# Properties

 Label 950.2.a.i.1.1 Level $950$ Weight $2$ Character 950.1 Self dual yes Analytic conductor $7.586$ Analytic rank $1$ 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: $$1$$ 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 1.1 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 $$-0.793683\pi$$
−0.797193 + 0.603725i $$0.793683\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.76156 1.12740
$$7$$ 0.761557 0.287842 0.143921 0.989589i $$-0.454029\pi$$
0.143921 + 0.989589i $$0.454029\pi$$
$$8$$ −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 $$-0.784889\pi$$
−0.780213 + 0.625514i $$0.784889\pi$$
$$14$$ −0.761557 −0.203535
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.62620 0.879482 0.439741 0.898125i $$-0.355070\pi$$
0.439741 + 0.898125i $$0.355070\pi$$
$$18$$ −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 $$-0.185171\pi$$
0.835512 + 0.549472i $$0.185171\pi$$
$$24$$ 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 $$-0.512481\pi$$
−0.0391998 + 0.999231i $$0.512481\pi$$
$$38$$ −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 $$-0.675346\pi$$
−0.523424 + 0.852072i $$0.675346\pi$$
$$44$$ −0.864641 −0.130350
$$45$$ 0 0
$$46$$ −8.01395 −1.18159
$$47$$ −1.25240 −0.182681 −0.0913404 0.995820i $$-0.529115\pi$$
−0.0913404 + 0.995820i $$0.529115\pi$$
$$48$$ −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 $$-0.552128\pi$$
−0.163033 + 0.986621i $$0.552128\pi$$
$$54$$ 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 $$-0.520085\pi$$
−0.0630563 + 0.998010i $$0.520085\pi$$
$$68$$ 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 $$-0.909481\pi$$
−0.959837 + 0.280557i $$0.909481\pi$$
$$74$$ 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 $$-0.495271\pi$$
0.0148576 + 0.999890i $$0.495271\pi$$
$$84$$ −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 $$-0.357891\pi$$
0.431765 + 0.901986i $$0.357891\pi$$
$$98$$ 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 $$-0.342471\pi$$
0.474936 + 0.880020i $$0.342471\pi$$
$$104$$ 5.62620 0.551694
$$105$$ 0 0
$$106$$ 2.37380 0.230564
$$107$$ −4.28467 −0.414215 −0.207107 0.978318i $$-0.566405\pi$$
−0.207107 + 0.978318i $$0.566405\pi$$
$$108$$ −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 $$-0.338615\pi$$
0.485563 + 0.874202i $$0.338615\pi$$
$$114$$ 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 $$-0.771607\pi$$
−0.753440 + 0.657517i $$0.771607\pi$$
$$128$$ −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 $$-0.539241\pi$$
−0.122967 + 0.992411i $$0.539241\pi$$
$$138$$ 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 $$-0.701734\pi$$
−0.592183 + 0.805804i $$0.701734\pi$$
$$158$$ 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 $$-0.675407\pi$$
−0.523587 + 0.851972i $$0.675407\pi$$
$$164$$ −2.65847 −0.207592
$$165$$ 0 0
$$166$$ −0.270718 −0.0210118
$$167$$ −9.84632 −0.761931 −0.380966 0.924589i $$-0.624408\pi$$
−0.380966 + 0.924589i $$0.624408\pi$$
$$168$$ 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 $$-0.536157\pi$$
−0.113346 + 0.993556i $$0.536157\pi$$
$$174$$ −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 $$-0.529156\pi$$
−0.0914683 + 0.995808i $$0.529156\pi$$
$$194$$ −8.50479 −0.610609
$$195$$ 0 0
$$196$$ −6.42003 −0.458574
$$197$$ 19.9109 1.41859 0.709295 0.704911i $$-0.249013\pi$$
0.709295 + 0.704911i $$0.249013\pi$$
$$198$$ 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 $$-0.350430\pi$$
0.452787 + 0.891619i $$0.350430\pi$$
$$224$$ −0.761557 −0.0508837
$$225$$ 0 0
$$226$$ −10.3232 −0.686689
$$227$$ −13.6016 −0.902771 −0.451386 0.892329i $$-0.649070\pi$$
−0.451386 + 0.892329i $$0.649070\pi$$
$$228$$ −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 $$-0.815664\pi$$
−0.836952 + 0.547277i $$0.815664\pi$$
$$234$$ 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 $$-0.498474\pi$$
0.00479319 + 0.999989i $$0.498474\pi$$
$$258$$ −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 $$-0.504954\pi$$
−0.0155634 + 0.999879i $$0.504954\pi$$
$$264$$ −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 $$-0.627234\pi$$
−0.389158 + 0.921171i $$0.627234\pi$$
$$278$$ −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 $$-0.317686\pi$$
0.541952 + 0.840410i $$0.317686\pi$$
$$284$$ −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 $$-0.518965\pi$$
−0.0595462 + 0.998226i $$0.518965\pi$$
$$294$$ −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 $$-0.656480\pi$$
−0.472035 + 0.881580i $$0.656480\pi$$
$$308$$ −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 $$-0.489910\pi$$
0.0316928 + 0.999498i $$0.489910\pi$$
$$314$$ 14.8401 0.837473
$$315$$ 0 0
$$316$$ −12.5693 −0.707081
$$317$$ 29.8882 1.67869 0.839344 0.543601i $$-0.182940\pi$$
0.839344 + 0.543601i $$0.182940\pi$$
$$318$$ −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 $$-0.754813\pi$$
−0.717718 + 0.696334i $$0.754813\pi$$
$$338$$ −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 $$-0.476264\pi$$
0.0744986 + 0.997221i $$0.476264\pi$$
$$348$$ 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 $$-0.428223\pi$$
0.223589 + 0.974684i $$0.428223\pi$$
$$354$$ 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 $$-0.533831\pi$$
−0.106084 + 0.994357i $$0.533831\pi$$
$$368$$ 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 $$-0.341941\pi$$
0.476402 + 0.879227i $$0.341941\pi$$
$$374$$ 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 $$-0.641794\pi$$
−0.430871 + 0.902413i $$0.641794\pi$$
$$384$$ 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 $$-0.632001\pi$$
−0.402909 + 0.915240i $$0.632001\pi$$
$$398$$ 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 $$-0.508685\pi$$
−0.0272809 + 0.999628i $$0.508685\pi$$
$$434$$ −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 $$-0.859629\pi$$
−0.904330 + 0.426835i $$0.859629\pi$$
$$444$$ 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 $$-0.625098\pi$$
−0.382967 + 0.923762i $$0.625098\pi$$
$$458$$ −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 $$-0.425140\pi$$
0.233018 + 0.972472i $$0.425140\pi$$
$$464$$ −7.35548 −0.341470
$$465$$ 0 0
$$466$$ 25.5510 1.18363
$$467$$ −32.7509 −1.51553 −0.757766 0.652526i $$-0.773709\pi$$
−0.757766 + 0.652526i $$0.773709\pi$$
$$468$$ −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 $$-0.419271\pi$$
0.250908 + 0.968011i $$0.419271\pi$$
$$488$$ 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 $$-0.643958\pi$$
−0.436996 + 0.899463i $$0.643958\pi$$
$$504$$ −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 $$-0.606326\pi$$
−0.327857 + 0.944728i $$0.606326\pi$$
$$524$$ 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 $$-0.532363\pi$$
−0.101496 + 0.994836i $$0.532363\pi$$
$$548$$ −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 $$-0.865434\pi$$
−0.911964 + 0.410271i $$0.865434\pi$$
$$558$$ −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 $$-0.616689\pi$$
−0.358434 + 0.933555i $$0.616689\pi$$
$$564$$ 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 $$-0.375341\pi$$
0.381693 + 0.924289i $$0.375341\pi$$
$$578$$ 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 $$-0.420699\pi$$
0.246563 + 0.969127i $$0.420699\pi$$
$$588$$ 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 $$-0.333035\pi$$
0.500811 + 0.865557i $$0.333035\pi$$
$$594$$ −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 $$-0.626131\pi$$
−0.385965 + 0.922513i $$0.626131\pi$$
$$608$$ −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 $$-0.342048\pi$$
0.476105 + 0.879389i $$0.342048\pi$$
$$614$$ 16.5414 0.667558
$$615$$ 0 0
$$616$$ 0.658473 0.0265307
$$617$$ 26.2707 1.05762 0.528810 0.848740i $$-0.322639\pi$$
0.528810 + 0.848740i $$0.322639\pi$$
$$618$$ 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 $$-0.337799\pi$$
0.487803 + 0.872954i $$0.337799\pi$$
$$644$$ 6.10308 0.240495
$$645$$ 0 0
$$646$$ −3.62620 −0.142671
$$647$$ 6.82611 0.268362 0.134181 0.990957i $$-0.457160\pi$$
0.134181 + 0.990957i $$0.457160\pi$$
$$648$$ 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 $$-0.543214\pi$$
−0.135344 + 0.990799i $$0.543214\pi$$
$$654$$ −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 $$-0.576177\pi$$
−0.237039 + 0.971500i $$0.576177\pi$$
$$674$$ 26.3511 1.01501
$$675$$ 0 0
$$676$$ 18.6541 0.717466
$$677$$ 35.9527 1.38178 0.690888 0.722962i $$-0.257220\pi$$
0.690888 + 0.722962i $$0.257220\pi$$
$$678$$ 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 $$-0.444857\pi$$
0.172371 + 0.985032i $$0.444857\pi$$
$$684$$ 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 $$-0.303688\pi$$
0.578373 + 0.815772i $$0.303688\pi$$
$$728$$ 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 $$-0.416978\pi$$
0.257874 + 0.966179i $$0.416978\pi$$
$$734$$ 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 $$-0.411066\pi$$
0.275775 + 0.961222i $$0.411066\pi$$
$$744$$ 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 $$-0.561595\pi$$
−0.192302 + 0.981336i $$0.561595\pi$$
$$758$$ −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 $$-0.735787\pi$$
−0.674840 + 0.737964i $$0.735787\pi$$
$$774$$ 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 $$-0.631860\pi$$
−0.402504 + 0.915418i $$0.631860\pi$$
$$788$$ 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 $$-0.719928\pi$$
−0.637250 + 0.770657i $$0.719928\pi$$
$$798$$ 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 $$-0.639850\pi$$
−0.425353 + 0.905028i $$0.639850\pi$$
$$824$$ −9.64015 −0.335831
$$825$$ 0 0
$$826$$ −3.42003 −0.118998
$$827$$ 11.6874 0.406411 0.203206 0.979136i $$-0.434864\pi$$
0.203206 + 0.979136i $$0.434864\pi$$
$$828$$ 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 $$-0.465916\pi$$
0.106875 + 0.994272i $$0.465916\pi$$
$$854$$ 8.27405 0.283132
$$855$$ 0 0
$$856$$ 4.28467 0.146447
$$857$$ 23.2158 0.793035 0.396517 0.918027i $$-0.370219\pi$$
0.396517 + 0.918027i $$0.370219\pi$$
$$858$$ 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 $$-0.460274\pi$$
0.124479 + 0.992222i $$0.460274\pi$$
$$864$$ 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 $$-0.378763\pi$$
0.371735 + 0.928339i $$0.378763\pi$$
$$878$$ 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 $$-0.885943\pi$$
−0.936487 + 0.350703i $$0.885943\pi$$
$$884$$ −20.4017 −0.686184
$$885$$ 0 0
$$886$$ 38.0679 1.27892
$$887$$ 6.41566 0.215417 0.107708 0.994183i $$-0.465649\pi$$
0.107708 + 0.994183i $$0.465649\pi$$
$$888$$ −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 $$-0.102849\pi$$
0.948253 + 0.317516i $$0.102849\pi$$
$$908$$ −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 $$-0.274121\pi$$
0.651545 + 0.758610i $$0.274121\pi$$
$$938$$ 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 $$-0.566216\pi$$
−0.206525 + 0.978441i $$0.566216\pi$$
$$948$$ 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 $$-0.125386\pi$$
0.923415 + 0.383804i $$0.125386\pi$$
$$954$$ 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 $$-0.321854\pi$$
0.530899 + 0.847435i $$0.321854\pi$$
$$968$$ 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 $$-0.569006\pi$$
−0.215096 + 0.976593i $$0.569006\pi$$
$$978$$ −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 $$-0.385544\pi$$
0.351875 + 0.936047i $$0.385544\pi$$
$$984$$ −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 $$-0.295930\pi$$
0.598082 + 0.801435i $$0.295930\pi$$
$$998$$ −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.i.1.1 3
3.2 odd 2 8550.2.a.cl.1.3 3
4.3 odd 2 7600.2.a.cd.1.3 3
5.2 odd 4 190.2.b.b.39.3 6
5.3 odd 4 190.2.b.b.39.4 yes 6
5.4 even 2 950.2.a.n.1.3 3
15.2 even 4 1710.2.d.d.1369.6 6
15.8 even 4 1710.2.d.d.1369.3 6
15.14 odd 2 8550.2.a.ck.1.1 3
20.3 even 4 1520.2.d.j.609.6 6
20.7 even 4 1520.2.d.j.609.1 6
20.19 odd 2 7600.2.a.bi.1.1 3

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