i $$-0.677022\pi$$
−0.527904 + 0.849304i $$0.677022\pi$$
$$570$$ 0 0
$$571$$ −4.18493 −0.175134 −0.0875669 0.996159i $$-0.527909\pi$$
−0.0875669 + 0.996159i $$0.527909\pi$$
$$572$$ 6.81507 0.284953
$$573$$ 16.6301 0.694734
$$574$$ −17.4452 −0.728149
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ −16.0000 −0.665512
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 20.4452 0.848211
$$582$$ 5.81507 0.241043
$$583$$ −10.6301 −0.440256
$$584$$ −4.81507 −0.199249
$$585$$ 0 0
$$586$$ 14.2603 0.589087
$$587$$ −10.1849 −0.420377 −0.210188 0.977661i $$-0.567408\pi$$
−0.210188 + 0.977661i $$0.567408\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 10.7397 0.442522
$$590$$ 0 0
$$591$$ −14.8151 −0.609411
$$592$$ −1.00000 −0.0410997
$$593$$ −7.63015 −0.313333 −0.156666 0.987652i $$-0.550075\pi$$
−0.156666 + 0.987652i $$0.550075\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 4.00000 0.163709
$$598$$ −32.8151 −1.34191
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −40.6301 −1.65734 −0.828669 0.559739i $$-0.810901\pi$$
−0.828669 + 0.559739i $$0.810901\pi$$
$$602$$ 17.4452 0.711014
$$603$$ 13.6301 0.555062
$$604$$ −6.81507 −0.277301
$$605$$ 0 0
$$606$$ 9.63015 0.391198
$$607$$ −5.26029 −0.213509 −0.106754 0.994285i $$-0.534046\pi$$
−0.106754 + 0.994285i $$0.534046\pi$$
$$608$$ −2.81507 −0.114166
$$609$$ 11.4452 0.463784
$$610$$ 0 0
$$611$$ −54.5206 −2.20567
$$612$$ −1.00000 −0.0404226
$$613$$ −47.0754 −1.90136 −0.950678 0.310179i $$-0.899611\pi$$
−0.950678 + 0.310179i $$0.899611\pi$$
$$614$$ 18.0000 0.726421
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ −45.4452 −1.82660 −0.913299 0.407290i $$-0.866474\pi$$
−0.913299 + 0.407290i $$0.866474\pi$$
$$620$$ 0 0
$$621$$ −4.81507 −0.193222
$$622$$ −21.4452 −0.859875
$$623$$ −3.55478 −0.142419
$$624$$ 6.81507 0.272821
$$625$$ 0 0
$$626$$ 30.8904 1.23463
$$627$$ −2.81507 −0.112423
$$628$$ 2.18493 0.0871881
$$629$$ 1.00000 0.0398726
$$630$$ 0 0
$$631$$ −40.7055 −1.62046 −0.810230 0.586112i $$-0.800658\pi$$
−0.810230 + 0.586112i $$0.800658\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 23.4452 0.931864
$$634$$ −7.81507 −0.310376
$$635$$ 0 0
$$636$$ −10.6301 −0.421513
$$637$$ −13.6301 −0.540046
$$638$$ −3.81507 −0.151040
$$639$$ 9.63015 0.380963
$$640$$ 0 0
$$641$$ −45.4452 −1.79498 −0.897489 0.441037i $$-0.854611\pi$$
−0.897489 + 0.441037i $$0.854611\pi$$
$$642$$ −18.8151 −0.742572
$$643$$ −17.0000 −0.670415 −0.335207 0.942144i $$-0.608806\pi$$
−0.335207 + 0.942144i $$0.608806\pi$$
$$644$$ −14.4452 −0.569221
$$645$$ 0 0
$$646$$ 2.81507 0.110758
$$647$$ 28.0754 1.10376 0.551878 0.833925i $$-0.313911\pi$$
0.551878 + 0.833925i $$0.313911\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −2.00000 −0.0785069
$$650$$ 0 0
$$651$$ −11.4452 −0.448573
$$652$$ 4.63015 0.181331
$$653$$ −4.00000 −0.156532 −0.0782660 0.996933i $$-0.524938\pi$$
−0.0782660 + 0.996933i $$0.524938\pi$$
$$654$$ −13.0000 −0.508340
$$655$$ 0 0
$$656$$ 5.81507 0.227040
$$657$$ −4.81507 −0.187854
$$658$$ −24.0000 −0.935617
$$659$$ 11.2603 0.438639 0.219319 0.975653i $$-0.429616\pi$$
0.219319 + 0.975653i $$0.429616\pi$$
$$660$$ 0 0
$$661$$ 46.2603 1.79932 0.899658 0.436595i $$-0.143816\pi$$
0.899658 + 0.436595i $$0.143816\pi$$
$$662$$ 19.2603 0.748572
$$663$$ −6.81507 −0.264675
$$664$$ −6.81507 −0.264476
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ 18.3699 0.711284
$$668$$ 11.1849 0.432758
$$669$$ −6.18493 −0.239123
$$670$$ 0 0
$$671$$ 3.81507 0.147279
$$672$$ 3.00000 0.115728
$$673$$ −16.8151 −0.648173 −0.324087 0.946027i $$-0.605057\pi$$
−0.324087 + 0.946027i $$0.605057\pi$$
$$674$$ 14.8151 0.570655
$$675$$ 0 0
$$676$$ 33.4452 1.28635
$$677$$ 6.07536 0.233495 0.116748 0.993162i $$-0.462753\pi$$
0.116748 + 0.993162i $$0.462753\pi$$
$$678$$ 3.81507 0.146517
$$679$$ 17.4452 0.669486
$$680$$ 0 0
$$681$$ −14.1849 −0.543568
$$682$$ 3.81507 0.146087
$$683$$ −48.3357 −1.84951 −0.924756 0.380560i $$-0.875731\pi$$
−0.924756 + 0.380560i $$0.875731\pi$$
$$684$$ −2.81507 −0.107637
$$685$$ 0 0
$$686$$ 15.0000 0.572703
$$687$$ −21.6301 −0.825242
$$688$$ −5.81507 −0.221698
$$689$$ −72.4452 −2.75994
$$690$$ 0 0
$$691$$ 19.0754 0.725661 0.362831 0.931855i $$-0.381810\pi$$
0.362831 + 0.931855i $$0.381810\pi$$
$$692$$ 1.00000 0.0380143
$$693$$ 3.00000 0.113961
$$694$$ 31.2603 1.18662
$$695$$ 0 0
$$696$$ −3.81507 −0.144610
$$697$$ −5.81507 −0.220262
$$698$$ −35.2603 −1.33462
$$699$$ 13.6301 0.515539
$$700$$ 0 0
$$701$$ 12.3699 0.467203 0.233601 0.972332i $$-0.424949\pi$$
0.233601 + 0.972332i $$0.424949\pi$$
$$702$$ 6.81507 0.257218
$$703$$ 2.81507 0.106172
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 23.8151 0.896292
$$707$$ 28.8904 1.08654
$$708$$ −2.00000 −0.0751646
$$709$$ −0.630146 −0.0236656 −0.0118328 0.999930i $$-0.503767\pi$$
−0.0118328 + 0.999930i $$0.503767\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 1.18493 0.0444070
$$713$$ −18.3699 −0.687956
$$714$$ −3.00000 −0.112272
$$715$$ 0 0
$$716$$ 16.0000 0.597948
$$717$$ 25.4452 0.950269
$$718$$ 27.6301 1.03115
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −11.0754 −0.412182
$$723$$ 26.0000 0.966950
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 10.0000 0.371135
$$727$$ 18.0000 0.667583 0.333792 0.942647i $$-0.391672\pi$$
0.333792 + 0.942647i $$0.391672\pi$$
$$728$$ 20.4452 0.757750
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 5.81507 0.215078
$$732$$ 3.81507 0.141009
$$733$$ −8.55478 −0.315978 −0.157989 0.987441i $$-0.550501\pi$$
−0.157989 + 0.987441i $$0.550501\pi$$
$$734$$ 21.0000 0.775124
$$735$$ 0 0
$$736$$ 4.81507 0.177486
$$737$$ −13.6301 −0.502073
$$738$$ 5.81507 0.214056
$$739$$ −19.0754 −0.701699 −0.350849 0.936432i $$-0.614107\pi$$
−0.350849 + 0.936432i $$0.614107\pi$$
$$740$$ 0 0
$$741$$ −19.1849 −0.704776
$$742$$ −31.8904 −1.17073
$$743$$ 27.4452 1.00687 0.503434 0.864034i $$-0.332070\pi$$
0.503434 + 0.864034i $$0.332070\pi$$
$$744$$ 3.81507 0.139867
$$745$$ 0 0
$$746$$ −25.2603 −0.924845
$$747$$ −6.81507 −0.249350
$$748$$ 1.00000 0.0365636
$$749$$ −56.4452 −2.06246
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 8.00000 0.291730
$$753$$ −9.63015 −0.350942
$$754$$ −26.0000 −0.946864
$$755$$ 0 0
$$756$$ 3.00000 0.109109
$$757$$ −13.1849 −0.479214 −0.239607 0.970870i $$-0.577019\pi$$
−0.239607 + 0.970870i $$0.577019\pi$$
$$758$$ 1.63015 0.0592096
$$759$$ 4.81507 0.174776
$$760$$ 0 0
$$761$$ −19.8151 −0.718296 −0.359148 0.933281i $$-0.616933\pi$$
−0.359148 + 0.933281i $$0.616933\pi$$
$$762$$ 10.4452 0.378390
$$763$$ −39.0000 −1.41189
$$764$$ −16.6301 −0.601658
$$765$$ 0 0
$$766$$ −1.55478 −0.0561765
$$767$$ −13.6301 −0.492156
$$768$$ −1.00000 −0.0360844
$$769$$ 6.73971 0.243040 0.121520 0.992589i $$-0.461223\pi$$
0.121520 + 0.992589i $$0.461223\pi$$
$$770$$ 0 0
$$771$$ 6.81507 0.245439
$$772$$ 2.00000 0.0719816
$$773$$ 28.6301 1.02975 0.514877 0.857264i $$-0.327837\pi$$
0.514877 + 0.857264i $$0.327837\pi$$
$$774$$ −5.81507 −0.209018
$$775$$ 0 0
$$776$$ −5.81507 −0.208749
$$777$$ −3.00000 −0.107624
$$778$$ 38.7055 1.38766
$$779$$ −16.3699 −0.586511
$$780$$ 0 0
$$781$$ −9.63015 −0.344594
$$782$$ −4.81507 −0.172187
$$783$$ −3.81507 −0.136340
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ −11.6301 −0.414834
$$787$$ 18.3699 0.654815 0.327407 0.944883i $$-0.393825\pi$$
0.327407 + 0.944883i $$0.393825\pi$$
$$788$$ 14.8151 0.527765
$$789$$ 1.44522 0.0514511
$$790$$ 0 0
$$791$$ 11.4452 0.406945
$$792$$ −1.00000 −0.0355335
$$793$$ 26.0000 0.923287
$$794$$ −15.6301 −0.554693
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −28.8904 −1.02335 −0.511676 0.859179i $$-0.670975\pi$$
−0.511676 + 0.859179i $$0.670975\pi$$
$$798$$ −8.44522 −0.298958
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 1.18493 0.0418673
$$802$$ 32.4452 1.14568
$$803$$ 4.81507 0.169920
$$804$$ −13.6301 −0.480698
$$805$$ 0 0
$$806$$ 26.0000 0.915811
$$807$$ 0.815073 0.0286919
$$808$$ −9.63015 −0.338787
$$809$$ 40.0754 1.40897 0.704487 0.709717i $$-0.251177\pi$$
0.704487 + 0.709717i $$0.251177\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −11.4452 −0.401648
$$813$$ 20.0000 0.701431
$$814$$ 1.00000 0.0350500
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ 16.3699 0.572709
$$818$$ 27.6301 0.966065
$$819$$ 20.4452 0.714414
$$820$$ 0 0
$$821$$ −42.0754 −1.46844 −0.734220 0.678911i $$-0.762452\pi$$
−0.734220 + 0.678911i $$0.762452\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ 12.0754 0.420921 0.210460 0.977602i $$-0.432504\pi$$
0.210460 + 0.977602i $$0.432504\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −6.00000 −0.208767
$$827$$ −25.0754 −0.871956 −0.435978 0.899957i $$-0.643597\pi$$
−0.435978 + 0.899957i $$0.643597\pi$$
$$828$$ 4.81507 0.167335
$$829$$ 31.5206 1.09476 0.547378 0.836886i $$-0.315626\pi$$
0.547378 + 0.836886i $$0.315626\pi$$
$$830$$ 0 0
$$831$$ −10.4452 −0.362341
$$832$$ −6.81507 −0.236270
$$833$$ −2.00000 −0.0692959
$$834$$ −12.1849 −0.421930
$$835$$ 0 0
$$836$$ 2.81507 0.0973613
$$837$$ 3.81507 0.131868
$$838$$ 6.44522 0.222646
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ −14.4452 −0.498111
$$842$$ 13.2603 0.456980
$$843$$ −12.4452 −0.428636
$$844$$ −23.4452 −0.807018
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 30.0000 1.03081
$$848$$ 10.6301 0.365041
$$849$$ −11.1849 −0.383866
$$850$$ 0 0
$$851$$ −4.81507 −0.165059
$$852$$ −9.63015 −0.329923
$$853$$ −42.0754 −1.44063 −0.720317 0.693646i $$-0.756003\pi$$
−0.720317 + 0.693646i $$0.756003\pi$$
$$854$$ 11.4452 0.391647
$$855$$ 0 0
$$856$$ 18.8151 0.643086
$$857$$ 12.2603 0.418804 0.209402 0.977830i $$-0.432848\pi$$
0.209402 + 0.977830i $$0.432848\pi$$
$$858$$ −6.81507 −0.232663
$$859$$ −0.445219 −0.0151907 −0.00759533 0.999971i $$-0.502418\pi$$
−0.00759533 + 0.999971i $$0.502418\pi$$
$$860$$ 0 0
$$861$$ 17.4452 0.594531
$$862$$ 21.0000 0.715263
$$863$$ 33.4452 1.13849 0.569244 0.822168i $$-0.307236\pi$$
0.569244 + 0.822168i $$0.307236\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 8.81507 0.299548
$$867$$ 16.0000 0.543388
$$868$$ 11.4452 0.388476
$$869$$ −12.0000 −0.407072
$$870$$ 0 0
$$871$$ −92.8904 −3.14747
$$872$$ 13.0000 0.440236
$$873$$ −5.81507 −0.196810
$$874$$ −13.5548 −0.458497
$$875$$ 0 0
$$876$$ 4.81507 0.162686
$$877$$ 24.7055 0.834246 0.417123 0.908850i $$-0.363038\pi$$
0.417123 + 0.908850i $$0.363038\pi$$
$$878$$ 13.4452 0.453754
$$879$$ −14.2603 −0.480988
$$880$$ 0 0
$$881$$ 53.8151 1.81308 0.906538 0.422124i $$-0.138715\pi$$
0.906538 + 0.422124i $$0.138715\pi$$
$$882$$ 2.00000 0.0673435
$$883$$ −27.0000 −0.908622 −0.454311 0.890843i $$-0.650115\pi$$
−0.454311 + 0.890843i $$0.650115\pi$$
$$884$$ 6.81507 0.229216
$$885$$ 0 0
$$886$$ 7.63015 0.256340
$$887$$ −39.0754 −1.31202 −0.656011 0.754751i $$-0.727758\pi$$
−0.656011 + 0.754751i $$0.727758\pi$$
$$888$$ 1.00000 0.0335578
$$889$$ 31.3357 1.05096
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 6.18493 0.207087
$$893$$ −22.5206 −0.753623
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ −3.00000 −0.100223
$$897$$ 32.8151 1.09566
$$898$$ −32.8904 −1.09757
$$899$$ −14.5548 −0.485429
$$900$$ 0 0
$$901$$ −10.6301 −0.354142
$$902$$ −5.81507 −0.193621
$$903$$ −17.4452 −0.580541
$$904$$ −3.81507 −0.126887
$$905$$ 0 0
$$906$$ 6.81507 0.226416
$$907$$ 35.1849 1.16830 0.584148 0.811647i $$-0.301429\pi$$
0.584148 + 0.811647i $$0.301429\pi$$
$$908$$ 14.1849 0.470743
$$909$$ −9.63015 −0.319412
$$910$$ 0 0
$$911$$ 30.5206 1.01119 0.505596 0.862770i $$-0.331273\pi$$
0.505596 + 0.862770i $$0.331273\pi$$
$$912$$ 2.81507 0.0932164
$$913$$ 6.81507 0.225546
$$914$$ −25.0754 −0.829419
$$915$$ 0 0
$$916$$ 21.6301 0.714680
$$917$$ −34.8904 −1.15218
$$918$$ 1.00000 0.0330049
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ −18.0000 −0.593120
$$922$$ 24.1849 0.796488
$$923$$ −65.6301 −2.16024
$$924$$ −3.00000 −0.0986928
$$925$$ 0 0
$$926$$ −3.63015 −0.119294
$$927$$ 0 0
$$928$$ 3.81507 0.125236
$$929$$ −7.44522 −0.244270 −0.122135 0.992514i $$-0.538974\pi$$
−0.122135 + 0.992514i $$0.538974\pi$$
$$930$$ 0 0
$$931$$ −5.63015 −0.184521
$$932$$ −13.6301 −0.446470
$$933$$ 21.4452 0.702085
$$934$$ −18.1849 −0.595029
$$935$$ 0 0
$$936$$ −6.81507 −0.222758
$$937$$ −9.63015 −0.314603 −0.157302 0.987551i $$-0.550279\pi$$
−0.157302 + 0.987551i $$0.550279\pi$$
$$938$$ −40.8904 −1.33512
$$939$$ −30.8904 −1.00807
$$940$$ 0 0
$$941$$ 41.2603 1.34505 0.672524 0.740076i $$-0.265210\pi$$
0.672524 + 0.740076i $$0.265210\pi$$
$$942$$ −2.18493 −0.0711888
$$943$$ 28.0000 0.911805
$$944$$ 2.00000 0.0650945
$$945$$ 0 0
$$946$$ 5.81507 0.189064
$$947$$ 57.4452 1.86672 0.933359 0.358943i $$-0.116863\pi$$
0.933359 + 0.358943i $$0.116863\pi$$
$$948$$ −12.0000 −0.389742
$$949$$ 32.8151 1.06522
$$950$$ 0 0
$$951$$ 7.81507 0.253421
$$952$$ 3.00000 0.0972306
$$953$$ 48.8904 1.58372 0.791858 0.610705i $$-0.209114\pi$$
0.791858 + 0.610705i $$0.209114\pi$$
$$954$$ 10.6301 0.344164
$$955$$ 0 0
$$956$$ −25.4452 −0.822957
$$957$$ 3.81507 0.123324
$$958$$ −0.815073 −0.0263338
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −16.4452 −0.530491
$$962$$ 6.81507 0.219727
$$963$$ 18.8151 0.606307
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 14.4452 0.464767
$$967$$ 13.6301 0.438316 0.219158 0.975689i $$-0.429669\pi$$
0.219158 + 0.975689i $$0.429669\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ −2.81507 −0.0904332
$$970$$ 0 0
$$971$$ 5.07536 0.162876 0.0814381 0.996678i $$-0.474049\pi$$
0.0814381 + 0.996678i $$0.474049\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −36.5548 −1.17189
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −3.81507 −0.122118
$$977$$ −21.0000 −0.671850 −0.335925 0.941889i $$-0.609049\pi$$
−0.335925 + 0.941889i $$0.609049\pi$$
$$978$$ −4.63015 −0.148056
$$979$$ −1.18493 −0.0378704
$$980$$ 0 0
$$981$$ 13.0000 0.415058
$$982$$ 12.8151 0.408945
$$983$$ 38.3357 1.22272 0.611359 0.791354i $$-0.290623\pi$$
0.611359 + 0.791354i $$0.290623\pi$$
$$984$$ −5.81507 −0.185378
$$985$$ 0 0
$$986$$ −3.81507 −0.121497
$$987$$ 24.0000 0.763928
$$988$$ 19.1849 0.610354
$$989$$ −28.0000 −0.890348
$$990$$ 0 0
$$991$$ −6.55478 −0.208219 −0.104110 0.994566i $$-0.533199\pi$$
−0.104110 + 0.994566i $$0.533199\pi$$
$$992$$ −3.81507 −0.121129
$$993$$ −19.2603 −0.611207
$$994$$ −28.8904 −0.916349
$$995$$ 0 0
$$996$$ 6.81507 0.215944
$$997$$ −1.92464 −0.0609538 −0.0304769 0.999535i $$-0.509703\pi$$
−0.0304769 + 0.999535i $$0.509703\pi$$
$$998$$ 14.4452 0.457255
$$999$$ 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 5550.2.a.bx.1.1 2
5.4 even 2 1110.2.a.q.1.2 2
15.14 odd 2 3330.2.a.bf.1.2 2
20.19 odd 2 8880.2.a.bh.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.2 2 5.4 even 2
3330.2.a.bf.1.2 2 15.14 odd 2
5550.2.a.bx.1.1 2 1.1 even 1 trivial
8880.2.a.bh.1.2 2 20.19 odd 2