Properties

 Label 950.2.a.n.1.3 Level $950$ Weight $2$ Character 950.1 Self dual yes Analytic conductor $7.586$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 950.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.76156 q^{3} +1.00000 q^{4} +2.76156 q^{6} -0.761557 q^{7} +1.00000 q^{8} +4.62620 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.76156 q^{3} +1.00000 q^{4} +2.76156 q^{6} -0.761557 q^{7} +1.00000 q^{8} +4.62620 q^{9} -0.864641 q^{11} +2.76156 q^{12} +5.62620 q^{13} -0.761557 q^{14} +1.00000 q^{16} -3.62620 q^{17} +4.62620 q^{18} +1.00000 q^{19} -2.10308 q^{21} -0.864641 q^{22} -8.01395 q^{23} +2.76156 q^{24} +5.62620 q^{26} +4.49084 q^{27} -0.761557 q^{28} -7.35548 q^{29} +8.11704 q^{31} +1.00000 q^{32} -2.38776 q^{33} -3.62620 q^{34} +4.62620 q^{36} +0.476886 q^{37} +1.00000 q^{38} +15.5371 q^{39} -2.65847 q^{41} -2.10308 q^{42} +6.86464 q^{43} -0.864641 q^{44} -8.01395 q^{46} +1.25240 q^{47} +2.76156 q^{48} -6.42003 q^{49} -10.0140 q^{51} +5.62620 q^{52} +2.37380 q^{53} +4.49084 q^{54} -0.761557 q^{56} +2.76156 q^{57} -7.35548 q^{58} +4.49084 q^{59} -10.8646 q^{61} +8.11704 q^{62} -3.52311 q^{63} +1.00000 q^{64} -2.38776 q^{66} +1.03228 q^{67} -3.62620 q^{68} -22.1310 q^{69} -10.1816 q^{71} +4.62620 q^{72} +16.4017 q^{73} +0.476886 q^{74} +1.00000 q^{76} +0.658473 q^{77} +15.5371 q^{78} -12.5693 q^{79} -1.47689 q^{81} -2.65847 q^{82} -0.270718 q^{83} -2.10308 q^{84} +6.86464 q^{86} -20.3126 q^{87} -0.864641 q^{88} +0.387755 q^{89} -4.28467 q^{91} -8.01395 q^{92} +22.4157 q^{93} +1.25240 q^{94} +2.76156 q^{96} -8.50479 q^{97} -6.42003 q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 4q^{7} + 3q^{8} + 5q^{9} + O(q^{10})$$ $$3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 4q^{7} + 3q^{8} + 5q^{9} + 2q^{12} + 8q^{13} + 4q^{14} + 3q^{16} - 2q^{17} + 5q^{18} + 3q^{19} - 10q^{21} + 2q^{24} + 8q^{26} + 2q^{27} + 4q^{28} - 8q^{29} + 4q^{31} + 3q^{32} + 8q^{33} - 2q^{34} + 5q^{36} + 14q^{37} + 3q^{38} + 10q^{39} + 2q^{41} - 10q^{42} + 18q^{43} - 14q^{47} + 2q^{48} - 3q^{49} - 6q^{51} + 8q^{52} + 16q^{53} + 2q^{54} + 4q^{56} + 2q^{57} - 8q^{58} + 2q^{59} - 30q^{61} + 4q^{62} + 2q^{63} + 3q^{64} + 8q^{66} + 2q^{67} - 2q^{68} - 22q^{69} - 8q^{71} + 5q^{72} + 10q^{73} + 14q^{74} + 3q^{76} - 8q^{77} + 10q^{78} - 17q^{81} + 2q^{82} - 6q^{83} - 10q^{84} + 18q^{86} + 6q^{87} - 14q^{89} + 6q^{91} + 4q^{93} - 14q^{94} + 2q^{96} + 10q^{97} - 3q^{98} - 12q^{99} + O(q^{100})$$

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
$$3$$ 2.76156 1.59439 0.797193 0.603725i $$-0.206317\pi$$
0.797193 + 0.603725i $$0.206317\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.76156 1.12740
$$7$$ −0.761557 −0.287842 −0.143921 0.989589i $$-0.545971\pi$$
−0.143921 + 0.989589i $$0.545971\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.62620 1.54207
$$10$$ 0 0
$$11$$ −0.864641 −0.260699 −0.130350 0.991468i $$-0.541610\pi$$
−0.130350 + 0.991468i $$0.541610\pi$$
$$12$$ 2.76156 0.797193
$$13$$ 5.62620 1.56043 0.780213 0.625514i $$-0.215111\pi$$
0.780213 + 0.625514i $$0.215111\pi$$
$$14$$ −0.761557 −0.203535
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.62620 −0.879482 −0.439741 0.898125i $$-0.644930\pi$$
−0.439741 + 0.898125i $$0.644930\pi$$
$$18$$ 4.62620 1.09041
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.10308 −0.458930
$$22$$ −0.864641 −0.184342
$$23$$ −8.01395 −1.67102 −0.835512 0.549472i $$-0.814829\pi$$
−0.835512 + 0.549472i $$0.814829\pi$$
$$24$$ 2.76156 0.563700
$$25$$ 0 0
$$26$$ 5.62620 1.10339
$$27$$ 4.49084 0.864262
$$28$$ −0.761557 −0.143921
$$29$$ −7.35548 −1.36588 −0.682939 0.730475i $$-0.739299\pi$$
−0.682939 + 0.730475i $$0.739299\pi$$
$$30$$ 0 0
$$31$$ 8.11704 1.45786 0.728931 0.684587i $$-0.240017\pi$$
0.728931 + 0.684587i $$0.240017\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.38776 −0.415655
$$34$$ −3.62620 −0.621888
$$35$$ 0 0
$$36$$ 4.62620 0.771033
$$37$$ 0.476886 0.0783995 0.0391998 0.999231i $$-0.487519\pi$$
0.0391998 + 0.999231i $$0.487519\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 15.5371 2.48792
$$40$$ 0 0
$$41$$ −2.65847 −0.415184 −0.207592 0.978216i $$-0.566563\pi$$
−0.207592 + 0.978216i $$0.566563\pi$$
$$42$$ −2.10308 −0.324513
$$43$$ 6.86464 1.04685 0.523424 0.852072i $$-0.324654\pi$$
0.523424 + 0.852072i $$0.324654\pi$$
$$44$$ −0.864641 −0.130350
$$45$$ 0 0
$$46$$ −8.01395 −1.18159
$$47$$ 1.25240 0.182681 0.0913404 0.995820i $$-0.470885\pi$$
0.0913404 + 0.995820i $$0.470885\pi$$
$$48$$ 2.76156 0.398596
$$49$$ −6.42003 −0.917147
$$50$$ 0 0
$$51$$ −10.0140 −1.40223
$$52$$ 5.62620 0.780213
$$53$$ 2.37380 0.326067 0.163033 0.986621i $$-0.447872\pi$$
0.163033 + 0.986621i $$0.447872\pi$$
$$54$$ 4.49084 0.611126
$$55$$ 0 0
$$56$$ −0.761557 −0.101767
$$57$$ 2.76156 0.365777
$$58$$ −7.35548 −0.965822
$$59$$ 4.49084 0.584657 0.292329 0.956318i $$-0.405570\pi$$
0.292329 + 0.956318i $$0.405570\pi$$
$$60$$ 0 0
$$61$$ −10.8646 −1.39107 −0.695537 0.718490i $$-0.744834\pi$$
−0.695537 + 0.718490i $$0.744834\pi$$
$$62$$ 8.11704 1.03086
$$63$$ −3.52311 −0.443871
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.38776 −0.293912
$$67$$ 1.03228 0.126113 0.0630563 0.998010i $$-0.479915\pi$$
0.0630563 + 0.998010i $$0.479915\pi$$
$$68$$ −3.62620 −0.439741
$$69$$ −22.1310 −2.66426
$$70$$ 0 0
$$71$$ −10.1816 −1.20833 −0.604166 0.796858i $$-0.706494\pi$$
−0.604166 + 0.796858i $$0.706494\pi$$
$$72$$ 4.62620 0.545203
$$73$$ 16.4017 1.91967 0.959837 0.280557i $$-0.0905191\pi$$
0.959837 + 0.280557i $$0.0905191\pi$$
$$74$$ 0.476886 0.0554368
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0.658473 0.0750400
$$78$$ 15.5371 1.75923
$$79$$ −12.5693 −1.41416 −0.707081 0.707133i $$-0.749988\pi$$
−0.707081 + 0.707133i $$0.749988\pi$$
$$80$$ 0 0
$$81$$ −1.47689 −0.164098
$$82$$ −2.65847 −0.293579
$$83$$ −0.270718 −0.0297152 −0.0148576 0.999890i $$-0.504729\pi$$
−0.0148576 + 0.999890i $$0.504729\pi$$
$$84$$ −2.10308 −0.229465
$$85$$ 0 0
$$86$$ 6.86464 0.740233
$$87$$ −20.3126 −2.17774
$$88$$ −0.864641 −0.0921710
$$89$$ 0.387755 0.0411020 0.0205510 0.999789i $$-0.493458\pi$$
0.0205510 + 0.999789i $$0.493458\pi$$
$$90$$ 0 0
$$91$$ −4.28467 −0.449156
$$92$$ −8.01395 −0.835512
$$93$$ 22.4157 2.32440
$$94$$ 1.25240 0.129175
$$95$$ 0 0
$$96$$ 2.76156 0.281850
$$97$$ −8.50479 −0.863531 −0.431765 0.901986i $$-0.642109\pi$$
−0.431765 + 0.901986i $$0.642109\pi$$
$$98$$ −6.42003 −0.648521
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 16.4157 1.63342 0.816710 0.577049i $$-0.195796\pi$$
0.816710 + 0.577049i $$0.195796\pi$$
$$102$$ −10.0140 −0.991529
$$103$$ −9.64015 −0.949872 −0.474936 0.880020i $$-0.657529\pi$$
−0.474936 + 0.880020i $$0.657529\pi$$
$$104$$ 5.62620 0.551694
$$105$$ 0 0
$$106$$ 2.37380 0.230564
$$107$$ 4.28467 0.414215 0.207107 0.978318i $$-0.433595\pi$$
0.207107 + 0.978318i $$0.433595\pi$$
$$108$$ 4.49084 0.432131
$$109$$ −13.4200 −1.28541 −0.642703 0.766116i $$-0.722187\pi$$
−0.642703 + 0.766116i $$0.722187\pi$$
$$110$$ 0 0
$$111$$ 1.31695 0.124999
$$112$$ −0.761557 −0.0719604
$$113$$ −10.3232 −0.971125 −0.485563 0.874202i $$-0.661385\pi$$
−0.485563 + 0.874202i $$0.661385\pi$$
$$114$$ 2.76156 0.258644
$$115$$ 0 0
$$116$$ −7.35548 −0.682939
$$117$$ 26.0279 2.40628
$$118$$ 4.49084 0.413415
$$119$$ 2.76156 0.253152
$$120$$ 0 0
$$121$$ −10.2524 −0.932036
$$122$$ −10.8646 −0.983638
$$123$$ −7.34153 −0.661963
$$124$$ 8.11704 0.728931
$$125$$ 0 0
$$126$$ −3.52311 −0.313864
$$127$$ 16.9817 1.50688 0.753440 0.657517i $$-0.228393\pi$$
0.753440 + 0.657517i $$0.228393\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 18.9571 1.66908
$$130$$ 0 0
$$131$$ 0.541436 0.0473055 0.0236528 0.999720i $$-0.492470\pi$$
0.0236528 + 0.999720i $$0.492470\pi$$
$$132$$ −2.38776 −0.207827
$$133$$ −0.761557 −0.0660354
$$134$$ 1.03228 0.0891750
$$135$$ 0 0
$$136$$ −3.62620 −0.310944
$$137$$ 2.87859 0.245935 0.122967 0.992411i $$-0.460759\pi$$
0.122967 + 0.992411i $$0.460759\pi$$
$$138$$ −22.1310 −1.88392
$$139$$ 3.58767 0.304302 0.152151 0.988357i $$-0.451380\pi$$
0.152151 + 0.988357i $$0.451380\pi$$
$$140$$ 0 0
$$141$$ 3.45856 0.291264
$$142$$ −10.1816 −0.854420
$$143$$ −4.86464 −0.406802
$$144$$ 4.62620 0.385517
$$145$$ 0 0
$$146$$ 16.4017 1.35742
$$147$$ −17.7293 −1.46229
$$148$$ 0.476886 0.0391998
$$149$$ −16.8401 −1.37959 −0.689796 0.724004i $$-0.742300\pi$$
−0.689796 + 0.724004i $$0.742300\pi$$
$$150$$ 0 0
$$151$$ −16.9817 −1.38195 −0.690975 0.722879i $$-0.742818\pi$$
−0.690975 + 0.722879i $$0.742818\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −16.7755 −1.35622
$$154$$ 0.658473 0.0530613
$$155$$ 0 0
$$156$$ 15.5371 1.24396
$$157$$ 14.8401 1.18437 0.592183 0.805804i $$-0.298266\pi$$
0.592183 + 0.805804i $$0.298266\pi$$
$$158$$ −12.5693 −0.999963
$$159$$ 6.55539 0.519876
$$160$$ 0 0
$$161$$ 6.10308 0.480990
$$162$$ −1.47689 −0.116035
$$163$$ 13.3694 1.04717 0.523587 0.851972i $$-0.324593\pi$$
0.523587 + 0.851972i $$0.324593\pi$$
$$164$$ −2.65847 −0.207592
$$165$$ 0 0
$$166$$ −0.270718 −0.0210118
$$167$$ 9.84632 0.761931 0.380966 0.924589i $$-0.375592\pi$$
0.380966 + 0.924589i $$0.375592\pi$$
$$168$$ −2.10308 −0.162256
$$169$$ 18.6541 1.43493
$$170$$ 0 0
$$171$$ 4.62620 0.353774
$$172$$ 6.86464 0.523424
$$173$$ 2.98168 0.226693 0.113346 0.993556i $$-0.463843\pi$$
0.113346 + 0.993556i $$0.463843\pi$$
$$174$$ −20.3126 −1.53989
$$175$$ 0 0
$$176$$ −0.864641 −0.0651748
$$177$$ 12.4017 0.932169
$$178$$ 0.387755 0.0290635
$$179$$ 11.7938 0.881512 0.440756 0.897627i $$-0.354710\pi$$
0.440756 + 0.897627i $$0.354710\pi$$
$$180$$ 0 0
$$181$$ 14.5693 1.08293 0.541465 0.840723i $$-0.317870\pi$$
0.541465 + 0.840723i $$0.317870\pi$$
$$182$$ −4.28467 −0.317601
$$183$$ −30.0033 −2.21791
$$184$$ −8.01395 −0.590796
$$185$$ 0 0
$$186$$ 22.4157 1.64360
$$187$$ 3.13536 0.229280
$$188$$ 1.25240 0.0913404
$$189$$ −3.42003 −0.248771
$$190$$ 0 0
$$191$$ 13.2384 0.957900 0.478950 0.877842i $$-0.341017\pi$$
0.478950 + 0.877842i $$0.341017\pi$$
$$192$$ 2.76156 0.199298
$$193$$ 2.54144 0.182937 0.0914683 0.995808i $$-0.470844\pi$$
0.0914683 + 0.995808i $$0.470844\pi$$
$$194$$ −8.50479 −0.610609
$$195$$ 0 0
$$196$$ −6.42003 −0.458574
$$197$$ −19.9109 −1.41859 −0.709295 0.704911i $$-0.750987\pi$$
−0.709295 + 0.704911i $$0.750987\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −20.3126 −1.43992 −0.719960 0.694015i $$-0.755840\pi$$
−0.719960 + 0.694015i $$0.755840\pi$$
$$200$$ 0 0
$$201$$ 2.85069 0.201072
$$202$$ 16.4157 1.15500
$$203$$ 5.60162 0.393157
$$204$$ −10.0140 −0.701117
$$205$$ 0 0
$$206$$ −9.64015 −0.671661
$$207$$ −37.0741 −2.57683
$$208$$ 5.62620 0.390107
$$209$$ −0.864641 −0.0598085
$$210$$ 0 0
$$211$$ 18.0419 1.24205 0.621026 0.783790i $$-0.286716\pi$$
0.621026 + 0.783790i $$0.286716\pi$$
$$212$$ 2.37380 0.163033
$$213$$ −28.1170 −1.92655
$$214$$ 4.28467 0.292894
$$215$$ 0 0
$$216$$ 4.49084 0.305563
$$217$$ −6.18159 −0.419634
$$218$$ −13.4200 −0.908919
$$219$$ 45.2943 3.06070
$$220$$ 0 0
$$221$$ −20.4017 −1.37237
$$222$$ 1.31695 0.0883877
$$223$$ −13.5231 −0.905575 −0.452787 0.891619i $$-0.649570\pi$$
−0.452787 + 0.891619i $$0.649570\pi$$
$$224$$ −0.761557 −0.0508837
$$225$$ 0 0
$$226$$ −10.3232 −0.686689
$$227$$ 13.6016 0.902771 0.451386 0.892329i $$-0.350930\pi$$
0.451386 + 0.892329i $$0.350930\pi$$
$$228$$ 2.76156 0.182889
$$229$$ 13.5877 0.897898 0.448949 0.893557i $$-0.351798\pi$$
0.448949 + 0.893557i $$0.351798\pi$$
$$230$$ 0 0
$$231$$ 1.81841 0.119643
$$232$$ −7.35548 −0.482911
$$233$$ 25.5510 1.67390 0.836952 0.547277i $$-0.184336\pi$$
0.836952 + 0.547277i $$0.184336\pi$$
$$234$$ 26.0279 1.70150
$$235$$ 0 0
$$236$$ 4.49084 0.292329
$$237$$ −34.7110 −2.25472
$$238$$ 2.76156 0.179005
$$239$$ −11.3309 −0.732935 −0.366468 0.930431i $$-0.619433\pi$$
−0.366468 + 0.930431i $$0.619433\pi$$
$$240$$ 0 0
$$241$$ −1.25240 −0.0806739 −0.0403370 0.999186i $$-0.512843\pi$$
−0.0403370 + 0.999186i $$0.512843\pi$$
$$242$$ −10.2524 −0.659049
$$243$$ −17.5510 −1.12590
$$244$$ −10.8646 −0.695537
$$245$$ 0 0
$$246$$ −7.34153 −0.468079
$$247$$ 5.62620 0.357986
$$248$$ 8.11704 0.515432
$$249$$ −0.747604 −0.0473775
$$250$$ 0 0
$$251$$ 10.5939 0.668682 0.334341 0.942452i $$-0.391486\pi$$
0.334341 + 0.942452i $$0.391486\pi$$
$$252$$ −3.52311 −0.221935
$$253$$ 6.92919 0.435635
$$254$$ 16.9817 1.06553
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −0.153681 −0.00958637 −0.00479319 0.999989i $$-0.501526\pi$$
−0.00479319 + 0.999989i $$0.501526\pi$$
$$258$$ 18.9571 1.18022
$$259$$ −0.363176 −0.0225666
$$260$$ 0 0
$$261$$ −34.0279 −2.10627
$$262$$ 0.541436 0.0334501
$$263$$ 0.504792 0.0311268 0.0155634 0.999879i $$-0.495046\pi$$
0.0155634 + 0.999879i $$0.495046\pi$$
$$264$$ −2.38776 −0.146956
$$265$$ 0 0
$$266$$ −0.761557 −0.0466941
$$267$$ 1.07081 0.0655324
$$268$$ 1.03228 0.0630563
$$269$$ −3.49521 −0.213107 −0.106553 0.994307i $$-0.533981\pi$$
−0.106553 + 0.994307i $$0.533981\pi$$
$$270$$ 0 0
$$271$$ 5.47252 0.332432 0.166216 0.986089i $$-0.446845\pi$$
0.166216 + 0.986089i $$0.446845\pi$$
$$272$$ −3.62620 −0.219871
$$273$$ −11.8324 −0.716127
$$274$$ 2.87859 0.173902
$$275$$ 0 0
$$276$$ −22.1310 −1.33213
$$277$$ 12.9538 0.778317 0.389158 0.921171i $$-0.372766\pi$$
0.389158 + 0.921171i $$0.372766\pi$$
$$278$$ 3.58767 0.215174
$$279$$ 37.5510 2.24812
$$280$$ 0 0
$$281$$ 0.153681 0.00916785 0.00458393 0.999989i $$-0.498541\pi$$
0.00458393 + 0.999989i $$0.498541\pi$$
$$282$$ 3.45856 0.205954
$$283$$ −18.2341 −1.08390 −0.541952 0.840410i $$-0.682314\pi$$
−0.541952 + 0.840410i $$0.682314\pi$$
$$284$$ −10.1816 −0.604166
$$285$$ 0 0
$$286$$ −4.86464 −0.287652
$$287$$ 2.02458 0.119507
$$288$$ 4.62620 0.272601
$$289$$ −3.85069 −0.226511
$$290$$ 0 0
$$291$$ −23.4865 −1.37680
$$292$$ 16.4017 0.959837
$$293$$ 2.03853 0.119092 0.0595462 0.998226i $$-0.481035\pi$$
0.0595462 + 0.998226i $$0.481035\pi$$
$$294$$ −17.7293 −1.03399
$$295$$ 0 0
$$296$$ 0.476886 0.0277184
$$297$$ −3.88296 −0.225312
$$298$$ −16.8401 −0.975519
$$299$$ −45.0881 −2.60751
$$300$$ 0 0
$$301$$ −5.22782 −0.301326
$$302$$ −16.9817 −0.977186
$$303$$ 45.3328 2.60430
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ −16.7755 −0.958992
$$307$$ 16.5414 0.944070 0.472035 0.881580i $$-0.343520\pi$$
0.472035 + 0.881580i $$0.343520\pi$$
$$308$$ 0.658473 0.0375200
$$309$$ −26.6218 −1.51446
$$310$$ 0 0
$$311$$ −21.4725 −1.21759 −0.608797 0.793326i $$-0.708348\pi$$
−0.608797 + 0.793326i $$0.708348\pi$$
$$312$$ 15.5371 0.879613
$$313$$ −1.12141 −0.0633856 −0.0316928 0.999498i $$-0.510090\pi$$
−0.0316928 + 0.999498i $$0.510090\pi$$
$$314$$ 14.8401 0.837473
$$315$$ 0 0
$$316$$ −12.5693 −0.707081
$$317$$ −29.8882 −1.67869 −0.839344 0.543601i $$-0.817060\pi$$
−0.839344 + 0.543601i $$0.817060\pi$$
$$318$$ 6.55539 0.367608
$$319$$ 6.35985 0.356083
$$320$$ 0 0
$$321$$ 11.8324 0.660418
$$322$$ 6.10308 0.340112
$$323$$ −3.62620 −0.201767
$$324$$ −1.47689 −0.0820492
$$325$$ 0 0
$$326$$ 13.3694 0.740464
$$327$$ −37.0602 −2.04943
$$328$$ −2.65847 −0.146790
$$329$$ −0.953771 −0.0525831
$$330$$ 0 0
$$331$$ 32.3126 1.77606 0.888030 0.459786i $$-0.152074\pi$$
0.888030 + 0.459786i $$0.152074\pi$$
$$332$$ −0.270718 −0.0148576
$$333$$ 2.20617 0.120897
$$334$$ 9.84632 0.538767
$$335$$ 0 0
$$336$$ −2.10308 −0.114733
$$337$$ 26.3511 1.43544 0.717718 0.696334i $$-0.245187\pi$$
0.717718 + 0.696334i $$0.245187\pi$$
$$338$$ 18.6541 1.01465
$$339$$ −28.5081 −1.54835
$$340$$ 0 0
$$341$$ −7.01832 −0.380063
$$342$$ 4.62620 0.250156
$$343$$ 10.2201 0.551835
$$344$$ 6.86464 0.370117
$$345$$ 0 0
$$346$$ 2.98168 0.160296
$$347$$ −2.77551 −0.148997 −0.0744986 0.997221i $$-0.523736\pi$$
−0.0744986 + 0.997221i $$0.523736\pi$$
$$348$$ −20.3126 −1.08887
$$349$$ 11.5510 0.618312 0.309156 0.951011i $$-0.399953\pi$$
0.309156 + 0.951011i $$0.399953\pi$$
$$350$$ 0 0
$$351$$ 25.2663 1.34862
$$352$$ −0.864641 −0.0460855
$$353$$ −8.40171 −0.447178 −0.223589 0.974684i $$-0.571777\pi$$
−0.223589 + 0.974684i $$0.571777\pi$$
$$354$$ 12.4017 0.659143
$$355$$ 0 0
$$356$$ 0.387755 0.0205510
$$357$$ 7.62620 0.403621
$$358$$ 11.7938 0.623323
$$359$$ 22.7895 1.20278 0.601391 0.798955i $$-0.294613\pi$$
0.601391 + 0.798955i $$0.294613\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.5693 0.765748
$$363$$ −28.3126 −1.48602
$$364$$ −4.28467 −0.224578
$$365$$ 0 0
$$366$$ −30.0033 −1.56830
$$367$$ 4.06455 0.212168 0.106084 0.994357i $$-0.466169\pi$$
0.106084 + 0.994357i $$0.466169\pi$$
$$368$$ −8.01395 −0.417756
$$369$$ −12.2986 −0.640241
$$370$$ 0 0
$$371$$ −1.80779 −0.0938556
$$372$$ 22.4157 1.16220
$$373$$ −18.4017 −0.952804 −0.476402 0.879227i $$-0.658059\pi$$
−0.476402 + 0.879227i $$0.658059\pi$$
$$374$$ 3.13536 0.162126
$$375$$ 0 0
$$376$$ 1.25240 0.0645874
$$377$$ −41.3834 −2.13135
$$378$$ −3.42003 −0.175907
$$379$$ 1.23844 0.0636145 0.0318073 0.999494i $$-0.489874\pi$$
0.0318073 + 0.999494i $$0.489874\pi$$
$$380$$ 0 0
$$381$$ 46.8959 2.40255
$$382$$ 13.2384 0.677338
$$383$$ 16.8646 0.861743 0.430871 0.902413i $$-0.358206\pi$$
0.430871 + 0.902413i $$0.358206\pi$$
$$384$$ 2.76156 0.140925
$$385$$ 0 0
$$386$$ 2.54144 0.129356
$$387$$ 31.7572 1.61431
$$388$$ −8.50479 −0.431765
$$389$$ 8.59392 0.435729 0.217865 0.975979i $$-0.430091\pi$$
0.217865 + 0.975979i $$0.430091\pi$$
$$390$$ 0 0
$$391$$ 29.0602 1.46964
$$392$$ −6.42003 −0.324261
$$393$$ 1.49521 0.0754233
$$394$$ −19.9109 −1.00310
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 16.0558 0.805818 0.402909 0.915240i $$-0.367999\pi$$
0.402909 + 0.915240i $$0.367999\pi$$
$$398$$ −20.3126 −1.01818
$$399$$ −2.10308 −0.105286
$$400$$ 0 0
$$401$$ −14.8925 −0.743698 −0.371849 0.928293i $$-0.621276\pi$$
−0.371849 + 0.928293i $$0.621276\pi$$
$$402$$ 2.85069 0.142179
$$403$$ 45.6681 2.27489
$$404$$ 16.4157 0.816710
$$405$$ 0 0
$$406$$ 5.60162 0.278004
$$407$$ −0.412335 −0.0204387
$$408$$ −10.0140 −0.495765
$$409$$ −18.3511 −0.907404 −0.453702 0.891153i $$-0.649897\pi$$
−0.453702 + 0.891153i $$0.649897\pi$$
$$410$$ 0 0
$$411$$ 7.94940 0.392115
$$412$$ −9.64015 −0.474936
$$413$$ −3.42003 −0.168289
$$414$$ −37.0741 −1.82209
$$415$$ 0 0
$$416$$ 5.62620 0.275847
$$417$$ 9.90754 0.485174
$$418$$ −0.864641 −0.0422910
$$419$$ 34.7509 1.69769 0.848847 0.528639i $$-0.177297\pi$$
0.848847 + 0.528639i $$0.177297\pi$$
$$420$$ 0 0
$$421$$ −40.1589 −1.95722 −0.978612 0.205713i $$-0.934049\pi$$
−0.978612 + 0.205713i $$0.934049\pi$$
$$422$$ 18.0419 0.878264
$$423$$ 5.79383 0.281706
$$424$$ 2.37380 0.115282
$$425$$ 0 0
$$426$$ −28.1170 −1.36227
$$427$$ 8.27405 0.400409
$$428$$ 4.28467 0.207107
$$429$$ −13.4340 −0.648599
$$430$$ 0 0
$$431$$ 34.9571 1.68382 0.841912 0.539615i $$-0.181430\pi$$
0.841912 + 0.539615i $$0.181430\pi$$
$$432$$ 4.49084 0.216066
$$433$$ 1.13536 0.0545619 0.0272809 0.999628i $$-0.491315\pi$$
0.0272809 + 0.999628i $$0.491315\pi$$
$$434$$ −6.18159 −0.296726
$$435$$ 0 0
$$436$$ −13.4200 −0.642703
$$437$$ −8.01395 −0.383359
$$438$$ 45.2943 2.16424
$$439$$ −6.80009 −0.324551 −0.162275 0.986746i $$-0.551883\pi$$
−0.162275 + 0.986746i $$0.551883\pi$$
$$440$$ 0 0
$$441$$ −29.7003 −1.41430
$$442$$ −20.4017 −0.970410
$$443$$ 38.0679 1.80866 0.904330 0.426835i $$-0.140371\pi$$
0.904330 + 0.426835i $$0.140371\pi$$
$$444$$ 1.31695 0.0624995
$$445$$ 0 0
$$446$$ −13.5231 −0.640338
$$447$$ −46.5048 −2.19960
$$448$$ −0.761557 −0.0359802
$$449$$ −18.5414 −0.875024 −0.437512 0.899212i $$-0.644140\pi$$
−0.437512 + 0.899212i $$0.644140\pi$$
$$450$$ 0 0
$$451$$ 2.29862 0.108238
$$452$$ −10.3232 −0.485563
$$453$$ −46.8959 −2.20336
$$454$$ 13.6016 0.638356
$$455$$ 0 0
$$456$$ 2.76156 0.129322
$$457$$ 16.3738 0.765934 0.382967 0.923762i $$-0.374902\pi$$
0.382967 + 0.923762i $$0.374902\pi$$
$$458$$ 13.5877 0.634910
$$459$$ −16.2847 −0.760103
$$460$$ 0 0
$$461$$ 1.70470 0.0793959 0.0396979 0.999212i $$-0.487360\pi$$
0.0396979 + 0.999212i $$0.487360\pi$$
$$462$$ 1.81841 0.0846002
$$463$$ −10.0279 −0.466036 −0.233018 0.972472i $$-0.574860\pi$$
−0.233018 + 0.972472i $$0.574860\pi$$
$$464$$ −7.35548 −0.341470
$$465$$ 0 0
$$466$$ 25.5510 1.18363
$$467$$ 32.7509 1.51553 0.757766 0.652526i $$-0.226291\pi$$
0.757766 + 0.652526i $$0.226291\pi$$
$$468$$ 26.0279 1.20314
$$469$$ −0.786137 −0.0363004
$$470$$ 0 0
$$471$$ 40.9817 1.88834
$$472$$ 4.49084 0.206708
$$473$$ −5.93545 −0.272912
$$474$$ −34.7110 −1.59433
$$475$$ 0 0
$$476$$ 2.76156 0.126576
$$477$$ 10.9817 0.502816
$$478$$ −11.3309 −0.518263
$$479$$ 27.2803 1.24647 0.623234 0.782035i $$-0.285818\pi$$
0.623234 + 0.782035i $$0.285818\pi$$
$$480$$ 0 0
$$481$$ 2.68305 0.122337
$$482$$ −1.25240 −0.0570451
$$483$$ 16.8540 0.766884
$$484$$ −10.2524 −0.466018
$$485$$ 0 0
$$486$$ −17.5510 −0.796130
$$487$$ −11.0741 −0.501817 −0.250908 0.968011i $$-0.580729\pi$$
−0.250908 + 0.968011i $$0.580729\pi$$
$$488$$ −10.8646 −0.491819
$$489$$ 36.9205 1.66960
$$490$$ 0 0
$$491$$ 8.11704 0.366317 0.183158 0.983083i $$-0.441368\pi$$
0.183158 + 0.983083i $$0.441368\pi$$
$$492$$ −7.34153 −0.330982
$$493$$ 26.6724 1.20127
$$494$$ 5.62620 0.253135
$$495$$ 0 0
$$496$$ 8.11704 0.364466
$$497$$ 7.75386 0.347808
$$498$$ −0.747604 −0.0335009
$$499$$ 0.295298 0.0132193 0.00660967 0.999978i $$-0.497896\pi$$
0.00660967 + 0.999978i $$0.497896\pi$$
$$500$$ 0 0
$$501$$ 27.1912 1.21481
$$502$$ 10.5939 0.472830
$$503$$ 19.6016 0.873993 0.436996 0.899463i $$-0.356042\pi$$
0.436996 + 0.899463i $$0.356042\pi$$
$$504$$ −3.52311 −0.156932
$$505$$ 0 0
$$506$$ 6.92919 0.308040
$$507$$ 51.5144 2.28783
$$508$$ 16.9817 0.753440
$$509$$ −1.79383 −0.0795102 −0.0397551 0.999209i $$-0.512658\pi$$
−0.0397551 + 0.999209i $$0.512658\pi$$
$$510$$ 0 0
$$511$$ −12.4908 −0.552562
$$512$$ 1.00000 0.0441942
$$513$$ 4.49084 0.198275
$$514$$ −0.153681 −0.00677859
$$515$$ 0 0
$$516$$ 18.9571 0.834540
$$517$$ −1.08287 −0.0476247
$$518$$ −0.363176 −0.0159570
$$519$$ 8.23407 0.361436
$$520$$ 0 0
$$521$$ −2.61850 −0.114719 −0.0573593 0.998354i $$-0.518268\pi$$
−0.0573593 + 0.998354i $$0.518268\pi$$
$$522$$ −34.0279 −1.48936
$$523$$ 14.9956 0.655713 0.327857 0.944728i $$-0.393674\pi$$
0.327857 + 0.944728i $$0.393674\pi$$
$$524$$ 0.541436 0.0236528
$$525$$ 0 0
$$526$$ 0.504792 0.0220100
$$527$$ −29.4340 −1.28216
$$528$$ −2.38776 −0.103914
$$529$$ 41.2234 1.79232
$$530$$ 0 0
$$531$$ 20.7755 0.901580
$$532$$ −0.761557 −0.0330177
$$533$$ −14.9571 −0.647864
$$534$$ 1.07081 0.0463384
$$535$$ 0 0
$$536$$ 1.03228 0.0445875
$$537$$ 32.5693 1.40547
$$538$$ −3.49521 −0.150689
$$539$$ 5.55102 0.239099
$$540$$ 0 0
$$541$$ 3.40608 0.146439 0.0732194 0.997316i $$-0.476673\pi$$
0.0732194 + 0.997316i $$0.476673\pi$$
$$542$$ 5.47252 0.235065
$$543$$ 40.2341 1.72661
$$544$$ −3.62620 −0.155472
$$545$$ 0 0
$$546$$ −11.8324 −0.506378
$$547$$ 4.74760 0.202993 0.101496 0.994836i $$-0.467637\pi$$
0.101496 + 0.994836i $$0.467637\pi$$
$$548$$ 2.87859 0.122967
$$549$$ −50.2620 −2.14513
$$550$$ 0 0
$$551$$ −7.35548 −0.313354
$$552$$ −22.1310 −0.941958
$$553$$ 9.57227 0.407054
$$554$$ 12.9538 0.550353
$$555$$ 0 0
$$556$$ 3.58767 0.152151
$$557$$ 43.0462 1.82393 0.911964 0.410271i $$-0.134566\pi$$
0.911964 + 0.410271i $$0.134566\pi$$
$$558$$ 37.5510 1.58966
$$559$$ 38.6218 1.63353
$$560$$ 0 0
$$561$$ 8.65847 0.365561
$$562$$ 0.153681 0.00648265
$$563$$ 17.0096 0.716869 0.358434 0.933555i $$-0.383311\pi$$
0.358434 + 0.933555i $$0.383311\pi$$
$$564$$ 3.45856 0.145632
$$565$$ 0 0
$$566$$ −18.2341 −0.766435
$$567$$ 1.12473 0.0472343
$$568$$ −10.1816 −0.427210
$$569$$ −19.7572 −0.828264 −0.414132 0.910217i $$-0.635915\pi$$
−0.414132 + 0.910217i $$0.635915\pi$$
$$570$$ 0 0
$$571$$ −11.3973 −0.476964 −0.238482 0.971147i $$-0.576650\pi$$
−0.238482 + 0.971147i $$0.576650\pi$$
$$572$$ −4.86464 −0.203401
$$573$$ 36.5587 1.52726
$$574$$ 2.02458 0.0845043
$$575$$ 0 0
$$576$$ 4.62620 0.192758
$$577$$ −18.3372 −0.763386 −0.381693 0.924289i $$-0.624659\pi$$
−0.381693 + 0.924289i $$0.624659\pi$$
$$578$$ −3.85069 −0.160167
$$579$$ 7.01832 0.291672
$$580$$ 0 0
$$581$$ 0.206167 0.00855327
$$582$$ −23.4865 −0.973546
$$583$$ −2.05249 −0.0850053
$$584$$ 16.4017 0.678708
$$585$$ 0 0
$$586$$ 2.03853 0.0842110
$$587$$ −11.9475 −0.493127 −0.246563 0.969127i $$-0.579301\pi$$
−0.246563 + 0.969127i $$0.579301\pi$$
$$588$$ −17.7293 −0.731143
$$589$$ 8.11704 0.334457
$$590$$ 0 0
$$591$$ −54.9850 −2.26178
$$592$$ 0.476886 0.0195999
$$593$$ −24.3911 −1.00162 −0.500811 0.865557i $$-0.666965\pi$$
−0.500811 + 0.865557i $$0.666965\pi$$
$$594$$ −3.88296 −0.159320
$$595$$ 0 0
$$596$$ −16.8401 −0.689796
$$597$$ −56.0943 −2.29579
$$598$$ −45.0881 −1.84379
$$599$$ −21.0708 −0.860930 −0.430465 0.902607i $$-0.641650\pi$$
−0.430465 + 0.902607i $$0.641650\pi$$
$$600$$ 0 0
$$601$$ 32.3878 1.32112 0.660562 0.750771i $$-0.270318\pi$$
0.660562 + 0.750771i $$0.270318\pi$$
$$602$$ −5.22782 −0.213070
$$603$$ 4.77551 0.194474
$$604$$ −16.9817 −0.690975
$$605$$ 0 0
$$606$$ 45.3328 1.84152
$$607$$ 19.0183 0.771930 0.385965 0.922513i $$-0.373869\pi$$
0.385965 + 0.922513i $$0.373869\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 15.4692 0.626843
$$610$$ 0 0
$$611$$ 7.04623 0.285060
$$612$$ −16.7755 −0.678110
$$613$$ −23.5756 −0.952210 −0.476105 0.879389i $$-0.657952\pi$$
−0.476105 + 0.879389i $$0.657952\pi$$
$$614$$ 16.5414 0.667558
$$615$$ 0 0
$$616$$ 0.658473 0.0265307
$$617$$ −26.2707 −1.05762 −0.528810 0.848740i $$-0.677361\pi$$
−0.528810 + 0.848740i $$0.677361\pi$$
$$618$$ −26.6218 −1.07089
$$619$$ 11.4985 0.462165 0.231083 0.972934i $$-0.425773\pi$$
0.231083 + 0.972934i $$0.425773\pi$$
$$620$$ 0 0
$$621$$ −35.9894 −1.44420
$$622$$ −21.4725 −0.860969
$$623$$ −0.295298 −0.0118309
$$624$$ 15.5371 0.621980
$$625$$ 0 0
$$626$$ −1.12141 −0.0448204
$$627$$ −2.38776 −0.0953578
$$628$$ 14.8401 0.592183
$$629$$ −1.72928 −0.0689510
$$630$$ 0 0
$$631$$ −45.8130 −1.82379 −0.911893 0.410427i $$-0.865380\pi$$
−0.911893 + 0.410427i $$0.865380\pi$$
$$632$$ −12.5693 −0.499982
$$633$$ 49.8236 1.98031
$$634$$ −29.8882 −1.18701
$$635$$ 0 0
$$636$$ 6.55539 0.259938
$$637$$ −36.1204 −1.43114
$$638$$ 6.35985 0.251789
$$639$$ −47.1020 −1.86333
$$640$$ 0 0
$$641$$ 1.36943 0.0540894 0.0270447 0.999634i $$-0.491390\pi$$
0.0270447 + 0.999634i $$0.491390\pi$$
$$642$$ 11.8324 0.466986
$$643$$ −24.7389 −0.975606 −0.487803 0.872954i $$-0.662201\pi$$
−0.487803 + 0.872954i $$0.662201\pi$$
$$644$$ 6.10308 0.240495
$$645$$ 0 0
$$646$$ −3.62620 −0.142671
$$647$$ −6.82611 −0.268362 −0.134181 0.990957i $$-0.542840\pi$$
−0.134181 + 0.990957i $$0.542840\pi$$
$$648$$ −1.47689 −0.0580175
$$649$$ −3.88296 −0.152420
$$650$$ 0 0
$$651$$ −17.0708 −0.669058
$$652$$ 13.3694 0.523587
$$653$$ 6.91713 0.270688 0.135344 0.990799i $$-0.456786\pi$$
0.135344 + 0.990799i $$0.456786\pi$$
$$654$$ −37.0602 −1.44917
$$655$$ 0 0
$$656$$ −2.65847 −0.103796
$$657$$ 75.8776 2.96027
$$658$$ −0.953771 −0.0371819
$$659$$ 9.44461 0.367910 0.183955 0.982935i $$-0.441110\pi$$
0.183955 + 0.982935i $$0.441110\pi$$
$$660$$ 0 0
$$661$$ −22.1955 −0.863306 −0.431653 0.902040i $$-0.642070\pi$$
−0.431653 + 0.902040i $$0.642070\pi$$
$$662$$ 32.3126 1.25586
$$663$$ −56.3405 −2.18808
$$664$$ −0.270718 −0.0105059
$$665$$ 0 0
$$666$$ 2.20617 0.0854873
$$667$$ 58.9465 2.28242
$$668$$ 9.84632 0.380966
$$669$$ −37.3449 −1.44384
$$670$$ 0 0
$$671$$ 9.39401 0.362652
$$672$$ −2.10308 −0.0811282
$$673$$ 12.2986 0.474077 0.237039 0.971500i $$-0.423823\pi$$
0.237039 + 0.971500i $$0.423823\pi$$
$$674$$ 26.3511 1.01501
$$675$$ 0 0
$$676$$ 18.6541 0.717466
$$677$$ −35.9527 −1.38178 −0.690888 0.722962i $$-0.742780\pi$$
−0.690888 + 0.722962i $$0.742780\pi$$
$$678$$ −28.5081 −1.09485
$$679$$ 6.47689 0.248560
$$680$$ 0 0
$$681$$ 37.5616 1.43937
$$682$$ −7.01832 −0.268745
$$683$$ −9.00958 −0.344742 −0.172371 0.985032i $$-0.555143\pi$$
−0.172371 + 0.985032i $$0.555143\pi$$
$$684$$ 4.62620 0.176887
$$685$$ 0 0
$$686$$ 10.2201 0.390206
$$687$$ 37.5231 1.43160
$$688$$ 6.86464 0.261712
$$689$$ 13.3555 0.508803
$$690$$ 0 0
$$691$$ −9.11078 −0.346590 −0.173295 0.984870i $$-0.555441\pi$$
−0.173295 + 0.984870i $$0.555441\pi$$
$$692$$ 2.98168 0.113346
$$693$$ 3.04623 0.115717
$$694$$ −2.77551 −0.105357
$$695$$ 0 0
$$696$$ −20.3126 −0.769946
$$697$$ 9.64015 0.365147
$$698$$ 11.5510 0.437213
$$699$$ 70.5606 2.66885
$$700$$ 0 0
$$701$$ −14.7476 −0.557009 −0.278505 0.960435i $$-0.589839\pi$$
−0.278505 + 0.960435i $$0.589839\pi$$
$$702$$ 25.2663 0.953617
$$703$$ 0.476886 0.0179861
$$704$$ −0.864641 −0.0325874
$$705$$ 0 0
$$706$$ −8.40171 −0.316202
$$707$$ −12.5015 −0.470166
$$708$$ 12.4017 0.466085
$$709$$ −8.63389 −0.324253 −0.162126 0.986770i $$-0.551835\pi$$
−0.162126 + 0.986770i $$0.551835\pi$$
$$710$$ 0 0
$$711$$ −58.1483 −2.18073
$$712$$ 0.387755 0.0145317
$$713$$ −65.0496 −2.43613
$$714$$ 7.62620 0.285403
$$715$$ 0 0
$$716$$ 11.7938 0.440756
$$717$$ −31.2909 −1.16858
$$718$$ 22.7895 0.850495
$$719$$ −38.2759 −1.42745 −0.713726 0.700425i $$-0.752994\pi$$
−0.713726 + 0.700425i $$0.752994\pi$$
$$720$$ 0 0
$$721$$ 7.34153 0.273413
$$722$$ 1.00000 0.0372161
$$723$$ −3.45856 −0.128625
$$724$$ 14.5693 0.541465
$$725$$ 0 0
$$726$$ −28.3126 −1.05078
$$727$$ −31.1893 −1.15675 −0.578373 0.815772i $$-0.696312\pi$$
−0.578373 + 0.815772i $$0.696312\pi$$
$$728$$ −4.28467 −0.158800
$$729$$ −44.0375 −1.63102
$$730$$ 0 0
$$731$$ −24.8925 −0.920684
$$732$$ −30.0033 −1.10895
$$733$$ −13.9634 −0.515748 −0.257874 0.966179i $$-0.583022\pi$$
−0.257874 + 0.966179i $$0.583022\pi$$
$$734$$ 4.06455 0.150025
$$735$$ 0 0
$$736$$ −8.01395 −0.295398
$$737$$ −0.892548 −0.0328774
$$738$$ −12.2986 −0.452719
$$739$$ 9.02165 0.331867 0.165933 0.986137i $$-0.446936\pi$$
0.165933 + 0.986137i $$0.446936\pi$$
$$740$$ 0 0
$$741$$ 15.5371 0.570768
$$742$$ −1.80779 −0.0663659
$$743$$ −15.0342 −0.551550 −0.275775 0.961222i $$-0.588934\pi$$
−0.275775 + 0.961222i $$0.588934\pi$$
$$744$$ 22.4157 0.821798
$$745$$ 0 0
$$746$$ −18.4017 −0.673734
$$747$$ −1.25240 −0.0458228
$$748$$ 3.13536 0.114640
$$749$$ −3.26302 −0.119228
$$750$$ 0 0
$$751$$ 29.6681 1.08260 0.541301 0.840829i $$-0.317932\pi$$
0.541301 + 0.840829i $$0.317932\pi$$
$$752$$ 1.25240 0.0456702
$$753$$ 29.2557 1.06614
$$754$$ −41.3834 −1.50709
$$755$$ 0 0
$$756$$ −3.42003 −0.124385
$$757$$ 10.5819 0.384604 0.192302 0.981336i $$-0.438405\pi$$
0.192302 + 0.981336i $$0.438405\pi$$
$$758$$ 1.23844 0.0449823
$$759$$ 19.1354 0.694570
$$760$$ 0 0
$$761$$ −0.979789 −0.0355173 −0.0177587 0.999842i $$-0.505653\pi$$
−0.0177587 + 0.999842i $$0.505653\pi$$
$$762$$ 46.8959 1.69886
$$763$$ 10.2201 0.369993
$$764$$ 13.2384 0.478950
$$765$$ 0 0
$$766$$ 16.8646 0.609344
$$767$$ 25.2663 0.912315
$$768$$ 2.76156 0.0996491
$$769$$ 43.1772 1.55701 0.778505 0.627638i $$-0.215978\pi$$
0.778505 + 0.627638i $$0.215978\pi$$
$$770$$ 0 0
$$771$$ −0.424399 −0.0152844
$$772$$ 2.54144 0.0914683
$$773$$ 37.5250 1.34968 0.674840 0.737964i $$-0.264213\pi$$
0.674840 + 0.737964i $$0.264213\pi$$
$$774$$ 31.7572 1.14149
$$775$$ 0 0
$$776$$ −8.50479 −0.305304
$$777$$ −1.00293 −0.0359799
$$778$$ 8.59392 0.308107
$$779$$ −2.65847 −0.0952497
$$780$$ 0 0
$$781$$ 8.80342 0.315011
$$782$$ 29.0602 1.03919
$$783$$ −33.0323 −1.18048
$$784$$ −6.42003 −0.229287
$$785$$ 0 0
$$786$$ 1.49521 0.0533323
$$787$$ 22.5833 0.805008 0.402504 0.915418i $$-0.368140\pi$$
0.402504 + 0.915418i $$0.368140\pi$$
$$788$$ −19.9109 −0.709295
$$789$$ 1.39401 0.0496282
$$790$$ 0 0
$$791$$ 7.86171 0.279530
$$792$$ −4.00000 −0.142134
$$793$$ −61.1266 −2.17067
$$794$$ 16.0558 0.569799
$$795$$ 0 0
$$796$$ −20.3126 −0.719960
$$797$$ 35.9806 1.27450 0.637250 0.770657i $$-0.280072\pi$$
0.637250 + 0.770657i $$0.280072\pi$$
$$798$$ −2.10308 −0.0744484
$$799$$ −4.54144 −0.160664
$$800$$ 0 0
$$801$$ 1.79383 0.0633820
$$802$$ −14.8925 −0.525874
$$803$$ −14.1816 −0.500457
$$804$$ 2.85069 0.100536
$$805$$ 0 0
$$806$$ 45.6681 1.60859
$$807$$ −9.65222 −0.339774
$$808$$ 16.4157 0.577501
$$809$$ −0.955660 −0.0335992 −0.0167996 0.999859i $$-0.505348\pi$$
−0.0167996 + 0.999859i $$0.505348\pi$$
$$810$$ 0 0
$$811$$ −7.53707 −0.264662 −0.132331 0.991206i $$-0.542246\pi$$
−0.132331 + 0.991206i $$0.542246\pi$$
$$812$$ 5.60162 0.196578
$$813$$ 15.1127 0.530024
$$814$$ −0.412335 −0.0144523
$$815$$ 0 0
$$816$$ −10.0140 −0.350558
$$817$$ 6.86464 0.240163
$$818$$ −18.3511 −0.641632
$$819$$ −19.8217 −0.692628
$$820$$ 0 0
$$821$$ 13.3082 0.464460 0.232230 0.972661i $$-0.425398\pi$$
0.232230 + 0.972661i $$0.425398\pi$$
$$822$$ 7.94940 0.277267
$$823$$ 24.4050 0.850706 0.425353 0.905028i $$-0.360150\pi$$
0.425353 + 0.905028i $$0.360150\pi$$
$$824$$ −9.64015 −0.335831
$$825$$ 0 0
$$826$$ −3.42003 −0.118998
$$827$$ −11.6874 −0.406411 −0.203206 0.979136i $$-0.565136\pi$$
−0.203206 + 0.979136i $$0.565136\pi$$
$$828$$ −37.0741 −1.28842
$$829$$ 25.6541 0.891004 0.445502 0.895281i $$-0.353025\pi$$
0.445502 + 0.895281i $$0.353025\pi$$
$$830$$ 0 0
$$831$$ 35.7726 1.24094
$$832$$ 5.62620 0.195053
$$833$$ 23.2803 0.806615
$$834$$ 9.90754 0.343070
$$835$$ 0 0
$$836$$ −0.864641 −0.0299042
$$837$$ 36.4523 1.25998
$$838$$ 34.7509 1.20045
$$839$$ −2.91713 −0.100710 −0.0503552 0.998731i $$-0.516035\pi$$
−0.0503552 + 0.998731i $$0.516035\pi$$
$$840$$ 0 0
$$841$$ 25.1031 0.865624
$$842$$ −40.1589 −1.38397
$$843$$ 0.424399 0.0146171
$$844$$ 18.0419 0.621026
$$845$$ 0 0
$$846$$ 5.79383 0.199196
$$847$$ 7.80779 0.268279
$$848$$ 2.37380 0.0815167
$$849$$ −50.3544 −1.72816
$$850$$ 0 0
$$851$$ −3.82174 −0.131008
$$852$$ −28.1170 −0.963274
$$853$$ −6.24281 −0.213750 −0.106875 0.994272i $$-0.534084\pi$$
−0.106875 + 0.994272i $$0.534084\pi$$
$$854$$ 8.27405 0.283132
$$855$$ 0 0
$$856$$ 4.28467 0.146447
$$857$$ −23.2158 −0.793035 −0.396517 0.918027i $$-0.629781\pi$$
−0.396517 + 0.918027i $$0.629781\pi$$
$$858$$ −13.4340 −0.458629
$$859$$ −7.13536 −0.243455 −0.121728 0.992564i $$-0.538843\pi$$
−0.121728 + 0.992564i $$0.538843\pi$$
$$860$$ 0 0
$$861$$ 5.59099 0.190541
$$862$$ 34.9571 1.19064
$$863$$ −7.31362 −0.248959 −0.124479 0.992222i $$-0.539726\pi$$
−0.124479 + 0.992222i $$0.539726\pi$$
$$864$$ 4.49084 0.152781
$$865$$ 0 0
$$866$$ 1.13536 0.0385811
$$867$$ −10.6339 −0.361146
$$868$$ −6.18159 −0.209817
$$869$$ 10.8680 0.368671
$$870$$ 0 0
$$871$$ 5.80779 0.196789
$$872$$ −13.4200 −0.454460
$$873$$ −39.3449 −1.33162
$$874$$ −8.01395 −0.271076
$$875$$ 0 0
$$876$$ 45.2943 1.53035
$$877$$ −22.0173 −0.743471 −0.371735 0.928339i $$-0.621237\pi$$
−0.371735 + 0.928339i $$0.621237\pi$$
$$878$$ −6.80009 −0.229492
$$879$$ 5.62953 0.189879
$$880$$ 0 0
$$881$$ 11.7572 0.396110 0.198055 0.980191i $$-0.436538\pi$$
0.198055 + 0.980191i $$0.436538\pi$$
$$882$$ −29.7003 −1.00006
$$883$$ 55.6560 1.87297 0.936487 0.350703i $$-0.114057\pi$$
0.936487 + 0.350703i $$0.114057\pi$$
$$884$$ −20.4017 −0.686184
$$885$$ 0 0
$$886$$ 38.0679 1.27892
$$887$$ −6.41566 −0.215417 −0.107708 0.994183i $$-0.534351\pi$$
−0.107708 + 0.994183i $$0.534351\pi$$
$$888$$ 1.31695 0.0441938
$$889$$ −12.9325 −0.433743
$$890$$ 0 0
$$891$$ 1.27698 0.0427803
$$892$$ −13.5231 −0.452787
$$893$$ 1.25240 0.0419098
$$894$$ −46.5048 −1.55535
$$895$$ 0 0
$$896$$ −0.761557 −0.0254418
$$897$$ −124.513 −4.15738
$$898$$ −18.5414 −0.618736
$$899$$ −59.7047 −1.99126
$$900$$ 0 0
$$901$$ −8.60788 −0.286770
$$902$$ 2.29862 0.0765358
$$903$$ −14.4369 −0.480430
$$904$$ −10.3232 −0.343345
$$905$$ 0 0
$$906$$ −46.8959 −1.55801
$$907$$ −57.1160 −1.89651 −0.948253 0.317516i $$-0.897151\pi$$
−0.948253 + 0.317516i $$0.897151\pi$$
$$908$$ 13.6016 0.451386
$$909$$ 75.9421 2.51884
$$910$$ 0 0
$$911$$ −26.6339 −0.882420 −0.441210 0.897404i $$-0.645451\pi$$
−0.441210 + 0.897404i $$0.645451\pi$$
$$912$$ 2.76156 0.0914443
$$913$$ 0.234074 0.00774672
$$914$$ 16.3738 0.541597
$$915$$ 0 0
$$916$$ 13.5877 0.448949
$$917$$ −0.412335 −0.0136165
$$918$$ −16.2847 −0.537474
$$919$$ 6.63246 0.218785 0.109392 0.993999i $$-0.465110\pi$$
0.109392 + 0.993999i $$0.465110\pi$$
$$920$$ 0 0
$$921$$ 45.6801 1.50521
$$922$$ 1.70470 0.0561414
$$923$$ −57.2836 −1.88551
$$924$$ 1.81841 0.0598214
$$925$$ 0 0
$$926$$ −10.0279 −0.329537
$$927$$ −44.5972 −1.46477
$$928$$ −7.35548 −0.241455
$$929$$ −50.0173 −1.64101 −0.820507 0.571637i $$-0.806309\pi$$
−0.820507 + 0.571637i $$0.806309\pi$$
$$930$$ 0 0
$$931$$ −6.42003 −0.210408
$$932$$ 25.5510 0.836952
$$933$$ −59.2976 −1.94132
$$934$$ 32.7509 1.07164
$$935$$ 0 0
$$936$$ 26.0279 0.850749
$$937$$ −39.8882 −1.30309 −0.651545 0.758610i $$-0.725879\pi$$
−0.651545 + 0.758610i $$0.725879\pi$$
$$938$$ −0.786137 −0.0256683
$$939$$ −3.09683 −0.101061
$$940$$ 0 0
$$941$$ −5.59829 −0.182499 −0.0912495 0.995828i $$-0.529086\pi$$
−0.0912495 + 0.995828i $$0.529086\pi$$
$$942$$ 40.9817 1.33526
$$943$$ 21.3049 0.693782
$$944$$ 4.49084 0.146164
$$945$$ 0 0
$$946$$ −5.93545 −0.192978
$$947$$ 12.7110 0.413051 0.206525 0.978441i $$-0.433784\pi$$
0.206525 + 0.978441i $$0.433784\pi$$
$$948$$ −34.7110 −1.12736
$$949$$ 92.2793 2.99551
$$950$$ 0 0
$$951$$ −82.5379 −2.67648
$$952$$ 2.76156 0.0895026
$$953$$ −57.0129 −1.84683 −0.923415 0.383804i $$-0.874614\pi$$
−0.923415 + 0.383804i $$0.874614\pi$$
$$954$$ 10.9817 0.355545
$$955$$ 0 0
$$956$$ −11.3309 −0.366468
$$957$$ 17.5631 0.567734
$$958$$ 27.2803 0.881387
$$959$$ −2.19221 −0.0707903
$$960$$ 0 0
$$961$$ 34.8863 1.12536
$$962$$ 2.68305 0.0865051
$$963$$ 19.8217 0.638747
$$964$$ −1.25240 −0.0403370
$$965$$ 0 0
$$966$$ 16.8540 0.542269
$$967$$ −33.0183 −1.06180 −0.530899 0.847435i $$-0.678146\pi$$
−0.530899 + 0.847435i $$0.678146\pi$$
$$968$$ −10.2524 −0.329524
$$969$$ −10.0140 −0.321695
$$970$$ 0 0
$$971$$ −3.04623 −0.0977581 −0.0488791 0.998805i $$-0.515565\pi$$
−0.0488791 + 0.998805i $$0.515565\pi$$
$$972$$ −17.5510 −0.562949
$$973$$ −2.73221 −0.0875907
$$974$$ −11.0741 −0.354838
$$975$$ 0 0
$$976$$ −10.8646 −0.347769
$$977$$ 13.4465 0.430192 0.215096 0.976593i $$-0.430994\pi$$
0.215096 + 0.976593i $$0.430994\pi$$
$$978$$ 36.9205 1.18059
$$979$$ −0.335269 −0.0107152
$$980$$ 0 0
$$981$$ −62.0837 −1.98218
$$982$$ 8.11704 0.259025
$$983$$ −22.0646 −0.703750 −0.351875 0.936047i $$-0.614456\pi$$
−0.351875 + 0.936047i $$0.614456\pi$$
$$984$$ −7.34153 −0.234039
$$985$$ 0 0
$$986$$ 26.6724 0.849423
$$987$$ −2.63389 −0.0838378
$$988$$ 5.62620 0.178993
$$989$$ −55.0129 −1.74931
$$990$$ 0 0
$$991$$ 51.9946 1.65166 0.825831 0.563917i $$-0.190706\pi$$
0.825831 + 0.563917i $$0.190706\pi$$
$$992$$ 8.11704 0.257716
$$993$$ 89.2330 2.83172
$$994$$ 7.75386 0.245938
$$995$$ 0 0
$$996$$ −0.747604 −0.0236887
$$997$$ −37.7693 −1.19616 −0.598082 0.801435i $$-0.704070\pi$$
−0.598082 + 0.801435i $$0.704070\pi$$
$$998$$ 0.295298 0.00934749
$$999$$ 2.14162 0.0677578
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 950.2.a.n.1.3 3
3.2 odd 2 8550.2.a.ck.1.1 3
4.3 odd 2 7600.2.a.bi.1.1 3
5.2 odd 4 190.2.b.b.39.4 yes 6
5.3 odd 4 190.2.b.b.39.3 6
5.4 even 2 950.2.a.i.1.1 3
15.2 even 4 1710.2.d.d.1369.3 6
15.8 even 4 1710.2.d.d.1369.6 6
15.14 odd 2 8550.2.a.cl.1.3 3
20.3 even 4 1520.2.d.j.609.1 6
20.7 even 4 1520.2.d.j.609.6 6
20.19 odd 2 7600.2.a.cd.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.3 6 5.3 odd 4
190.2.b.b.39.4 yes 6 5.2 odd 4
950.2.a.i.1.1 3 5.4 even 2
950.2.a.n.1.3 3 1.1 even 1 trivial
1520.2.d.j.609.1 6 20.3 even 4
1520.2.d.j.609.6 6 20.7 even 4
1710.2.d.d.1369.3 6 15.2 even 4
1710.2.d.d.1369.6 6 15.8 even 4
7600.2.a.bi.1.1 3 4.3 odd 2
7600.2.a.cd.1.3 3 20.19 odd 2
8550.2.a.ck.1.1 3 3.2 odd 2
8550.2.a.cl.1.3 3 15.14 odd 2