# Properties

 Label 1150.2.b.j.599.5 Level $1150$ Weight $2$ Character 1150.599 Analytic conductor $9.183$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.18279623245$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.77580864.1 Defining polynomial: $$x^{6} + 19 x^{4} + 105 x^{2} + 144$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 599.5 Root $$-1.43163i$$ of defining polynomial Character $$\chi$$ $$=$$ 1150.599 Dual form 1150.2.b.j.599.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.43163i q^{3} -1.00000 q^{4} +1.43163 q^{6} +3.08719i q^{7} -1.00000i q^{8} +0.950444 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.43163i q^{3} -1.00000 q^{4} +1.43163 q^{6} +3.08719i q^{7} -1.00000i q^{8} +0.950444 q^{9} -6.46926 q^{11} +1.43163i q^{12} -3.95044i q^{13} -3.08719 q^{14} +1.00000 q^{16} -3.43163i q^{17} +0.950444i q^{18} -3.08719 q^{19} +4.41970 q^{21} -6.46926i q^{22} +1.00000i q^{23} -1.43163 q^{24} +3.95044 q^{26} -5.65556i q^{27} -3.08719i q^{28} -0.863254 q^{29} -5.95044 q^{31} +1.00000i q^{32} +9.26157i q^{33} +3.43163 q^{34} -0.950444 q^{36} -7.03763i q^{37} -3.08719i q^{38} -5.65556 q^{39} +5.60601 q^{41} +4.41970i q^{42} -8.00000i q^{43} +6.46926 q^{44} -1.00000 q^{46} +3.90089i q^{47} -1.43163i q^{48} -2.53074 q^{49} -4.91281 q^{51} +3.95044i q^{52} +6.00000i q^{53} +5.65556 q^{54} +3.08719 q^{56} +4.41970i q^{57} -0.863254i q^{58} -6.86325 q^{59} -13.5069 q^{61} -5.95044i q^{62} +2.93420i q^{63} -1.00000 q^{64} -9.26157 q^{66} -10.0753i q^{67} +3.43163i q^{68} +1.43163 q^{69} +2.56837 q^{71} -0.950444i q^{72} -5.90089i q^{73} +7.03763 q^{74} +3.08719 q^{76} -19.9718i q^{77} -5.65556i q^{78} -15.8018 q^{79} -5.24533 q^{81} +5.60601i q^{82} -9.03763i q^{83} -4.41970 q^{84} +8.00000 q^{86} +1.23586i q^{87} +6.46926i q^{88} -16.7641 q^{89} +12.1958 q^{91} -1.00000i q^{92} +8.51882i q^{93} -3.90089 q^{94} +1.43163 q^{96} -14.2949i q^{97} -2.53074i q^{98} -6.14867 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} + 2q^{6} - 20q^{9} + O(q^{10})$$ $$6q - 6q^{4} + 2q^{6} - 20q^{9} + 6q^{11} - 6q^{14} + 6q^{16} - 6q^{19} - 44q^{21} - 2q^{24} - 2q^{26} + 8q^{29} - 10q^{31} + 14q^{34} + 20q^{36} - 28q^{39} + 2q^{41} - 6q^{44} - 6q^{46} - 60q^{49} - 42q^{51} + 28q^{54} + 6q^{56} - 28q^{59} + 2q^{61} - 6q^{64} - 18q^{66} + 2q^{69} + 22q^{71} + 4q^{74} + 6q^{76} + 8q^{79} + 14q^{81} + 44q^{84} + 48q^{86} - 36q^{89} + 2q^{91} + 28q^{94} + 2q^{96} - 114q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.43163i − 0.826550i −0.910606 0.413275i $$-0.864385\pi$$
0.910606 0.413275i $$-0.135615\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.43163 0.584459
$$7$$ 3.08719i 1.16685i 0.812168 + 0.583424i $$0.198287\pi$$
−0.812168 + 0.583424i $$0.801713\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0.950444 0.316815
$$10$$ 0 0
$$11$$ −6.46926 −1.95056 −0.975278 0.220983i $$-0.929074\pi$$
−0.975278 + 0.220983i $$0.929074\pi$$
$$12$$ 1.43163i 0.413275i
$$13$$ − 3.95044i − 1.09566i −0.836591 0.547828i $$-0.815455\pi$$
0.836591 0.547828i $$-0.184545\pi$$
$$14$$ −3.08719 −0.825086
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 3.43163i − 0.832292i −0.909298 0.416146i $$-0.863381\pi$$
0.909298 0.416146i $$-0.136619\pi$$
$$18$$ 0.950444i 0.224022i
$$19$$ −3.08719 −0.708250 −0.354125 0.935198i $$-0.615221\pi$$
−0.354125 + 0.935198i $$0.615221\pi$$
$$20$$ 0 0
$$21$$ 4.41970 0.964459
$$22$$ − 6.46926i − 1.37925i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.43163 −0.292230
$$25$$ 0 0
$$26$$ 3.95044 0.774746
$$27$$ − 5.65556i − 1.08841i
$$28$$ − 3.08719i − 0.583424i
$$29$$ −0.863254 −0.160302 −0.0801511 0.996783i $$-0.525540\pi$$
−0.0801511 + 0.996783i $$0.525540\pi$$
$$30$$ 0 0
$$31$$ −5.95044 −1.06873 −0.534366 0.845253i $$-0.679449\pi$$
−0.534366 + 0.845253i $$0.679449\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 9.26157i 1.61223i
$$34$$ 3.43163 0.588519
$$35$$ 0 0
$$36$$ −0.950444 −0.158407
$$37$$ − 7.03763i − 1.15698i −0.815690 0.578490i $$-0.803642\pi$$
0.815690 0.578490i $$-0.196358\pi$$
$$38$$ − 3.08719i − 0.500808i
$$39$$ −5.65556 −0.905615
$$40$$ 0 0
$$41$$ 5.60601 0.875511 0.437756 0.899094i $$-0.355774\pi$$
0.437756 + 0.899094i $$0.355774\pi$$
$$42$$ 4.41970i 0.681975i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 6.46926 0.975278
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 3.90089i 0.569003i 0.958676 + 0.284501i $$0.0918281\pi$$
−0.958676 + 0.284501i $$0.908172\pi$$
$$48$$ − 1.43163i − 0.206638i
$$49$$ −2.53074 −0.361534
$$50$$ 0 0
$$51$$ −4.91281 −0.687931
$$52$$ 3.95044i 0.547828i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 5.65556 0.769625
$$55$$ 0 0
$$56$$ 3.08719 0.412543
$$57$$ 4.41970i 0.585404i
$$58$$ − 0.863254i − 0.113351i
$$59$$ −6.86325 −0.893520 −0.446760 0.894654i $$-0.647422\pi$$
−0.446760 + 0.894654i $$0.647422\pi$$
$$60$$ 0 0
$$61$$ −13.5069 −1.72938 −0.864690 0.502305i $$-0.832485\pi$$
−0.864690 + 0.502305i $$0.832485\pi$$
$$62$$ − 5.95044i − 0.755707i
$$63$$ 2.93420i 0.369674i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −9.26157 −1.14002
$$67$$ − 10.0753i − 1.23089i −0.788180 0.615445i $$-0.788976\pi$$
0.788180 0.615445i $$-0.211024\pi$$
$$68$$ 3.43163i 0.416146i
$$69$$ 1.43163 0.172348
$$70$$ 0 0
$$71$$ 2.56837 0.304810 0.152405 0.988318i $$-0.451298\pi$$
0.152405 + 0.988318i $$0.451298\pi$$
$$72$$ − 0.950444i − 0.112011i
$$73$$ − 5.90089i − 0.690647i −0.938484 0.345323i $$-0.887769\pi$$
0.938484 0.345323i $$-0.112231\pi$$
$$74$$ 7.03763 0.818108
$$75$$ 0 0
$$76$$ 3.08719 0.354125
$$77$$ − 19.9718i − 2.27600i
$$78$$ − 5.65556i − 0.640366i
$$79$$ −15.8018 −1.77784 −0.888919 0.458064i $$-0.848543\pi$$
−0.888919 + 0.458064i $$0.848543\pi$$
$$80$$ 0 0
$$81$$ −5.24533 −0.582814
$$82$$ 5.60601i 0.619080i
$$83$$ − 9.03763i − 0.992009i −0.868320 0.496005i $$-0.834800\pi$$
0.868320 0.496005i $$-0.165200\pi$$
$$84$$ −4.41970 −0.482229
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 1.23586i 0.132498i
$$88$$ 6.46926i 0.689625i
$$89$$ −16.7641 −1.77700 −0.888498 0.458881i $$-0.848250\pi$$
−0.888498 + 0.458881i $$0.848250\pi$$
$$90$$ 0 0
$$91$$ 12.1958 1.27846
$$92$$ − 1.00000i − 0.104257i
$$93$$ 8.51882i 0.883360i
$$94$$ −3.90089 −0.402346
$$95$$ 0 0
$$96$$ 1.43163 0.146115
$$97$$ − 14.2949i − 1.45143i −0.687998 0.725713i $$-0.741510\pi$$
0.687998 0.725713i $$-0.258490\pi$$
$$98$$ − 2.53074i − 0.255643i
$$99$$ −6.14867 −0.617964
$$100$$ 0 0
$$101$$ 0.863254 0.0858970 0.0429485 0.999077i $$-0.486325\pi$$
0.0429485 + 0.999077i $$0.486325\pi$$
$$102$$ − 4.91281i − 0.486441i
$$103$$ − 1.53074i − 0.150828i −0.997152 0.0754141i $$-0.975972\pi$$
0.997152 0.0754141i $$-0.0240279\pi$$
$$104$$ −3.95044 −0.387373
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 17.6274i 1.70410i 0.523457 + 0.852052i $$0.324642\pi$$
−0.523457 + 0.852052i $$0.675358\pi$$
$$108$$ 5.65556i 0.544207i
$$109$$ 2.91281 0.278997 0.139498 0.990222i $$-0.455451\pi$$
0.139498 + 0.990222i $$0.455451\pi$$
$$110$$ 0 0
$$111$$ −10.0753 −0.956302
$$112$$ 3.08719i 0.291712i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ −4.41970 −0.413943
$$115$$ 0 0
$$116$$ 0.863254 0.0801511
$$117$$ − 3.75467i − 0.347120i
$$118$$ − 6.86325i − 0.631814i
$$119$$ 10.5941 0.971158
$$120$$ 0 0
$$121$$ 30.8513 2.80467
$$122$$ − 13.5069i − 1.22286i
$$123$$ − 8.02571i − 0.723654i
$$124$$ 5.95044 0.534366
$$125$$ 0 0
$$126$$ −2.93420 −0.261399
$$127$$ 20.9385i 1.85799i 0.370088 + 0.928997i $$0.379327\pi$$
−0.370088 + 0.928997i $$0.620673\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −11.4530 −1.00838
$$130$$ 0 0
$$131$$ 10.7641 0.940467 0.470234 0.882542i $$-0.344170\pi$$
0.470234 + 0.882542i $$0.344170\pi$$
$$132$$ − 9.26157i − 0.806116i
$$133$$ − 9.53074i − 0.826420i
$$134$$ 10.0753 0.870370
$$135$$ 0 0
$$136$$ −3.43163 −0.294260
$$137$$ 3.26157i 0.278655i 0.990246 + 0.139327i $$0.0444940\pi$$
−0.990246 + 0.139327i $$0.955506\pi$$
$$138$$ 1.43163i 0.121868i
$$139$$ 16.9385 1.43671 0.718353 0.695678i $$-0.244896\pi$$
0.718353 + 0.695678i $$0.244896\pi$$
$$140$$ 0 0
$$141$$ 5.58462 0.470310
$$142$$ 2.56837i 0.215533i
$$143$$ 25.5565i 2.13714i
$$144$$ 0.950444 0.0792037
$$145$$ 0 0
$$146$$ 5.90089 0.488361
$$147$$ 3.62308i 0.298826i
$$148$$ 7.03763i 0.578490i
$$149$$ −3.26157 −0.267198 −0.133599 0.991035i $$-0.542653\pi$$
−0.133599 + 0.991035i $$0.542653\pi$$
$$150$$ 0 0
$$151$$ 0.294881 0.0239971 0.0119986 0.999928i $$-0.496181\pi$$
0.0119986 + 0.999928i $$0.496181\pi$$
$$152$$ 3.08719i 0.250404i
$$153$$ − 3.26157i − 0.263682i
$$154$$ 19.9718 1.60938
$$155$$ 0 0
$$156$$ 5.65556 0.452807
$$157$$ 7.13675i 0.569574i 0.958591 + 0.284787i $$0.0919229\pi$$
−0.958591 + 0.284787i $$0.908077\pi$$
$$158$$ − 15.8018i − 1.25712i
$$159$$ 8.58976 0.681212
$$160$$ 0 0
$$161$$ −3.08719 −0.243305
$$162$$ − 5.24533i − 0.412112i
$$163$$ 8.12482i 0.636385i 0.948026 + 0.318193i $$0.103076\pi$$
−0.948026 + 0.318193i $$0.896924\pi$$
$$164$$ −5.60601 −0.437756
$$165$$ 0 0
$$166$$ 9.03763 0.701456
$$167$$ − 7.80178i − 0.603719i −0.953352 0.301860i $$-0.902393\pi$$
0.953352 0.301860i $$-0.0976074\pi$$
$$168$$ − 4.41970i − 0.340988i
$$169$$ −2.60601 −0.200462
$$170$$ 0 0
$$171$$ −2.93420 −0.224384
$$172$$ 8.00000i 0.609994i
$$173$$ 19.3325i 1.46982i 0.678163 + 0.734912i $$0.262777\pi$$
−0.678163 + 0.734912i $$0.737223\pi$$
$$174$$ −1.23586 −0.0936902
$$175$$ 0 0
$$176$$ −6.46926 −0.487639
$$177$$ 9.82562i 0.738539i
$$178$$ − 16.7641i − 1.25653i
$$179$$ −2.17438 −0.162521 −0.0812604 0.996693i $$-0.525895\pi$$
−0.0812604 + 0.996693i $$0.525895\pi$$
$$180$$ 0 0
$$181$$ −7.26157 −0.539748 −0.269874 0.962896i $$-0.586982\pi$$
−0.269874 + 0.962896i $$0.586982\pi$$
$$182$$ 12.1958i 0.904011i
$$183$$ 19.3368i 1.42942i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ −8.51882 −0.624630
$$187$$ 22.2001i 1.62343i
$$188$$ − 3.90089i − 0.284501i
$$189$$ 17.4598 1.27001
$$190$$ 0 0
$$191$$ 8.58976 0.621533 0.310767 0.950486i $$-0.399414\pi$$
0.310767 + 0.950486i $$0.399414\pi$$
$$192$$ 1.43163i 0.103319i
$$193$$ − 2.44787i − 0.176202i −0.996112 0.0881008i $$-0.971920\pi$$
0.996112 0.0881008i $$-0.0280798\pi$$
$$194$$ 14.2949 1.02631
$$195$$ 0 0
$$196$$ 2.53074 0.180767
$$197$$ − 10.7428i − 0.765389i −0.923875 0.382695i $$-0.874996\pi$$
0.923875 0.382695i $$-0.125004\pi$$
$$198$$ − 6.14867i − 0.436967i
$$199$$ 11.3111 0.801824 0.400912 0.916116i $$-0.368693\pi$$
0.400912 + 0.916116i $$0.368693\pi$$
$$200$$ 0 0
$$201$$ −14.4240 −1.01739
$$202$$ 0.863254i 0.0607384i
$$203$$ − 2.66503i − 0.187048i
$$204$$ 4.91281 0.343966
$$205$$ 0 0
$$206$$ 1.53074 0.106652
$$207$$ 0.950444i 0.0660604i
$$208$$ − 3.95044i − 0.273914i
$$209$$ 19.9718 1.38148
$$210$$ 0 0
$$211$$ 23.1129 1.59116 0.795579 0.605850i $$-0.207167\pi$$
0.795579 + 0.605850i $$0.207167\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 3.67695i − 0.251941i
$$214$$ −17.6274 −1.20498
$$215$$ 0 0
$$216$$ −5.65556 −0.384812
$$217$$ − 18.3701i − 1.24705i
$$218$$ 2.91281i 0.197280i
$$219$$ −8.44787 −0.570854
$$220$$ 0 0
$$221$$ −13.5565 −0.911906
$$222$$ − 10.0753i − 0.676208i
$$223$$ 5.72651i 0.383475i 0.981446 + 0.191738i $$0.0614123\pi$$
−0.981446 + 0.191738i $$0.938588\pi$$
$$224$$ −3.08719 −0.206272
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 17.6274i − 1.16997i −0.811044 0.584986i $$-0.801100\pi$$
0.811044 0.584986i $$-0.198900\pi$$
$$228$$ − 4.41970i − 0.292702i
$$229$$ 16.0753 1.06228 0.531142 0.847283i $$-0.321763\pi$$
0.531142 + 0.847283i $$0.321763\pi$$
$$230$$ 0 0
$$231$$ −28.5922 −1.88123
$$232$$ 0.863254i 0.0566754i
$$233$$ 6.44787i 0.422414i 0.977441 + 0.211207i $$0.0677394\pi$$
−0.977441 + 0.211207i $$0.932261\pi$$
$$234$$ 3.75467 0.245451
$$235$$ 0 0
$$236$$ 6.86325 0.446760
$$237$$ 22.6222i 1.46947i
$$238$$ 10.5941i 0.686712i
$$239$$ −4.34876 −0.281298 −0.140649 0.990060i $$-0.544919\pi$$
−0.140649 + 0.990060i $$0.544919\pi$$
$$240$$ 0 0
$$241$$ 0.764142 0.0492227 0.0246114 0.999697i $$-0.492165\pi$$
0.0246114 + 0.999697i $$0.492165\pi$$
$$242$$ 30.8513i 1.98320i
$$243$$ − 9.45734i − 0.606688i
$$244$$ 13.5069 0.864690
$$245$$ 0 0
$$246$$ 8.02571 0.511701
$$247$$ 12.1958i 0.775998i
$$248$$ 5.95044i 0.377854i
$$249$$ −12.9385 −0.819945
$$250$$ 0 0
$$251$$ −22.5402 −1.42273 −0.711363 0.702825i $$-0.751922\pi$$
−0.711363 + 0.702825i $$0.751922\pi$$
$$252$$ − 2.93420i − 0.184837i
$$253$$ − 6.46926i − 0.406719i
$$254$$ −20.9385 −1.31380
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 9.90089i − 0.617600i −0.951127 0.308800i $$-0.900073\pi$$
0.951127 0.308800i $$-0.0999274\pi$$
$$258$$ − 11.4530i − 0.713034i
$$259$$ 21.7265 1.35002
$$260$$ 0 0
$$261$$ −0.820475 −0.0507861
$$262$$ 10.7641i 0.665011i
$$263$$ 7.25725i 0.447501i 0.974646 + 0.223751i $$0.0718301\pi$$
−0.974646 + 0.223751i $$0.928170\pi$$
$$264$$ 9.26157 0.570010
$$265$$ 0 0
$$266$$ 9.53074 0.584367
$$267$$ 24.0000i 1.46878i
$$268$$ 10.0753i 0.615445i
$$269$$ 6.93852 0.423049 0.211525 0.977373i $$-0.432157\pi$$
0.211525 + 0.977373i $$0.432157\pi$$
$$270$$ 0 0
$$271$$ −10.2992 −0.625632 −0.312816 0.949814i $$-0.601272\pi$$
−0.312816 + 0.949814i $$0.601272\pi$$
$$272$$ − 3.43163i − 0.208073i
$$273$$ − 17.4598i − 1.05671i
$$274$$ −3.26157 −0.197039
$$275$$ 0 0
$$276$$ −1.43163 −0.0861738
$$277$$ − 17.8018i − 1.06961i −0.844977 0.534803i $$-0.820386\pi$$
0.844977 0.534803i $$-0.179614\pi$$
$$278$$ 16.9385i 1.01590i
$$279$$ −5.65556 −0.338590
$$280$$ 0 0
$$281$$ −18.9385 −1.12978 −0.564889 0.825167i $$-0.691081\pi$$
−0.564889 + 0.825167i $$0.691081\pi$$
$$282$$ 5.58462i 0.332559i
$$283$$ − 12.6889i − 0.754275i −0.926157 0.377138i $$-0.876908\pi$$
0.926157 0.377138i $$-0.123092\pi$$
$$284$$ −2.56837 −0.152405
$$285$$ 0 0
$$286$$ −25.5565 −1.51118
$$287$$ 17.3068i 1.02159i
$$288$$ 0.950444i 0.0560054i
$$289$$ 5.22394 0.307290
$$290$$ 0 0
$$291$$ −20.4649 −1.19968
$$292$$ 5.90089i 0.345323i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ −3.62308 −0.211302
$$295$$ 0 0
$$296$$ −7.03763 −0.409054
$$297$$ 36.5873i 2.12301i
$$298$$ − 3.26157i − 0.188938i
$$299$$ 3.95044 0.228460
$$300$$ 0 0
$$301$$ 24.6975 1.42354
$$302$$ 0.294881i 0.0169685i
$$303$$ − 1.23586i − 0.0709982i
$$304$$ −3.08719 −0.177062
$$305$$ 0 0
$$306$$ 3.26157 0.186451
$$307$$ − 22.9599i − 1.31039i −0.755459 0.655196i $$-0.772586\pi$$
0.755459 0.655196i $$-0.227414\pi$$
$$308$$ 19.9718i 1.13800i
$$309$$ −2.19145 −0.124667
$$310$$ 0 0
$$311$$ 18.0753 1.02495 0.512477 0.858701i $$-0.328728\pi$$
0.512477 + 0.858701i $$0.328728\pi$$
$$312$$ 5.65556i 0.320183i
$$313$$ − 15.1625i − 0.857033i −0.903534 0.428516i $$-0.859036\pi$$
0.903534 0.428516i $$-0.140964\pi$$
$$314$$ −7.13675 −0.402750
$$315$$ 0 0
$$316$$ 15.8018 0.888919
$$317$$ 14.0257i 0.787762i 0.919161 + 0.393881i $$0.128868\pi$$
−0.919161 + 0.393881i $$0.871132\pi$$
$$318$$ 8.58976i 0.481690i
$$319$$ 5.58462 0.312678
$$320$$ 0 0
$$321$$ 25.2359 1.40853
$$322$$ − 3.08719i − 0.172042i
$$323$$ 10.5941i 0.589471i
$$324$$ 5.24533 0.291407
$$325$$ 0 0
$$326$$ −8.12482 −0.449992
$$327$$ − 4.17006i − 0.230605i
$$328$$ − 5.60601i − 0.309540i
$$329$$ −12.0428 −0.663940
$$330$$ 0 0
$$331$$ −24.7403 −1.35985 −0.679925 0.733282i $$-0.737988\pi$$
−0.679925 + 0.733282i $$0.737988\pi$$
$$332$$ 9.03763i 0.496005i
$$333$$ − 6.68888i − 0.366548i
$$334$$ 7.80178 0.426894
$$335$$ 0 0
$$336$$ 4.41970 0.241115
$$337$$ 14.7146i 0.801555i 0.916176 + 0.400777i $$0.131260\pi$$
−0.916176 + 0.400777i $$0.868740\pi$$
$$338$$ − 2.60601i − 0.141748i
$$339$$ 8.58976 0.466532
$$340$$ 0 0
$$341$$ 38.4950 2.08462
$$342$$ − 2.93420i − 0.158663i
$$343$$ 13.7975i 0.744993i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −19.3325 −1.03932
$$347$$ 9.43163i 0.506316i 0.967425 + 0.253158i $$0.0814693\pi$$
−0.967425 + 0.253158i $$0.918531\pi$$
$$348$$ − 1.23586i − 0.0662490i
$$349$$ −18.3488 −0.982187 −0.491093 0.871107i $$-0.663403\pi$$
−0.491093 + 0.871107i $$0.663403\pi$$
$$350$$ 0 0
$$351$$ −22.3420 −1.19253
$$352$$ − 6.46926i − 0.344813i
$$353$$ 33.1129i 1.76242i 0.472723 + 0.881211i $$0.343271\pi$$
−0.472723 + 0.881211i $$0.656729\pi$$
$$354$$ −9.82562 −0.522226
$$355$$ 0 0
$$356$$ 16.7641 0.888498
$$357$$ − 15.1668i − 0.802711i
$$358$$ − 2.17438i − 0.114920i
$$359$$ −33.0376 −1.74366 −0.871830 0.489809i $$-0.837067\pi$$
−0.871830 + 0.489809i $$0.837067\pi$$
$$360$$ 0 0
$$361$$ −9.46926 −0.498382
$$362$$ − 7.26157i − 0.381660i
$$363$$ − 44.1676i − 2.31820i
$$364$$ −12.1958 −0.639232
$$365$$ 0 0
$$366$$ −19.3368 −1.01075
$$367$$ − 2.27349i − 0.118675i −0.998238 0.0593376i $$-0.981101\pi$$
0.998238 0.0593376i $$-0.0188989\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 5.32819 0.277375
$$370$$ 0 0
$$371$$ −18.5231 −0.961673
$$372$$ − 8.51882i − 0.441680i
$$373$$ − 23.9762i − 1.24144i −0.784033 0.620719i $$-0.786841\pi$$
0.784033 0.620719i $$-0.213159\pi$$
$$374$$ −22.2001 −1.14794
$$375$$ 0 0
$$376$$ 3.90089 0.201173
$$377$$ 3.41024i 0.175636i
$$378$$ 17.4598i 0.898035i
$$379$$ 13.8795 0.712942 0.356471 0.934306i $$-0.383980\pi$$
0.356471 + 0.934306i $$0.383980\pi$$
$$380$$ 0 0
$$381$$ 29.9762 1.53572
$$382$$ 8.58976i 0.439490i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −1.43163 −0.0730574
$$385$$ 0 0
$$386$$ 2.44787 0.124593
$$387$$ − 7.60355i − 0.386510i
$$388$$ 14.2949i 0.725713i
$$389$$ 26.6436 1.35089 0.675443 0.737412i $$-0.263952\pi$$
0.675443 + 0.737412i $$0.263952\pi$$
$$390$$ 0 0
$$391$$ 3.43163 0.173545
$$392$$ 2.53074i 0.127822i
$$393$$ − 15.4102i − 0.777344i
$$394$$ 10.7428 0.541212
$$395$$ 0 0
$$396$$ 6.14867 0.308982
$$397$$ − 8.66749i − 0.435009i −0.976059 0.217504i $$-0.930208\pi$$
0.976059 0.217504i $$-0.0697916\pi$$
$$398$$ 11.3111i 0.566975i
$$399$$ −13.6445 −0.683078
$$400$$ 0 0
$$401$$ 39.9762 1.99631 0.998157 0.0606854i $$-0.0193286\pi$$
0.998157 + 0.0606854i $$0.0193286\pi$$
$$402$$ − 14.4240i − 0.719405i
$$403$$ 23.5069i 1.17096i
$$404$$ −0.863254 −0.0429485
$$405$$ 0 0
$$406$$ 2.66503 0.132263
$$407$$ 45.5283i 2.25675i
$$408$$ 4.91281i 0.243220i
$$409$$ −30.1248 −1.48958 −0.744788 0.667301i $$-0.767450\pi$$
−0.744788 + 0.667301i $$0.767450\pi$$
$$410$$ 0 0
$$411$$ 4.66935 0.230322
$$412$$ 1.53074i 0.0754141i
$$413$$ − 21.1882i − 1.04260i
$$414$$ −0.950444 −0.0467118
$$415$$ 0 0
$$416$$ 3.95044 0.193686
$$417$$ − 24.2496i − 1.18751i
$$418$$ 19.9718i 0.976854i
$$419$$ −25.8770 −1.26418 −0.632088 0.774897i $$-0.717802\pi$$
−0.632088 + 0.774897i $$0.717802\pi$$
$$420$$ 0 0
$$421$$ −10.7146 −0.522197 −0.261098 0.965312i $$-0.584085\pi$$
−0.261098 + 0.965312i $$0.584085\pi$$
$$422$$ 23.1129i 1.12512i
$$423$$ 3.70757i 0.180268i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 3.67695 0.178149
$$427$$ − 41.6983i − 2.01792i
$$428$$ − 17.6274i − 0.852052i
$$429$$ 36.5873 1.76645
$$430$$ 0 0
$$431$$ 32.2496 1.55341 0.776705 0.629864i $$-0.216889\pi$$
0.776705 + 0.629864i $$0.216889\pi$$
$$432$$ − 5.65556i − 0.272103i
$$433$$ − 20.6393i − 0.991862i −0.868362 0.495931i $$-0.834827\pi$$
0.868362 0.495931i $$-0.165173\pi$$
$$434$$ 18.3701 0.881795
$$435$$ 0 0
$$436$$ −2.91281 −0.139498
$$437$$ − 3.08719i − 0.147680i
$$438$$ − 8.44787i − 0.403655i
$$439$$ 16.2239 0.774326 0.387163 0.922011i $$-0.373455\pi$$
0.387163 + 0.922011i $$0.373455\pi$$
$$440$$ 0 0
$$441$$ −2.40533 −0.114539
$$442$$ − 13.5565i − 0.644815i
$$443$$ 15.6770i 0.744834i 0.928065 + 0.372417i $$0.121471\pi$$
−0.928065 + 0.372417i $$0.878529\pi$$
$$444$$ 10.0753 0.478151
$$445$$ 0 0
$$446$$ −5.72651 −0.271158
$$447$$ 4.66935i 0.220853i
$$448$$ − 3.08719i − 0.145856i
$$449$$ −22.5727 −1.06527 −0.532636 0.846345i $$-0.678798\pi$$
−0.532636 + 0.846345i $$0.678798\pi$$
$$450$$ 0 0
$$451$$ −36.2667 −1.70773
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 0.422160i − 0.0198348i
$$454$$ 17.6274 0.827295
$$455$$ 0 0
$$456$$ 4.41970 0.206972
$$457$$ − 1.45302i − 0.0679693i −0.999422 0.0339846i $$-0.989180\pi$$
0.999422 0.0339846i $$-0.0108197\pi$$
$$458$$ 16.0753i 0.751148i
$$459$$ −19.4078 −0.905878
$$460$$ 0 0
$$461$$ −29.7027 −1.38339 −0.691695 0.722189i $$-0.743136\pi$$
−0.691695 + 0.722189i $$0.743136\pi$$
$$462$$ − 28.5922i − 1.33023i
$$463$$ − 22.9624i − 1.06715i −0.845752 0.533576i $$-0.820848\pi$$
0.845752 0.533576i $$-0.179152\pi$$
$$464$$ −0.863254 −0.0400756
$$465$$ 0 0
$$466$$ −6.44787 −0.298692
$$467$$ 9.48550i 0.438937i 0.975620 + 0.219468i $$0.0704323\pi$$
−0.975620 + 0.219468i $$0.929568\pi$$
$$468$$ 3.75467i 0.173560i
$$469$$ 31.1043 1.43626
$$470$$ 0 0
$$471$$ 10.2172 0.470782
$$472$$ 6.86325i 0.315907i
$$473$$ 51.7541i 2.37966i
$$474$$ −22.6222 −1.03907
$$475$$ 0 0
$$476$$ −10.5941 −0.485579
$$477$$ 5.70266i 0.261107i
$$478$$ − 4.34876i − 0.198908i
$$479$$ 30.5659 1.39659 0.698296 0.715809i $$-0.253942\pi$$
0.698296 + 0.715809i $$0.253942\pi$$
$$480$$ 0 0
$$481$$ −27.8018 −1.26765
$$482$$ 0.764142i 0.0348057i
$$483$$ 4.41970i 0.201104i
$$484$$ −30.8513 −1.40233
$$485$$ 0 0
$$486$$ 9.45734 0.428994
$$487$$ − 21.1367i − 0.957797i −0.877870 0.478899i $$-0.841036\pi$$
0.877870 0.478899i $$-0.158964\pi$$
$$488$$ 13.5069i 0.611428i
$$489$$ 11.6317 0.526004
$$490$$ 0 0
$$491$$ −28.0514 −1.26594 −0.632971 0.774175i $$-0.718165\pi$$
−0.632971 + 0.774175i $$0.718165\pi$$
$$492$$ 8.02571i 0.361827i
$$493$$ 2.96237i 0.133418i
$$494$$ −12.1958 −0.548714
$$495$$ 0 0
$$496$$ −5.95044 −0.267183
$$497$$ 7.92905i 0.355667i
$$498$$ − 12.9385i − 0.579789i
$$499$$ −3.80178 −0.170191 −0.0850954 0.996373i $$-0.527120\pi$$
−0.0850954 + 0.996373i $$0.527120\pi$$
$$500$$ 0 0
$$501$$ −11.1692 −0.499005
$$502$$ − 22.5402i − 1.00602i
$$503$$ − 19.0872i − 0.851056i −0.904945 0.425528i $$-0.860088\pi$$
0.904945 0.425528i $$-0.139912\pi$$
$$504$$ 2.93420 0.130700
$$505$$ 0 0
$$506$$ 6.46926 0.287594
$$507$$ 3.73083i 0.165692i
$$508$$ − 20.9385i − 0.928997i
$$509$$ 9.41024 0.417101 0.208551 0.978012i $$-0.433125\pi$$
0.208551 + 0.978012i $$0.433125\pi$$
$$510$$ 0 0
$$511$$ 18.2172 0.805880
$$512$$ 1.00000i 0.0441942i
$$513$$ 17.4598i 0.770869i
$$514$$ 9.90089 0.436709
$$515$$ 0 0
$$516$$ 11.4530 0.504191
$$517$$ − 25.2359i − 1.10987i
$$518$$ 21.7265i 0.954608i
$$519$$ 27.6770 1.21488
$$520$$ 0 0
$$521$$ 25.8018 1.13040 0.565198 0.824955i $$-0.308800\pi$$
0.565198 + 0.824955i $$0.308800\pi$$
$$522$$ − 0.820475i − 0.0359112i
$$523$$ 19.1129i 0.835749i 0.908505 + 0.417874i $$0.137225\pi$$
−0.908505 + 0.417874i $$0.862775\pi$$
$$524$$ −10.7641 −0.470234
$$525$$ 0 0
$$526$$ −7.25725 −0.316431
$$527$$ 20.4197i 0.889496i
$$528$$ 9.26157i 0.403058i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −6.52314 −0.283080
$$532$$ 9.53074i 0.413210i
$$533$$ − 22.1462i − 0.959259i
$$534$$ −24.0000 −1.03858
$$535$$ 0 0
$$536$$ −10.0753 −0.435185
$$537$$ 3.11290i 0.134332i
$$538$$ 6.93852i 0.299141i
$$539$$ 16.3720 0.705193
$$540$$ 0 0
$$541$$ 30.8394 1.32589 0.662945 0.748668i $$-0.269306\pi$$
0.662945 + 0.748668i $$0.269306\pi$$
$$542$$ − 10.2992i − 0.442389i
$$543$$ 10.3959i 0.446129i
$$544$$ 3.43163 0.147130
$$545$$ 0 0
$$546$$ 17.4598 0.747210
$$547$$ 13.8513i 0.592240i 0.955151 + 0.296120i $$0.0956929\pi$$
−0.955151 + 0.296120i $$0.904307\pi$$
$$548$$ − 3.26157i − 0.139327i
$$549$$ −12.8375 −0.547893
$$550$$ 0 0
$$551$$ 2.66503 0.113534
$$552$$ − 1.43163i − 0.0609341i
$$553$$ − 48.7831i − 2.07447i
$$554$$ 17.8018 0.756325
$$555$$ 0 0
$$556$$ −16.9385 −0.718353
$$557$$ − 28.7641i − 1.21878i −0.792872 0.609388i $$-0.791415\pi$$
0.792872 0.609388i $$-0.208585\pi$$
$$558$$ − 5.65556i − 0.239419i
$$559$$ −31.6036 −1.33669
$$560$$ 0 0
$$561$$ 31.7823 1.34185
$$562$$ − 18.9385i − 0.798873i
$$563$$ − 36.1505i − 1.52356i −0.647834 0.761782i $$-0.724325\pi$$
0.647834 0.761782i $$-0.275675\pi$$
$$564$$ −5.58462 −0.235155
$$565$$ 0 0
$$566$$ 12.6889 0.533353
$$567$$ − 16.1933i − 0.680055i
$$568$$ − 2.56837i − 0.107767i
$$569$$ 10.3488 0.433843 0.216921 0.976189i $$-0.430399\pi$$
0.216921 + 0.976189i $$0.430399\pi$$
$$570$$ 0 0
$$571$$ 19.6060 0.820486 0.410243 0.911976i $$-0.365444\pi$$
0.410243 + 0.911976i $$0.365444\pi$$
$$572$$ − 25.5565i − 1.06857i
$$573$$ − 12.2973i − 0.513729i
$$574$$ −17.3068 −0.722372
$$575$$ 0 0
$$576$$ −0.950444 −0.0396018
$$577$$ − 34.1505i − 1.42171i −0.703341 0.710853i $$-0.748309\pi$$
0.703341 0.710853i $$-0.251691\pi$$
$$578$$ 5.22394i 0.217287i
$$579$$ −3.50444 −0.145639
$$580$$ 0 0
$$581$$ 27.9009 1.15752
$$582$$ − 20.4649i − 0.848299i
$$583$$ − 38.8156i − 1.60758i
$$584$$ −5.90089 −0.244180
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ − 5.19062i − 0.214240i −0.994246 0.107120i $$-0.965837\pi$$
0.994246 0.107120i $$-0.0341629\pi$$
$$588$$ − 3.62308i − 0.149413i
$$589$$ 18.3701 0.756929
$$590$$ 0 0
$$591$$ −15.3796 −0.632633
$$592$$ − 7.03763i − 0.289245i
$$593$$ 2.09911i 0.0862002i 0.999071 + 0.0431001i $$0.0137234\pi$$
−0.999071 + 0.0431001i $$0.986277\pi$$
$$594$$ −36.5873 −1.50120
$$595$$ 0 0
$$596$$ 3.26157 0.133599
$$597$$ − 16.1933i − 0.662748i
$$598$$ 3.95044i 0.161546i
$$599$$ −11.1581 −0.455909 −0.227955 0.973672i $$-0.573204\pi$$
−0.227955 + 0.973672i $$0.573204\pi$$
$$600$$ 0 0
$$601$$ 9.57784 0.390688 0.195344 0.980735i $$-0.437418\pi$$
0.195344 + 0.980735i $$0.437418\pi$$
$$602$$ 24.6975i 1.00660i
$$603$$ − 9.57597i − 0.389964i
$$604$$ −0.294881 −0.0119986
$$605$$ 0 0
$$606$$ 1.23586 0.0502033
$$607$$ − 24.6975i − 1.00244i −0.865320 0.501221i $$-0.832885\pi$$
0.865320 0.501221i $$-0.167115\pi$$
$$608$$ − 3.08719i − 0.125202i
$$609$$ −3.81533 −0.154605
$$610$$ 0 0
$$611$$ 15.4102 0.623431
$$612$$ 3.26157i 0.131841i
$$613$$ 26.3488i 1.06422i 0.846676 + 0.532108i $$0.178600\pi$$
−0.846676 + 0.532108i $$0.821400\pi$$
$$614$$ 22.9599 0.926587
$$615$$ 0 0
$$616$$ −19.9718 −0.804688
$$617$$ 5.94612i 0.239382i 0.992811 + 0.119691i $$0.0381904\pi$$
−0.992811 + 0.119691i $$0.961810\pi$$
$$618$$ − 2.19145i − 0.0881530i
$$619$$ −39.6856 −1.59510 −0.797549 0.603254i $$-0.793871\pi$$
−0.797549 + 0.603254i $$0.793871\pi$$
$$620$$ 0 0
$$621$$ 5.65556 0.226950
$$622$$ 18.0753i 0.724752i
$$623$$ − 51.7541i − 2.07348i
$$624$$ −5.65556 −0.226404
$$625$$ 0 0
$$626$$ 15.1625 0.606014
$$627$$ − 28.5922i − 1.14186i
$$628$$ − 7.13675i − 0.284787i
$$629$$ −24.1505 −0.962945
$$630$$ 0 0
$$631$$ −23.0138 −0.916164 −0.458082 0.888910i $$-0.651463\pi$$
−0.458082 + 0.888910i $$0.651463\pi$$
$$632$$ 15.8018i 0.628561i
$$633$$ − 33.0891i − 1.31517i
$$634$$ −14.0257 −0.557032
$$635$$ 0 0
$$636$$ −8.58976 −0.340606
$$637$$ 9.99754i 0.396117i
$$638$$ 5.58462i 0.221097i
$$639$$ 2.44109 0.0965682
$$640$$ 0 0
$$641$$ 25.4617 1.00568 0.502838 0.864381i $$-0.332289\pi$$
0.502838 + 0.864381i $$0.332289\pi$$
$$642$$ 25.2359i 0.995980i
$$643$$ 16.4479i 0.648641i 0.945947 + 0.324320i $$0.105136\pi$$
−0.945947 + 0.324320i $$0.894864\pi$$
$$644$$ 3.08719 0.121652
$$645$$ 0 0
$$646$$ −10.5941 −0.416819
$$647$$ − 34.9147i − 1.37264i −0.727301 0.686319i $$-0.759226\pi$$
0.727301 0.686319i $$-0.240774\pi$$
$$648$$ 5.24533i 0.206056i
$$649$$ 44.4002 1.74286
$$650$$ 0 0
$$651$$ −26.2992 −1.03075
$$652$$ − 8.12482i − 0.318193i
$$653$$ 34.2754i 1.34130i 0.741775 + 0.670649i $$0.233984\pi$$
−0.741775 + 0.670649i $$0.766016\pi$$
$$654$$ 4.17006 0.163062
$$655$$ 0 0
$$656$$ 5.60601 0.218878
$$657$$ − 5.60846i − 0.218807i
$$658$$ − 12.0428i − 0.469476i
$$659$$ −35.2548 −1.37333 −0.686666 0.726973i $$-0.740926\pi$$
−0.686666 + 0.726973i $$0.740926\pi$$
$$660$$ 0 0
$$661$$ 28.5684 1.11118 0.555590 0.831456i $$-0.312492\pi$$
0.555590 + 0.831456i $$0.312492\pi$$
$$662$$ − 24.7403i − 0.961559i
$$663$$ 19.4078i 0.753736i
$$664$$ −9.03763 −0.350728
$$665$$ 0 0
$$666$$ 6.68888 0.259189
$$667$$ − 0.863254i − 0.0334253i
$$668$$ 7.80178i 0.301860i
$$669$$ 8.19822 0.316962
$$670$$ 0 0
$$671$$ 87.3796 3.37325
$$672$$ 4.41970i 0.170494i
$$673$$ 23.8770i 0.920392i 0.887817 + 0.460196i $$0.152221\pi$$
−0.887817 + 0.460196i $$0.847779\pi$$
$$674$$ −14.7146 −0.566785
$$675$$ 0 0
$$676$$ 2.60601 0.100231
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 8.58976i 0.329888i
$$679$$ 44.1310 1.69359
$$680$$ 0 0
$$681$$ −25.2359 −0.967040
$$682$$ 38.4950i 1.47405i
$$683$$ − 34.6480i − 1.32577i −0.748722 0.662884i $$-0.769332\pi$$
0.748722 0.662884i $$-0.230668\pi$$
$$684$$ 2.93420 0.112192
$$685$$ 0 0
$$686$$ −13.7975 −0.526789
$$687$$ − 23.0138i − 0.878031i
$$688$$ − 8.00000i − 0.304997i
$$689$$ 23.7027 0.903000
$$690$$ 0 0
$$691$$ −18.1744 −0.691386 −0.345693 0.938348i $$-0.612356\pi$$
−0.345693 + 0.938348i $$0.612356\pi$$
$$692$$ − 19.3325i − 0.734912i
$$693$$ − 18.9821i − 0.721071i
$$694$$ −9.43163 −0.358020
$$695$$ 0 0
$$696$$ 1.23586 0.0468451
$$697$$ − 19.2377i − 0.728681i
$$698$$ − 18.3488i − 0.694511i
$$699$$ 9.23095 0.349146
$$700$$ 0 0
$$701$$ −40.6907 −1.53687 −0.768434 0.639929i $$-0.778964\pi$$
−0.768434 + 0.639929i $$0.778964\pi$$
$$702$$ − 22.3420i − 0.843244i
$$703$$ 21.7265i 0.819431i
$$704$$ 6.46926 0.243819
$$705$$ 0 0
$$706$$ −33.1129 −1.24622
$$707$$ 2.66503i 0.100229i
$$708$$ − 9.82562i − 0.369269i
$$709$$ 26.4454 0.993178 0.496589 0.867986i $$-0.334586\pi$$
0.496589 + 0.867986i $$0.334586\pi$$
$$710$$ 0 0
$$711$$ −15.0187 −0.563245
$$712$$ 16.7641i 0.628263i
$$713$$ − 5.95044i − 0.222846i
$$714$$ 15.1668 0.567602
$$715$$ 0 0
$$716$$ 2.17438 0.0812604
$$717$$ 6.22580i 0.232507i
$$718$$ − 33.0376i − 1.23295i
$$719$$ 2.24778 0.0838281 0.0419140 0.999121i $$-0.486654\pi$$
0.0419140 + 0.999121i $$0.486654\pi$$
$$720$$ 0 0
$$721$$ 4.72568 0.175994
$$722$$ − 9.46926i − 0.352409i
$$723$$ − 1.09397i − 0.0406850i
$$724$$ 7.26157 0.269874
$$725$$ 0 0
$$726$$ 44.1676 1.63921
$$727$$ 21.4830i 0.796762i 0.917220 + 0.398381i $$0.130428\pi$$
−0.917220 + 0.398381i $$0.869572\pi$$
$$728$$ − 12.1958i − 0.452005i
$$729$$ −29.2754 −1.08427
$$730$$ 0 0
$$731$$ −27.4530 −1.01539
$$732$$ − 19.3368i − 0.714710i
$$733$$ − 11.3778i − 0.420247i −0.977675 0.210123i $$-0.932613\pi$$
0.977675 0.210123i $$-0.0673866\pi$$
$$734$$ 2.27349 0.0839161
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 65.1795i 2.40092i
$$738$$ 5.32819i 0.196134i
$$739$$ −21.8770 −0.804760 −0.402380 0.915473i $$-0.631817\pi$$
−0.402380 + 0.915473i $$0.631817\pi$$
$$740$$ 0 0
$$741$$ 17.4598 0.641402
$$742$$ − 18.5231i − 0.680006i
$$743$$ − 5.36068i − 0.196664i −0.995154 0.0983322i $$-0.968649\pi$$
0.995154 0.0983322i $$-0.0313508\pi$$
$$744$$ 8.51882 0.312315
$$745$$ 0 0
$$746$$ 23.9762 0.877829
$$747$$ − 8.58976i − 0.314283i
$$748$$ − 22.2001i − 0.811716i
$$749$$ −54.4191 −1.98843
$$750$$ 0 0
$$751$$ 24.3915 0.890060 0.445030 0.895516i $$-0.353193\pi$$
0.445030 + 0.895516i $$0.353193\pi$$
$$752$$ 3.90089i 0.142251i
$$753$$ 32.2692i 1.17595i
$$754$$ −3.41024 −0.124194
$$755$$ 0 0
$$756$$ −17.4598 −0.635007
$$757$$ 41.9437i 1.52447i 0.647301 + 0.762234i $$0.275898\pi$$
−0.647301 + 0.762234i $$0.724102\pi$$
$$758$$ 13.8795i 0.504126i
$$759$$ −9.26157 −0.336174
$$760$$ 0 0
$$761$$ −18.3745 −0.666074 −0.333037 0.942914i $$-0.608073\pi$$
−0.333037 + 0.942914i $$0.608073\pi$$
$$762$$ 29.9762i 1.08592i
$$763$$ 8.99240i 0.325547i
$$764$$ −8.58976 −0.310767
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 27.1129i 0.978990i
$$768$$ − 1.43163i − 0.0516594i
$$769$$ −42.4993 −1.53256 −0.766282 0.642505i $$-0.777895\pi$$
−0.766282 + 0.642505i $$0.777895\pi$$
$$770$$ 0 0
$$771$$ −14.1744 −0.510478
$$772$$ 2.44787i 0.0881008i
$$773$$ 15.0376i 0.540866i 0.962739 + 0.270433i $$0.0871669\pi$$
−0.962739 + 0.270433i $$0.912833\pi$$
$$774$$ 7.60355 0.273304
$$775$$ 0 0
$$776$$ −14.2949 −0.513156
$$777$$ − 31.1043i − 1.11586i
$$778$$ 26.6436i 0.955221i
$$779$$ −17.3068 −0.620081
$$780$$ 0 0
$$781$$ −16.6155 −0.594548
$$782$$ 3.43163i 0.122715i
$$783$$ 4.88219i 0.174475i
$$784$$ −2.53074 −0.0903836
$$785$$ 0 0
$$786$$ 15.4102 0.549665
$$787$$ − 8.39154i − 0.299126i −0.988752 0.149563i $$-0.952213\pi$$
0.988752 0.149563i $$-0.0477867\pi$$
$$788$$ 10.7428i 0.382695i
$$789$$ 10.3897 0.369882
$$790$$ 0 0
$$791$$ −18.5231 −0.658607
$$792$$ 6.14867i 0.218483i
$$793$$ 53.3582i 1.89481i
$$794$$ 8.66749 0.307598
$$795$$ 0 0
$$796$$ −11.3111 −0.400912
$$797$$ − 22.0514i − 0.781101i −0.920581 0.390551i $$-0.872285\pi$$
0.920581 0.390551i $$-0.127715\pi$$
$$798$$ − 13.6445i − 0.483009i
$$799$$ 13.3864 0.473576
$$800$$ 0 0
$$801$$ −15.9334 −0.562978
$$802$$ 39.9762i 1.41161i
$$803$$ 38.1744i 1.34714i
$$804$$ 14.4240 0.508696
$$805$$ 0 0
$$806$$ −23.5069 −0.827995
$$807$$ − 9.93337i − 0.349671i
$$808$$ − 0.863254i − 0.0303692i
$$809$$ −2.04524 −0.0719066 −0.0359533 0.999353i $$-0.511447\pi$$
−0.0359533 + 0.999353i $$0.511447\pi$$
$$810$$ 0 0
$$811$$ 4.24965 0.149225 0.0746126 0.997213i $$-0.476228\pi$$
0.0746126 + 0.997213i $$0.476228\pi$$
$$812$$ 2.66503i 0.0935242i
$$813$$ 14.7446i 0.517116i
$$814$$ −45.5283 −1.59577
$$815$$ 0 0
$$816$$ −4.91281 −0.171983
$$817$$ 24.6975i 0.864057i
$$818$$ − 30.1248i − 1.05329i
$$819$$ 11.5914 0.405036
$$820$$ 0 0
$$821$$ −29.2548 −1.02100 −0.510500 0.859878i $$-0.670540\pi$$
−0.510500 + 0.859878i $$0.670540\pi$$
$$822$$ 4.66935i 0.162862i
$$823$$ − 13.5846i − 0.473530i −0.971567 0.236765i $$-0.923913\pi$$
0.971567 0.236765i $$-0.0760871\pi$$
$$824$$ −1.53074 −0.0533258
$$825$$ 0 0
$$826$$ 21.1882 0.737231
$$827$$ 24.7880i 0.861963i 0.902361 + 0.430981i $$0.141833\pi$$
−0.902361 + 0.430981i $$0.858167\pi$$
$$828$$ − 0.950444i − 0.0330302i
$$829$$ −26.1505 −0.908246 −0.454123 0.890939i $$-0.650047\pi$$
−0.454123 + 0.890939i $$0.650047\pi$$
$$830$$ 0 0
$$831$$ −25.4855 −0.884082
$$832$$ 3.95044i 0.136957i
$$833$$ 8.68455i 0.300902i
$$834$$ 24.2496 0.839697
$$835$$ 0 0
$$836$$ −19.9718 −0.690740
$$837$$ 33.6531i 1.16322i
$$838$$ − 25.8770i − 0.893908i
$$839$$ −6.37260 −0.220007 −0.110003 0.993931i $$-0.535086\pi$$
−0.110003 + 0.993931i $$0.535086\pi$$
$$840$$ 0 0
$$841$$ −28.2548 −0.974303
$$842$$ − 10.7146i − 0.369249i
$$843$$ 27.1129i 0.933818i
$$844$$ −23.1129 −0.795579
$$845$$ 0 0
$$846$$ −3.70757 −0.127469
$$847$$ 95.2439i 3.27262i
$$848$$ 6.00000i 0.206041i
$$849$$ −18.1657 −0.623447
$$850$$ 0 0
$$851$$ 7.03763 0.241247
$$852$$ 3.67695i 0.125970i
$$853$$ 33.6293i 1.15144i 0.817646 + 0.575722i $$0.195279\pi$$
−0.817646 + 0.575722i $$0.804721\pi$$
$$854$$ 41.6983 1.42689
$$855$$ 0 0
$$856$$ 17.6274 0.602492
$$857$$ 11.1795i 0.381885i 0.981601 + 0.190943i $$0.0611545\pi$$
−0.981601 + 0.190943i $$0.938846\pi$$
$$858$$ 36.5873i 1.24907i
$$859$$ −47.9009 −1.63436 −0.817179 0.576385i $$-0.804463\pi$$
−0.817179 + 0.576385i $$0.804463\pi$$
$$860$$ 0 0
$$861$$ 24.7769 0.844394
$$862$$ 32.2496i 1.09843i
$$863$$ 30.1180i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$864$$ 5.65556 0.192406
$$865$$ 0 0
$$866$$ 20.6393 0.701353
$$867$$ − 7.47873i − 0.253991i
$$868$$ 18.3701i 0.623523i
$$869$$ 102.226 3.46777
$$870$$ 0 0
$$871$$ −39.8018 −1.34863
$$872$$ − 2.91281i − 0.0986402i
$$873$$ − 13.5865i − 0.459833i
$$874$$ 3.08719 0.104426
$$875$$ 0 0
$$876$$ 8.44787 0.285427
$$877$$ − 22.3940i − 0.756191i −0.925767 0.378096i $$-0.876579\pi$$
0.925767 0.378096i $$-0.123421\pi$$
$$878$$ 16.2239i 0.547531i
$$879$$ 8.58976 0.289726
$$880$$ 0 0
$$881$$ 21.1129 0.711312 0.355656 0.934617i $$-0.384258\pi$$
0.355656 + 0.934617i $$0.384258\pi$$
$$882$$ − 2.40533i − 0.0809915i
$$883$$ − 49.1061i − 1.65255i −0.563265 0.826276i $$-0.690455\pi$$
0.563265 0.826276i $$-0.309545\pi$$
$$884$$ 13.5565 0.455953
$$885$$ 0 0
$$886$$ −15.6770 −0.526678
$$887$$ − 43.0566i − 1.44570i −0.691006 0.722849i $$-0.742832\pi$$
0.691006 0.722849i $$-0.257168\pi$$
$$888$$ 10.0753i 0.338104i
$$889$$ −64.6412 −2.16800
$$890$$ 0 0
$$891$$ 33.9334 1.13681
$$892$$ − 5.72651i − 0.191738i
$$893$$ − 12.0428i − 0.402996i
$$894$$ −4.66935 −0.156166
$$895$$ 0 0
$$896$$ 3.08719 0.103136
$$897$$ − 5.65556i − 0.188834i
$$898$$ − 22.5727i − 0.753261i
$$899$$ 5.13675 0.171320
$$900$$ 0 0
$$901$$ 20.5898 0.685944
$$902$$ − 36.2667i − 1.20755i
$$903$$ − 35.3576i − 1.17663i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0.422160 0.0140253
$$907$$ − 37.9762i − 1.26098i −0.776198 0.630489i $$-0.782854\pi$$
0.776198 0.630489i $$-0.217146\pi$$
$$908$$ 17.6274i 0.584986i
$$909$$ 0.820475 0.0272134
$$910$$ 0 0
$$911$$ −15.7504 −0.521833 −0.260916 0.965361i $$-0.584025\pi$$
−0.260916 + 0.965361i $$0.584025\pi$$
$$912$$ 4.41970i 0.146351i
$$913$$ 58.4668i 1.93497i
$$914$$ 1.45302 0.0480615
$$915$$ 0 0
$$916$$ −16.0753 −0.531142
$$917$$ 33.2309i 1.09738i
$$918$$ − 19.4078i − 0.640552i
$$919$$ −30.4240 −1.00360 −0.501798 0.864985i $$-0.667328\pi$$
−0.501798 + 0.864985i $$0.667328\pi$$
$$920$$ 0 0
$$921$$ −32.8700 −1.08310
$$922$$ − 29.7027i − 0.978205i
$$923$$ − 10.1462i − 0.333967i
$$924$$ 28.5922 0.940615
$$925$$ 0 0
$$926$$ 22.9624 0.754590
$$927$$ − 1.45488i − 0.0477846i
$$928$$ − 0.863254i − 0.0283377i
$$929$$ 37.8018 1.24024 0.620118 0.784509i $$-0.287085\pi$$
0.620118 + 0.784509i $$0.287085\pi$$
$$930$$ 0 0
$$931$$ 7.81287 0.256057
$$932$$ − 6.44787i − 0.211207i
$$933$$ − 25.8770i − 0.847176i
$$934$$ −9.48550 −0.310375
$$935$$ 0 0
$$936$$ −3.75467 −0.122725
$$937$$ − 32.6907i − 1.06796i −0.845497 0.533980i $$-0.820696\pi$$
0.845497 0.533980i $$-0.179304\pi$$
$$938$$ 31.1043i 1.01559i
$$939$$ −21.7070 −0.708381
$$940$$ 0 0
$$941$$ −46.4882 −1.51547 −0.757736 0.652561i $$-0.773694\pi$$
−0.757736 + 0.652561i $$0.773694\pi$$
$$942$$ 10.2172i 0.332893i
$$943$$ 5.60601i 0.182557i
$$944$$ −6.86325 −0.223380
$$945$$ 0 0
$$946$$ −51.7541 −1.68267
$$947$$ − 37.2052i − 1.20901i −0.796602 0.604504i $$-0.793371\pi$$
0.796602 0.604504i $$-0.206629\pi$$
$$948$$ − 22.6222i − 0.734737i
$$949$$ −23.3111 −0.756711
$$950$$ 0 0
$$951$$ 20.0796 0.651125
$$952$$ − 10.5941i − 0.343356i
$$953$$ − 49.1104i − 1.59084i −0.606056 0.795422i $$-0.707249\pi$$
0.606056 0.795422i $$-0.292751\pi$$
$$954$$ −5.70266 −0.184631
$$955$$ 0 0
$$956$$ 4.34876 0.140649
$$957$$ − 7.99509i − 0.258445i
$$958$$ 30.5659i 0.987540i
$$959$$ −10.0691 −0.325148
$$960$$ 0 0
$$961$$ 4.40778 0.142187
$$962$$ − 27.8018i − 0.896365i
$$963$$ 16.7538i 0.539885i
$$964$$ −0.764142 −0.0246114
$$965$$ 0 0
$$966$$ −4.41970 −0.142202
$$967$$ 1.96751i 0.0632709i 0.999499 + 0.0316355i $$0.0100716\pi$$
−0.999499 + 0.0316355i $$0.989928\pi$$
$$968$$ − 30.8513i − 0.991599i
$$969$$ 15.1668 0.487227
$$970$$ 0 0
$$971$$ 26.1205 0.838247 0.419123 0.907929i $$-0.362337\pi$$
0.419123 + 0.907929i $$0.362337\pi$$
$$972$$ 9.45734i 0.303344i
$$973$$ 52.2924i 1.67642i
$$974$$ 21.1367 0.677265
$$975$$ 0 0
$$976$$ −13.5069 −0.432345
$$977$$ − 0.639319i − 0.0204536i −0.999948 0.0102268i $$-0.996745\pi$$
0.999948 0.0102268i $$-0.00325535\pi$$
$$978$$ 11.6317i 0.371941i
$$979$$ 108.452 3.46613
$$980$$ 0 0
$$981$$ 2.76846 0.0883902
$$982$$ − 28.0514i − 0.895157i
$$983$$ 6.46926i 0.206337i 0.994664 + 0.103169i $$0.0328981\pi$$
−0.994664 + 0.103169i $$0.967102\pi$$
$$984$$ −8.02571 −0.255850
$$985$$ 0 0
$$986$$ −2.96237 −0.0943410
$$987$$ 17.2408i 0.548780i
$$988$$ − 12.1958i − 0.387999i
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 37.2334 1.18276 0.591379 0.806394i $$-0.298584\pi$$
0.591379 + 0.806394i $$0.298584\pi$$
$$992$$ − 5.95044i − 0.188927i
$$993$$ 35.4189i 1.12398i
$$994$$ −7.92905 −0.251494
$$995$$ 0 0
$$996$$ 12.9385 0.409973
$$997$$ 9.65124i 0.305658i 0.988253 + 0.152829i $$0.0488384\pi$$
−0.988253 + 0.152829i $$0.951162\pi$$
$$998$$ − 3.80178i − 0.120343i
$$999$$ −39.8018 −1.25927
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 1150.2.b.j.599.5 6
5.2 odd 4 1150.2.a.q.1.2 3
5.3 odd 4 230.2.a.d.1.2 3
5.4 even 2 inner 1150.2.b.j.599.2 6
15.8 even 4 2070.2.a.z.1.2 3
20.3 even 4 1840.2.a.r.1.2 3
20.7 even 4 9200.2.a.cf.1.2 3
40.3 even 4 7360.2.a.ce.1.2 3
40.13 odd 4 7360.2.a.bz.1.2 3
115.68 even 4 5290.2.a.r.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.2 3 5.3 odd 4
1150.2.a.q.1.2 3 5.2 odd 4
1150.2.b.j.599.2 6 5.4 even 2 inner
1150.2.b.j.599.5 6 1.1 even 1 trivial
1840.2.a.r.1.2 3 20.3 even 4
2070.2.a.z.1.2 3 15.8 even 4
5290.2.a.r.1.2 3 115.68 even 4
7360.2.a.bz.1.2 3 40.13 odd 4
7360.2.a.ce.1.2 3 40.3 even 4
9200.2.a.cf.1.2 3 20.7 even 4