# Properties

 Label 1805.2.a.m.1.2 Level $1805$ Weight $2$ Character 1805.1 Self dual yes Analytic conductor $14.413$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7168.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 6x^{2} + 7$$ x^4 - 6*x^2 + 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.25928$$ of defining polynomial Character $$\chi$$ $$=$$ 1805.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +1.00000 q^{5} +2.24264 q^{6} -0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} +O(q^{10})$$ $$q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +1.00000 q^{5} +2.24264 q^{6} -0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} -1.25928 q^{10} +2.00000 q^{11} +0.737669 q^{12} -4.29945 q^{13} +1.04322 q^{14} -1.78089 q^{15} -3.00000 q^{16} +7.65685 q^{17} -0.216058 q^{18} -0.414214 q^{20} +1.47534 q^{21} -2.51856 q^{22} -0.828427 q^{23} -5.41421 q^{24} +1.00000 q^{25} +5.41421 q^{26} +5.03712 q^{27} +0.343146 q^{28} -8.59890 q^{29} +2.24264 q^{30} -3.56178 q^{31} -2.30250 q^{32} -3.56178 q^{33} -9.64212 q^{34} -0.828427 q^{35} -0.0710678 q^{36} -0.737669 q^{37} +7.65685 q^{39} +3.04017 q^{40} -1.85786 q^{42} -4.82843 q^{43} -0.828427 q^{44} +0.171573 q^{45} +1.04322 q^{46} +10.4853 q^{47} +5.34267 q^{48} -6.31371 q^{49} -1.25928 q^{50} -13.6360 q^{51} +1.78089 q^{52} +7.86123 q^{53} -6.34315 q^{54} +2.00000 q^{55} -2.51856 q^{56} +10.8284 q^{58} +7.12356 q^{59} +0.737669 q^{60} -2.82843 q^{61} +4.48528 q^{62} -0.142136 q^{63} +8.89949 q^{64} -4.29945 q^{65} +4.48528 q^{66} -6.81801 q^{67} -3.17157 q^{68} +1.47534 q^{69} +1.04322 q^{70} -13.6360 q^{71} +0.521611 q^{72} -11.6569 q^{73} +0.928932 q^{74} -1.78089 q^{75} -1.65685 q^{77} -9.64212 q^{78} -7.12356 q^{79} -3.00000 q^{80} -9.48528 q^{81} +8.82843 q^{83} -0.611105 q^{84} +7.65685 q^{85} +6.08034 q^{86} +15.3137 q^{87} +6.08034 q^{88} +15.7225 q^{89} -0.216058 q^{90} +3.56178 q^{91} +0.343146 q^{92} +6.34315 q^{93} -13.2039 q^{94} +4.10051 q^{96} -4.29945 q^{97} +7.95073 q^{98} +0.343146 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 4 q^{5} - 8 q^{6} + 8 q^{7} + 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 + 4 * q^5 - 8 * q^6 + 8 * q^7 + 12 * q^9 $$4 q + 4 q^{4} + 4 q^{5} - 8 q^{6} + 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} + 8 q^{17} + 4 q^{20} + 8 q^{23} - 16 q^{24} + 4 q^{25} + 16 q^{26} + 24 q^{28} - 8 q^{30} + 8 q^{35} + 28 q^{36} + 8 q^{39} - 64 q^{42} - 8 q^{43} + 8 q^{44} + 12 q^{45} + 8 q^{47} + 20 q^{49} - 48 q^{54} + 8 q^{55} + 32 q^{58} - 16 q^{62} + 56 q^{63} - 4 q^{64} - 16 q^{66} - 24 q^{68} - 24 q^{73} + 32 q^{74} + 16 q^{77} - 12 q^{80} - 4 q^{81} + 24 q^{83} + 8 q^{85} + 16 q^{87} + 24 q^{92} + 48 q^{93} + 56 q^{96} + 24 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 + 4 * q^5 - 8 * q^6 + 8 * q^7 + 12 * q^9 + 8 * q^11 - 12 * q^16 + 8 * q^17 + 4 * q^20 + 8 * q^23 - 16 * q^24 + 4 * q^25 + 16 * q^26 + 24 * q^28 - 8 * q^30 + 8 * q^35 + 28 * q^36 + 8 * q^39 - 64 * q^42 - 8 * q^43 + 8 * q^44 + 12 * q^45 + 8 * q^47 + 20 * q^49 - 48 * q^54 + 8 * q^55 + 32 * q^58 - 16 * q^62 + 56 * q^63 - 4 * q^64 - 16 * q^66 - 24 * q^68 - 24 * q^73 + 32 * q^74 + 16 * q^77 - 12 * q^80 - 4 * q^81 + 24 * q^83 + 8 * q^85 + 16 * q^87 + 24 * q^92 + 48 * q^93 + 56 * q^96 + 24 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.25928 −0.890446 −0.445223 0.895420i $$-0.646876\pi$$
−0.445223 + 0.895420i $$0.646876\pi$$
$$3$$ −1.78089 −1.02820 −0.514099 0.857731i $$-0.671874\pi$$
−0.514099 + 0.857731i $$0.671874\pi$$
$$4$$ −0.414214 −0.207107
$$5$$ 1.00000 0.447214
$$6$$ 2.24264 0.915554
$$7$$ −0.828427 −0.313116 −0.156558 0.987669i $$-0.550040\pi$$
−0.156558 + 0.987669i $$0.550040\pi$$
$$8$$ 3.04017 1.07486
$$9$$ 0.171573 0.0571910
$$10$$ −1.25928 −0.398219
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0.737669 0.212947
$$13$$ −4.29945 −1.19245 −0.596227 0.802816i $$-0.703334\pi$$
−0.596227 + 0.802816i $$0.703334\pi$$
$$14$$ 1.04322 0.278813
$$15$$ −1.78089 −0.459824
$$16$$ −3.00000 −0.750000
$$17$$ 7.65685 1.85706 0.928530 0.371257i $$-0.121073\pi$$
0.928530 + 0.371257i $$0.121073\pi$$
$$18$$ −0.216058 −0.0509254
$$19$$ 0 0
$$20$$ −0.414214 −0.0926210
$$21$$ 1.47534 0.321945
$$22$$ −2.51856 −0.536959
$$23$$ −0.828427 −0.172739 −0.0863695 0.996263i $$-0.527527\pi$$
−0.0863695 + 0.996263i $$0.527527\pi$$
$$24$$ −5.41421 −1.10517
$$25$$ 1.00000 0.200000
$$26$$ 5.41421 1.06181
$$27$$ 5.03712 0.969394
$$28$$ 0.343146 0.0648485
$$29$$ −8.59890 −1.59678 −0.798388 0.602143i $$-0.794314\pi$$
−0.798388 + 0.602143i $$0.794314\pi$$
$$30$$ 2.24264 0.409448
$$31$$ −3.56178 −0.639715 −0.319857 0.947466i $$-0.603635\pi$$
−0.319857 + 0.947466i $$0.603635\pi$$
$$32$$ −2.30250 −0.407029
$$33$$ −3.56178 −0.620027
$$34$$ −9.64212 −1.65361
$$35$$ −0.828427 −0.140030
$$36$$ −0.0710678 −0.0118446
$$37$$ −0.737669 −0.121272 −0.0606360 0.998160i $$-0.519313\pi$$
−0.0606360 + 0.998160i $$0.519313\pi$$
$$38$$ 0 0
$$39$$ 7.65685 1.22608
$$40$$ 3.04017 0.480693
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ −1.85786 −0.286675
$$43$$ −4.82843 −0.736328 −0.368164 0.929761i $$-0.620014\pi$$
−0.368164 + 0.929761i $$0.620014\pi$$
$$44$$ −0.828427 −0.124890
$$45$$ 0.171573 0.0255766
$$46$$ 1.04322 0.153815
$$47$$ 10.4853 1.52944 0.764718 0.644365i $$-0.222878\pi$$
0.764718 + 0.644365i $$0.222878\pi$$
$$48$$ 5.34267 0.771148
$$49$$ −6.31371 −0.901958
$$50$$ −1.25928 −0.178089
$$51$$ −13.6360 −1.90943
$$52$$ 1.78089 0.246965
$$53$$ 7.86123 1.07982 0.539912 0.841722i $$-0.318458\pi$$
0.539912 + 0.841722i $$0.318458\pi$$
$$54$$ −6.34315 −0.863193
$$55$$ 2.00000 0.269680
$$56$$ −2.51856 −0.336557
$$57$$ 0 0
$$58$$ 10.8284 1.42184
$$59$$ 7.12356 0.927409 0.463705 0.885990i $$-0.346520\pi$$
0.463705 + 0.885990i $$0.346520\pi$$
$$60$$ 0.737669 0.0952327
$$61$$ −2.82843 −0.362143 −0.181071 0.983470i $$-0.557957\pi$$
−0.181071 + 0.983470i $$0.557957\pi$$
$$62$$ 4.48528 0.569631
$$63$$ −0.142136 −0.0179074
$$64$$ 8.89949 1.11244
$$65$$ −4.29945 −0.533281
$$66$$ 4.48528 0.552100
$$67$$ −6.81801 −0.832953 −0.416476 0.909147i $$-0.636735\pi$$
−0.416476 + 0.909147i $$0.636735\pi$$
$$68$$ −3.17157 −0.384610
$$69$$ 1.47534 0.177610
$$70$$ 1.04322 0.124689
$$71$$ −13.6360 −1.61830 −0.809149 0.587603i $$-0.800072\pi$$
−0.809149 + 0.587603i $$0.800072\pi$$
$$72$$ 0.521611 0.0614724
$$73$$ −11.6569 −1.36433 −0.682166 0.731198i $$-0.738962\pi$$
−0.682166 + 0.731198i $$0.738962\pi$$
$$74$$ 0.928932 0.107986
$$75$$ −1.78089 −0.205640
$$76$$ 0 0
$$77$$ −1.65685 −0.188816
$$78$$ −9.64212 −1.09176
$$79$$ −7.12356 −0.801464 −0.400732 0.916195i $$-0.631244\pi$$
−0.400732 + 0.916195i $$0.631244\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ −9.48528 −1.05392
$$82$$ 0 0
$$83$$ 8.82843 0.969046 0.484523 0.874779i $$-0.338993\pi$$
0.484523 + 0.874779i $$0.338993\pi$$
$$84$$ −0.611105 −0.0666770
$$85$$ 7.65685 0.830502
$$86$$ 6.08034 0.655660
$$87$$ 15.3137 1.64180
$$88$$ 6.08034 0.648167
$$89$$ 15.7225 1.66658 0.833289 0.552838i $$-0.186455\pi$$
0.833289 + 0.552838i $$0.186455\pi$$
$$90$$ −0.216058 −0.0227745
$$91$$ 3.56178 0.373376
$$92$$ 0.343146 0.0357754
$$93$$ 6.34315 0.657754
$$94$$ −13.2039 −1.36188
$$95$$ 0 0
$$96$$ 4.10051 0.418506
$$97$$ −4.29945 −0.436543 −0.218272 0.975888i $$-0.570042\pi$$
−0.218272 + 0.975888i $$0.570042\pi$$
$$98$$ 7.95073 0.803145
$$99$$ 0.343146 0.0344874
$$100$$ −0.414214 −0.0414214
$$101$$ 0.485281 0.0482873 0.0241437 0.999708i $$-0.492314\pi$$
0.0241437 + 0.999708i $$0.492314\pi$$
$$102$$ 17.1716 1.70024
$$103$$ 5.34267 0.526429 0.263215 0.964737i $$-0.415217\pi$$
0.263215 + 0.964737i $$0.415217\pi$$
$$104$$ −13.0711 −1.28172
$$105$$ 1.47534 0.143978
$$106$$ −9.89949 −0.961524
$$107$$ 13.9416 1.34778 0.673891 0.738830i $$-0.264622\pi$$
0.673891 + 0.738830i $$0.264622\pi$$
$$108$$ −2.08644 −0.200768
$$109$$ 18.6731 1.78856 0.894281 0.447505i $$-0.147687\pi$$
0.894281 + 0.447505i $$0.147687\pi$$
$$110$$ −2.51856 −0.240135
$$111$$ 1.31371 0.124692
$$112$$ 2.48528 0.234837
$$113$$ 9.33657 0.878311 0.439155 0.898411i $$-0.355278\pi$$
0.439155 + 0.898411i $$0.355278\pi$$
$$114$$ 0 0
$$115$$ −0.828427 −0.0772512
$$116$$ 3.56178 0.330703
$$117$$ −0.737669 −0.0681975
$$118$$ −8.97056 −0.825807
$$119$$ −6.34315 −0.581475
$$120$$ −5.41421 −0.494248
$$121$$ −7.00000 −0.636364
$$122$$ 3.56178 0.322469
$$123$$ 0 0
$$124$$ 1.47534 0.132489
$$125$$ 1.00000 0.0894427
$$126$$ 0.178989 0.0159456
$$127$$ 17.5034 1.55317 0.776586 0.630011i $$-0.216950\pi$$
0.776586 + 0.630011i $$0.216950\pi$$
$$128$$ −6.60195 −0.583536
$$129$$ 8.59890 0.757091
$$130$$ 5.41421 0.474858
$$131$$ 15.3137 1.33796 0.668982 0.743278i $$-0.266730\pi$$
0.668982 + 0.743278i $$0.266730\pi$$
$$132$$ 1.47534 0.128412
$$133$$ 0 0
$$134$$ 8.58579 0.741699
$$135$$ 5.03712 0.433526
$$136$$ 23.2781 1.99608
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ −1.85786 −0.158152
$$139$$ 19.6569 1.66727 0.833636 0.552314i $$-0.186255\pi$$
0.833636 + 0.552314i $$0.186255\pi$$
$$140$$ 0.343146 0.0290011
$$141$$ −18.6731 −1.57256
$$142$$ 17.1716 1.44101
$$143$$ −8.59890 −0.719076
$$144$$ −0.514719 −0.0428932
$$145$$ −8.59890 −0.714100
$$146$$ 14.6792 1.21486
$$147$$ 11.2440 0.927392
$$148$$ 0.305553 0.0251163
$$149$$ −14.8284 −1.21479 −0.607396 0.794399i $$-0.707786\pi$$
−0.607396 + 0.794399i $$0.707786\pi$$
$$150$$ 2.24264 0.183111
$$151$$ 10.6853 0.869561 0.434781 0.900536i $$-0.356826\pi$$
0.434781 + 0.900536i $$0.356826\pi$$
$$152$$ 0 0
$$153$$ 1.31371 0.106207
$$154$$ 2.08644 0.168130
$$155$$ −3.56178 −0.286089
$$156$$ −3.17157 −0.253929
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 8.97056 0.713660
$$159$$ −14.0000 −1.11027
$$160$$ −2.30250 −0.182029
$$161$$ 0.686292 0.0540873
$$162$$ 11.9446 0.938458
$$163$$ 7.17157 0.561721 0.280860 0.959749i $$-0.409380\pi$$
0.280860 + 0.959749i $$0.409380\pi$$
$$164$$ 0 0
$$165$$ −3.56178 −0.277284
$$166$$ −11.1175 −0.862882
$$167$$ 16.8923 1.30716 0.653581 0.756857i $$-0.273266\pi$$
0.653581 + 0.756857i $$0.273266\pi$$
$$168$$ 4.48528 0.346047
$$169$$ 5.48528 0.421945
$$170$$ −9.64212 −0.739517
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ −2.82411 −0.214713 −0.107357 0.994221i $$-0.534239\pi$$
−0.107357 + 0.994221i $$0.534239\pi$$
$$174$$ −19.2842 −1.46194
$$175$$ −0.828427 −0.0626232
$$176$$ −6.00000 −0.452267
$$177$$ −12.6863 −0.953560
$$178$$ −19.7990 −1.48400
$$179$$ −17.1978 −1.28542 −0.642712 0.766108i $$-0.722191\pi$$
−0.642712 + 0.766108i $$0.722191\pi$$
$$180$$ −0.0710678 −0.00529708
$$181$$ 10.0742 0.748812 0.374406 0.927265i $$-0.377847\pi$$
0.374406 + 0.927265i $$0.377847\pi$$
$$182$$ −4.48528 −0.332471
$$183$$ 5.03712 0.372355
$$184$$ −2.51856 −0.185671
$$185$$ −0.737669 −0.0542345
$$186$$ −7.98780 −0.585694
$$187$$ 15.3137 1.11985
$$188$$ −4.34315 −0.316756
$$189$$ −4.17289 −0.303533
$$190$$ 0 0
$$191$$ −2.34315 −0.169544 −0.0847720 0.996400i $$-0.527016\pi$$
−0.0847720 + 0.996400i $$0.527016\pi$$
$$192$$ −15.8490 −1.14381
$$193$$ 20.0219 1.44121 0.720605 0.693346i $$-0.243864\pi$$
0.720605 + 0.693346i $$0.243864\pi$$
$$194$$ 5.41421 0.388718
$$195$$ 7.65685 0.548319
$$196$$ 2.61522 0.186802
$$197$$ 9.31371 0.663574 0.331787 0.943354i $$-0.392348\pi$$
0.331787 + 0.943354i $$0.392348\pi$$
$$198$$ −0.432117 −0.0307092
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 3.04017 0.214973
$$201$$ 12.1421 0.856440
$$202$$ −0.611105 −0.0429972
$$203$$ 7.12356 0.499976
$$204$$ 5.64823 0.395455
$$205$$ 0 0
$$206$$ −6.72792 −0.468757
$$207$$ −0.142136 −0.00987911
$$208$$ 12.8984 0.894340
$$209$$ 0 0
$$210$$ −1.85786 −0.128205
$$211$$ 13.6360 0.938743 0.469371 0.883001i $$-0.344481\pi$$
0.469371 + 0.883001i $$0.344481\pi$$
$$212$$ −3.25623 −0.223639
$$213$$ 24.2843 1.66393
$$214$$ −17.5563 −1.20013
$$215$$ −4.82843 −0.329296
$$216$$ 15.3137 1.04197
$$217$$ 2.95068 0.200305
$$218$$ −23.5147 −1.59262
$$219$$ 20.7596 1.40280
$$220$$ −0.828427 −0.0558525
$$221$$ −32.9203 −2.21446
$$222$$ −1.65433 −0.111031
$$223$$ −0.305553 −0.0204613 −0.0102307 0.999948i $$-0.503257\pi$$
−0.0102307 + 0.999948i $$0.503257\pi$$
$$224$$ 1.90746 0.127447
$$225$$ 0.171573 0.0114382
$$226$$ −11.7574 −0.782088
$$227$$ 16.8923 1.12118 0.560589 0.828094i $$-0.310575\pi$$
0.560589 + 0.828094i $$0.310575\pi$$
$$228$$ 0 0
$$229$$ −4.48528 −0.296396 −0.148198 0.988958i $$-0.547347\pi$$
−0.148198 + 0.988958i $$0.547347\pi$$
$$230$$ 1.04322 0.0687880
$$231$$ 2.95068 0.194140
$$232$$ −26.1421 −1.71632
$$233$$ −9.31371 −0.610161 −0.305081 0.952327i $$-0.598683\pi$$
−0.305081 + 0.952327i $$0.598683\pi$$
$$234$$ 0.928932 0.0607262
$$235$$ 10.4853 0.683984
$$236$$ −2.95068 −0.192073
$$237$$ 12.6863 0.824063
$$238$$ 7.98780 0.517772
$$239$$ 9.65685 0.624650 0.312325 0.949975i $$-0.398892\pi$$
0.312325 + 0.949975i $$0.398892\pi$$
$$240$$ 5.34267 0.344868
$$241$$ −24.3214 −1.56668 −0.783339 0.621595i $$-0.786485\pi$$
−0.783339 + 0.621595i $$0.786485\pi$$
$$242$$ 8.81496 0.566647
$$243$$ 1.78089 0.114244
$$244$$ 1.17157 0.0750023
$$245$$ −6.31371 −0.403368
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −10.8284 −0.687606
$$249$$ −15.7225 −0.996371
$$250$$ −1.25928 −0.0796439
$$251$$ 20.9706 1.32365 0.661825 0.749658i $$-0.269782\pi$$
0.661825 + 0.749658i $$0.269782\pi$$
$$252$$ 0.0588745 0.00370875
$$253$$ −1.65685 −0.104166
$$254$$ −22.0416 −1.38301
$$255$$ −13.6360 −0.853921
$$256$$ −9.48528 −0.592830
$$257$$ −20.0219 −1.24893 −0.624466 0.781052i $$-0.714684\pi$$
−0.624466 + 0.781052i $$0.714684\pi$$
$$258$$ −10.8284 −0.674148
$$259$$ 0.611105 0.0379722
$$260$$ 1.78089 0.110446
$$261$$ −1.47534 −0.0913212
$$262$$ −19.2842 −1.19138
$$263$$ −16.1421 −0.995367 −0.497683 0.867359i $$-0.665816\pi$$
−0.497683 + 0.867359i $$0.665816\pi$$
$$264$$ −10.8284 −0.666444
$$265$$ 7.86123 0.482912
$$266$$ 0 0
$$267$$ −28.0000 −1.71357
$$268$$ 2.82411 0.172510
$$269$$ −8.59890 −0.524284 −0.262142 0.965029i $$-0.584429\pi$$
−0.262142 + 0.965029i $$0.584429\pi$$
$$270$$ −6.34315 −0.386032
$$271$$ 30.9706 1.88133 0.940664 0.339340i $$-0.110204\pi$$
0.940664 + 0.339340i $$0.110204\pi$$
$$272$$ −22.9706 −1.39279
$$273$$ −6.34315 −0.383905
$$274$$ 2.51856 0.152152
$$275$$ 2.00000 0.120605
$$276$$ −0.611105 −0.0367842
$$277$$ −17.3137 −1.04028 −0.520140 0.854081i $$-0.674120\pi$$
−0.520140 + 0.854081i $$0.674120\pi$$
$$278$$ −24.7535 −1.48462
$$279$$ −0.611105 −0.0365859
$$280$$ −2.51856 −0.150513
$$281$$ −14.2471 −0.849912 −0.424956 0.905214i $$-0.639711\pi$$
−0.424956 + 0.905214i $$0.639711\pi$$
$$282$$ 23.5147 1.40028
$$283$$ −9.79899 −0.582489 −0.291245 0.956649i $$-0.594069\pi$$
−0.291245 + 0.956649i $$0.594069\pi$$
$$284$$ 5.64823 0.335161
$$285$$ 0 0
$$286$$ 10.8284 0.640298
$$287$$ 0 0
$$288$$ −0.395047 −0.0232784
$$289$$ 41.6274 2.44867
$$290$$ 10.8284 0.635867
$$291$$ 7.65685 0.448853
$$292$$ 4.82843 0.282562
$$293$$ 17.9355 1.04780 0.523901 0.851779i $$-0.324476\pi$$
0.523901 + 0.851779i $$0.324476\pi$$
$$294$$ −14.1594 −0.825792
$$295$$ 7.12356 0.414750
$$296$$ −2.24264 −0.130351
$$297$$ 10.0742 0.584567
$$298$$ 18.6731 1.08171
$$299$$ 3.56178 0.205983
$$300$$ 0.737669 0.0425894
$$301$$ 4.00000 0.230556
$$302$$ −13.4558 −0.774297
$$303$$ −0.864233 −0.0496489
$$304$$ 0 0
$$305$$ −2.82843 −0.161955
$$306$$ −1.65433 −0.0945716
$$307$$ −3.25623 −0.185843 −0.0929214 0.995673i $$-0.529621\pi$$
−0.0929214 + 0.995673i $$0.529621\pi$$
$$308$$ 0.686292 0.0391051
$$309$$ −9.51472 −0.541273
$$310$$ 4.48528 0.254747
$$311$$ −20.3431 −1.15355 −0.576777 0.816902i $$-0.695690\pi$$
−0.576777 + 0.816902i $$0.695690\pi$$
$$312$$ 23.2781 1.31787
$$313$$ 24.6274 1.39202 0.696012 0.718030i $$-0.254956\pi$$
0.696012 + 0.718030i $$0.254956\pi$$
$$314$$ −22.6670 −1.27918
$$315$$ −0.142136 −0.00800844
$$316$$ 2.95068 0.165989
$$317$$ 5.77479 0.324345 0.162172 0.986762i $$-0.448150\pi$$
0.162172 + 0.986762i $$0.448150\pi$$
$$318$$ 17.6299 0.988637
$$319$$ −17.1978 −0.962892
$$320$$ 8.89949 0.497497
$$321$$ −24.8284 −1.38579
$$322$$ −0.864233 −0.0481618
$$323$$ 0 0
$$324$$ 3.92893 0.218274
$$325$$ −4.29945 −0.238491
$$326$$ −9.03102 −0.500182
$$327$$ −33.2548 −1.83900
$$328$$ 0 0
$$329$$ −8.68629 −0.478891
$$330$$ 4.48528 0.246907
$$331$$ 17.8089 0.978866 0.489433 0.872041i $$-0.337204\pi$$
0.489433 + 0.872041i $$0.337204\pi$$
$$332$$ −3.65685 −0.200696
$$333$$ −0.126564 −0.00693567
$$334$$ −21.2721 −1.16396
$$335$$ −6.81801 −0.372508
$$336$$ −4.42602 −0.241459
$$337$$ 13.5095 0.735907 0.367954 0.929844i $$-0.380059\pi$$
0.367954 + 0.929844i $$0.380059\pi$$
$$338$$ −6.90751 −0.375719
$$339$$ −16.6274 −0.903077
$$340$$ −3.17157 −0.172003
$$341$$ −7.12356 −0.385763
$$342$$ 0 0
$$343$$ 11.0294 0.595534
$$344$$ −14.6792 −0.791452
$$345$$ 1.47534 0.0794296
$$346$$ 3.55635 0.191191
$$347$$ 19.1716 1.02918 0.514592 0.857435i $$-0.327943\pi$$
0.514592 + 0.857435i $$0.327943\pi$$
$$348$$ −6.34315 −0.340028
$$349$$ −29.3137 −1.56913 −0.784563 0.620049i $$-0.787113\pi$$
−0.784563 + 0.620049i $$0.787113\pi$$
$$350$$ 1.04322 0.0557626
$$351$$ −21.6569 −1.15596
$$352$$ −4.60500 −0.245448
$$353$$ 7.65685 0.407533 0.203767 0.979019i $$-0.434682\pi$$
0.203767 + 0.979019i $$0.434682\pi$$
$$354$$ 15.9756 0.849093
$$355$$ −13.6360 −0.723725
$$356$$ −6.51246 −0.345160
$$357$$ 11.2965 0.597872
$$358$$ 21.6569 1.14460
$$359$$ −2.68629 −0.141777 −0.0708885 0.997484i $$-0.522583\pi$$
−0.0708885 + 0.997484i $$0.522583\pi$$
$$360$$ 0.521611 0.0274913
$$361$$ 0 0
$$362$$ −12.6863 −0.666777
$$363$$ 12.4662 0.654308
$$364$$ −1.47534 −0.0773287
$$365$$ −11.6569 −0.610148
$$366$$ −6.34315 −0.331562
$$367$$ 3.17157 0.165555 0.0827774 0.996568i $$-0.473621\pi$$
0.0827774 + 0.996568i $$0.473621\pi$$
$$368$$ 2.48528 0.129554
$$369$$ 0 0
$$370$$ 0.928932 0.0482929
$$371$$ −6.51246 −0.338110
$$372$$ −2.62742 −0.136225
$$373$$ 10.8119 0.559819 0.279910 0.960026i $$-0.409695\pi$$
0.279910 + 0.960026i $$0.409695\pi$$
$$374$$ −19.2842 −0.997165
$$375$$ −1.78089 −0.0919648
$$376$$ 31.8771 1.64393
$$377$$ 36.9706 1.90408
$$378$$ 5.25483 0.270279
$$379$$ 17.1978 0.883392 0.441696 0.897165i $$-0.354377\pi$$
0.441696 + 0.897165i $$0.354377\pi$$
$$380$$ 0 0
$$381$$ −31.1716 −1.59697
$$382$$ 2.95068 0.150970
$$383$$ −32.6147 −1.66653 −0.833267 0.552871i $$-0.813532\pi$$
−0.833267 + 0.552871i $$0.813532\pi$$
$$384$$ 11.7574 0.599990
$$385$$ −1.65685 −0.0844411
$$386$$ −25.2132 −1.28332
$$387$$ −0.828427 −0.0421113
$$388$$ 1.78089 0.0904110
$$389$$ −20.6274 −1.04585 −0.522926 0.852378i $$-0.675160\pi$$
−0.522926 + 0.852378i $$0.675160\pi$$
$$390$$ −9.64212 −0.488248
$$391$$ −6.34315 −0.320787
$$392$$ −19.1948 −0.969482
$$393$$ −27.2720 −1.37569
$$394$$ −11.7286 −0.590877
$$395$$ −7.12356 −0.358425
$$396$$ −0.142136 −0.00714258
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 5.03712 0.252488
$$399$$ 0 0
$$400$$ −3.00000 −0.150000
$$401$$ 10.0742 0.503084 0.251542 0.967846i $$-0.419062\pi$$
0.251542 + 0.967846i $$0.419062\pi$$
$$402$$ −15.2904 −0.762613
$$403$$ 15.3137 0.762830
$$404$$ −0.201010 −0.0100006
$$405$$ −9.48528 −0.471327
$$406$$ −8.97056 −0.445202
$$407$$ −1.47534 −0.0731298
$$408$$ −41.4558 −2.05237
$$409$$ −12.7718 −0.631524 −0.315762 0.948838i $$-0.602260\pi$$
−0.315762 + 0.948838i $$0.602260\pi$$
$$410$$ 0 0
$$411$$ 3.56178 0.175690
$$412$$ −2.21301 −0.109027
$$413$$ −5.90135 −0.290387
$$414$$ 0.178989 0.00879681
$$415$$ 8.82843 0.433370
$$416$$ 9.89949 0.485363
$$417$$ −35.0067 −1.71429
$$418$$ 0 0
$$419$$ −24.9706 −1.21989 −0.609946 0.792443i $$-0.708809\pi$$
−0.609946 + 0.792443i $$0.708809\pi$$
$$420$$ −0.611105 −0.0298189
$$421$$ 10.0742 0.490988 0.245494 0.969398i $$-0.421050\pi$$
0.245494 + 0.969398i $$0.421050\pi$$
$$422$$ −17.1716 −0.835899
$$423$$ 1.79899 0.0874699
$$424$$ 23.8995 1.16066
$$425$$ 7.65685 0.371412
$$426$$ −30.5807 −1.48164
$$427$$ 2.34315 0.113393
$$428$$ −5.77479 −0.279135
$$429$$ 15.3137 0.739353
$$430$$ 6.08034 0.293220
$$431$$ −3.56178 −0.171565 −0.0857825 0.996314i $$-0.527339\pi$$
−0.0857825 + 0.996314i $$0.527339\pi$$
$$432$$ −15.1114 −0.727046
$$433$$ −4.91056 −0.235986 −0.117993 0.993014i $$-0.537646\pi$$
−0.117993 + 0.993014i $$0.537646\pi$$
$$434$$ −3.71573 −0.178361
$$435$$ 15.3137 0.734236
$$436$$ −7.73467 −0.370423
$$437$$ 0 0
$$438$$ −26.1421 −1.24912
$$439$$ −2.95068 −0.140828 −0.0704141 0.997518i $$-0.522432\pi$$
−0.0704141 + 0.997518i $$0.522432\pi$$
$$440$$ 6.08034 0.289869
$$441$$ −1.08326 −0.0515839
$$442$$ 41.4558 1.97185
$$443$$ 20.1421 0.956982 0.478491 0.878093i $$-0.341184\pi$$
0.478491 + 0.878093i $$0.341184\pi$$
$$444$$ −0.544156 −0.0258245
$$445$$ 15.7225 0.745316
$$446$$ 0.384776 0.0182197
$$447$$ 26.4078 1.24905
$$448$$ −7.37258 −0.348322
$$449$$ 8.59890 0.405807 0.202904 0.979199i $$-0.434962\pi$$
0.202904 + 0.979199i $$0.434962\pi$$
$$450$$ −0.216058 −0.0101851
$$451$$ 0 0
$$452$$ −3.86733 −0.181904
$$453$$ −19.0294 −0.894081
$$454$$ −21.2721 −0.998348
$$455$$ 3.56178 0.166979
$$456$$ 0 0
$$457$$ 5.31371 0.248565 0.124282 0.992247i $$-0.460337\pi$$
0.124282 + 0.992247i $$0.460337\pi$$
$$458$$ 5.64823 0.263924
$$459$$ 38.5685 1.80022
$$460$$ 0.343146 0.0159993
$$461$$ −10.6863 −0.497710 −0.248855 0.968541i $$-0.580054\pi$$
−0.248855 + 0.968541i $$0.580054\pi$$
$$462$$ −3.71573 −0.172871
$$463$$ −32.8284 −1.52567 −0.762833 0.646595i $$-0.776192\pi$$
−0.762833 + 0.646595i $$0.776192\pi$$
$$464$$ 25.7967 1.19758
$$465$$ 6.34315 0.294156
$$466$$ 11.7286 0.543315
$$467$$ 38.4853 1.78089 0.890443 0.455094i $$-0.150394\pi$$
0.890443 + 0.455094i $$0.150394\pi$$
$$468$$ 0.305553 0.0141242
$$469$$ 5.64823 0.260811
$$470$$ −13.2039 −0.609051
$$471$$ −32.0560 −1.47706
$$472$$ 21.6569 0.996838
$$473$$ −9.65685 −0.444023
$$474$$ −15.9756 −0.733783
$$475$$ 0 0
$$476$$ 2.62742 0.120427
$$477$$ 1.34877 0.0617561
$$478$$ −12.1607 −0.556217
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 4.10051 0.187162
$$481$$ 3.17157 0.144611
$$482$$ 30.6274 1.39504
$$483$$ −1.22221 −0.0556125
$$484$$ 2.89949 0.131795
$$485$$ −4.29945 −0.195228
$$486$$ −2.24264 −0.101728
$$487$$ 29.6640 1.34421 0.672103 0.740458i $$-0.265391\pi$$
0.672103 + 0.740458i $$0.265391\pi$$
$$488$$ −8.59890 −0.389254
$$489$$ −12.7718 −0.577560
$$490$$ 7.95073 0.359177
$$491$$ 32.2843 1.45697 0.728484 0.685062i $$-0.240225\pi$$
0.728484 + 0.685062i $$0.240225\pi$$
$$492$$ 0 0
$$493$$ −65.8405 −2.96531
$$494$$ 0 0
$$495$$ 0.343146 0.0154233
$$496$$ 10.6853 0.479786
$$497$$ 11.2965 0.506715
$$498$$ 19.7990 0.887214
$$499$$ 1.02944 0.0460839 0.0230420 0.999734i $$-0.492665\pi$$
0.0230420 + 0.999734i $$0.492665\pi$$
$$500$$ −0.414214 −0.0185242
$$501$$ −30.0833 −1.34402
$$502$$ −26.4078 −1.17864
$$503$$ 2.48528 0.110813 0.0554066 0.998464i $$-0.482354\pi$$
0.0554066 + 0.998464i $$0.482354\pi$$
$$504$$ −0.432117 −0.0192480
$$505$$ 0.485281 0.0215947
$$506$$ 2.08644 0.0927537
$$507$$ −9.76869 −0.433843
$$508$$ −7.25013 −0.321672
$$509$$ 32.9203 1.45917 0.729583 0.683893i $$-0.239714\pi$$
0.729583 + 0.683893i $$0.239714\pi$$
$$510$$ 17.1716 0.760370
$$511$$ 9.65685 0.427194
$$512$$ 25.1485 1.11142
$$513$$ 0 0
$$514$$ 25.2132 1.11211
$$515$$ 5.34267 0.235426
$$516$$ −3.56178 −0.156799
$$517$$ 20.9706 0.922284
$$518$$ −0.769553 −0.0338122
$$519$$ 5.02944 0.220768
$$520$$ −13.0711 −0.573204
$$521$$ 17.1978 0.753450 0.376725 0.926325i $$-0.377050\pi$$
0.376725 + 0.926325i $$0.377050\pi$$
$$522$$ 1.85786 0.0813165
$$523$$ −15.4169 −0.674135 −0.337067 0.941481i $$-0.609435\pi$$
−0.337067 + 0.941481i $$0.609435\pi$$
$$524$$ −6.34315 −0.277102
$$525$$ 1.47534 0.0643890
$$526$$ 20.3275 0.886320
$$527$$ −27.2720 −1.18799
$$528$$ 10.6853 0.465020
$$529$$ −22.3137 −0.970161
$$530$$ −9.89949 −0.430007
$$531$$ 1.22221 0.0530394
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 35.2598 1.52584
$$535$$ 13.9416 0.602747
$$536$$ −20.7279 −0.895310
$$537$$ 30.6274 1.32167
$$538$$ 10.8284 0.466847
$$539$$ −12.6274 −0.543901
$$540$$ −2.08644 −0.0897862
$$541$$ 22.1421 0.951965 0.475982 0.879455i $$-0.342093\pi$$
0.475982 + 0.879455i $$0.342093\pi$$
$$542$$ −39.0006 −1.67522
$$543$$ −17.9411 −0.769927
$$544$$ −17.6299 −0.755877
$$545$$ 18.6731 0.799870
$$546$$ 7.98780 0.341846
$$547$$ −11.8551 −0.506889 −0.253444 0.967350i $$-0.581563\pi$$
−0.253444 + 0.967350i $$0.581563\pi$$
$$548$$ 0.828427 0.0353887
$$549$$ −0.485281 −0.0207113
$$550$$ −2.51856 −0.107392
$$551$$ 0 0
$$552$$ 4.48528 0.190906
$$553$$ 5.90135 0.250951
$$554$$ 21.8028 0.926313
$$555$$ 1.31371 0.0557638
$$556$$ −8.14214 −0.345303
$$557$$ −28.6274 −1.21298 −0.606491 0.795090i $$-0.707424\pi$$
−0.606491 + 0.795090i $$0.707424\pi$$
$$558$$ 0.769553 0.0325778
$$559$$ 20.7596 0.878037
$$560$$ 2.48528 0.105022
$$561$$ −27.2720 −1.15143
$$562$$ 17.9411 0.756801
$$563$$ 26.1023 1.10008 0.550040 0.835139i $$-0.314613\pi$$
0.550040 + 0.835139i $$0.314613\pi$$
$$564$$ 7.73467 0.325688
$$565$$ 9.33657 0.392793
$$566$$ 12.3397 0.518675
$$567$$ 7.85786 0.329999
$$568$$ −41.4558 −1.73945
$$569$$ 15.7225 0.659120 0.329560 0.944135i $$-0.393100\pi$$
0.329560 + 0.944135i $$0.393100\pi$$
$$570$$ 0 0
$$571$$ 13.3137 0.557161 0.278581 0.960413i $$-0.410136\pi$$
0.278581 + 0.960413i $$0.410136\pi$$
$$572$$ 3.56178 0.148926
$$573$$ 4.17289 0.174325
$$574$$ 0 0
$$575$$ −0.828427 −0.0345478
$$576$$ 1.52691 0.0636213
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ −52.4206 −2.18041
$$579$$ −35.6569 −1.48185
$$580$$ 3.56178 0.147895
$$581$$ −7.31371 −0.303424
$$582$$ −9.64212 −0.399679
$$583$$ 15.7225 0.651158
$$584$$ −35.4388 −1.46647
$$585$$ −0.737669 −0.0304989
$$586$$ −22.5858 −0.933010
$$587$$ 18.4853 0.762969 0.381485 0.924375i $$-0.375413\pi$$
0.381485 + 0.924375i $$0.375413\pi$$
$$588$$ −4.65743 −0.192069
$$589$$ 0 0
$$590$$ −8.97056 −0.369312
$$591$$ −16.5867 −0.682286
$$592$$ 2.21301 0.0909541
$$593$$ −9.31371 −0.382468 −0.191234 0.981544i $$-0.561249\pi$$
−0.191234 + 0.981544i $$0.561249\pi$$
$$594$$ −12.6863 −0.520525
$$595$$ −6.34315 −0.260044
$$596$$ 6.14214 0.251592
$$597$$ 7.12356 0.291548
$$598$$ −4.48528 −0.183417
$$599$$ 34.3956 1.40537 0.702683 0.711503i $$-0.251985\pi$$
0.702683 + 0.711503i $$0.251985\pi$$
$$600$$ −5.41421 −0.221034
$$601$$ 17.1978 0.701513 0.350757 0.936467i $$-0.385924\pi$$
0.350757 + 0.936467i $$0.385924\pi$$
$$602$$ −5.03712 −0.205298
$$603$$ −1.16979 −0.0476374
$$604$$ −4.42602 −0.180092
$$605$$ −7.00000 −0.284590
$$606$$ 1.08831 0.0442096
$$607$$ −33.2258 −1.34859 −0.674297 0.738460i $$-0.735553\pi$$
−0.674297 + 0.738460i $$0.735553\pi$$
$$608$$ 0 0
$$609$$ −12.6863 −0.514074
$$610$$ 3.56178 0.144212
$$611$$ −45.0810 −1.82378
$$612$$ −0.544156 −0.0219962
$$613$$ −14.2843 −0.576936 −0.288468 0.957489i $$-0.593146\pi$$
−0.288468 + 0.957489i $$0.593146\pi$$
$$614$$ 4.10051 0.165483
$$615$$ 0 0
$$616$$ −5.03712 −0.202951
$$617$$ 30.2843 1.21920 0.609599 0.792710i $$-0.291330\pi$$
0.609599 + 0.792710i $$0.291330\pi$$
$$618$$ 11.9817 0.481975
$$619$$ −18.9706 −0.762491 −0.381246 0.924474i $$-0.624505\pi$$
−0.381246 + 0.924474i $$0.624505\pi$$
$$620$$ 1.47534 0.0592510
$$621$$ −4.17289 −0.167452
$$622$$ 25.6177 1.02718
$$623$$ −13.0249 −0.521832
$$624$$ −22.9706 −0.919558
$$625$$ 1.00000 0.0400000
$$626$$ −31.0128 −1.23952
$$627$$ 0 0
$$628$$ −7.45584 −0.297521
$$629$$ −5.64823 −0.225210
$$630$$ 0.178989 0.00713108
$$631$$ 18.2843 0.727885 0.363943 0.931421i $$-0.381430\pi$$
0.363943 + 0.931421i $$0.381430\pi$$
$$632$$ −21.6569 −0.861463
$$633$$ −24.2843 −0.965213
$$634$$ −7.27208 −0.288811
$$635$$ 17.5034 0.694600
$$636$$ 5.79899 0.229945
$$637$$ 27.1455 1.07554
$$638$$ 21.6569 0.857403
$$639$$ −2.33957 −0.0925520
$$640$$ −6.60195 −0.260965
$$641$$ 21.3707 0.844092 0.422046 0.906575i $$-0.361312\pi$$
0.422046 + 0.906575i $$0.361312\pi$$
$$642$$ 31.2659 1.23397
$$643$$ 9.51472 0.375224 0.187612 0.982243i $$-0.439925\pi$$
0.187612 + 0.982243i $$0.439925\pi$$
$$644$$ −0.284271 −0.0112019
$$645$$ 8.59890 0.338581
$$646$$ 0 0
$$647$$ −27.1716 −1.06822 −0.534112 0.845413i $$-0.679354\pi$$
−0.534112 + 0.845413i $$0.679354\pi$$
$$648$$ −28.8369 −1.13282
$$649$$ 14.2471 0.559249
$$650$$ 5.41421 0.212363
$$651$$ −5.25483 −0.205953
$$652$$ −2.97056 −0.116336
$$653$$ 27.6569 1.08230 0.541148 0.840927i $$-0.317990\pi$$
0.541148 + 0.840927i $$0.317990\pi$$
$$654$$ 41.8772 1.63753
$$655$$ 15.3137 0.598356
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ 10.9385 0.426426
$$659$$ 20.1485 0.784873 0.392437 0.919779i $$-0.371632\pi$$
0.392437 + 0.919779i $$0.371632\pi$$
$$660$$ 1.47534 0.0574275
$$661$$ −2.95068 −0.114768 −0.0573840 0.998352i $$-0.518276\pi$$
−0.0573840 + 0.998352i $$0.518276\pi$$
$$662$$ −22.4264 −0.871627
$$663$$ 58.6274 2.27690
$$664$$ 26.8399 1.04159
$$665$$ 0 0
$$666$$ 0.159380 0.00617583
$$667$$ 7.12356 0.275826
$$668$$ −6.99700 −0.270722
$$669$$ 0.544156 0.0210383
$$670$$ 8.58579 0.331698
$$671$$ −5.65685 −0.218380
$$672$$ −3.39697 −0.131041
$$673$$ −50.8557 −1.96034 −0.980172 0.198146i $$-0.936508\pi$$
−0.980172 + 0.198146i $$0.936508\pi$$
$$674$$ −17.0122 −0.655285
$$675$$ 5.03712 0.193879
$$676$$ −2.27208 −0.0873876
$$677$$ −2.21301 −0.0850528 −0.0425264 0.999095i $$-0.513541\pi$$
−0.0425264 + 0.999095i $$0.513541\pi$$
$$678$$ 20.9386 0.804141
$$679$$ 3.56178 0.136689
$$680$$ 23.2781 0.892676
$$681$$ −30.0833 −1.15279
$$682$$ 8.97056 0.343501
$$683$$ 11.8551 0.453624 0.226812 0.973939i $$-0.427170\pi$$
0.226812 + 0.973939i $$0.427170\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ −13.8892 −0.530290
$$687$$ 7.98780 0.304753
$$688$$ 14.4853 0.552246
$$689$$ −33.7990 −1.28764
$$690$$ −1.85786 −0.0707277
$$691$$ 16.6274 0.632537 0.316268 0.948670i $$-0.397570\pi$$
0.316268 + 0.948670i $$0.397570\pi$$
$$692$$ 1.16979 0.0444686
$$693$$ −0.284271 −0.0107986
$$694$$ −24.1424 −0.916432
$$695$$ 19.6569 0.745627
$$696$$ 46.5563 1.76471
$$697$$ 0 0
$$698$$ 36.9142 1.39722
$$699$$ 16.5867 0.627367
$$700$$ 0.343146 0.0129697
$$701$$ −24.4853 −0.924796 −0.462398 0.886672i $$-0.653011\pi$$
−0.462398 + 0.886672i $$0.653011\pi$$
$$702$$ 27.2720 1.02932
$$703$$ 0 0
$$704$$ 17.7990 0.670825
$$705$$ −18.6731 −0.703271
$$706$$ −9.64212 −0.362886
$$707$$ −0.402020 −0.0151195
$$708$$ 5.25483 0.197489
$$709$$ −5.31371 −0.199561 −0.0997803 0.995009i $$-0.531814\pi$$
−0.0997803 + 0.995009i $$0.531814\pi$$
$$710$$ 17.1716 0.644438
$$711$$ −1.22221 −0.0458365
$$712$$ 47.7990 1.79134
$$713$$ 2.95068 0.110504
$$714$$ −14.2254 −0.532372
$$715$$ −8.59890 −0.321581
$$716$$ 7.12356 0.266220
$$717$$ −17.1978 −0.642264
$$718$$ 3.38279 0.126245
$$719$$ 31.9411 1.19120 0.595601 0.803280i $$-0.296914\pi$$
0.595601 + 0.803280i $$0.296914\pi$$
$$720$$ −0.514719 −0.0191824
$$721$$ −4.42602 −0.164833
$$722$$ 0 0
$$723$$ 43.3137 1.61085
$$724$$ −4.17289 −0.155084
$$725$$ −8.59890 −0.319355
$$726$$ −15.6985 −0.582625
$$727$$ 6.48528 0.240526 0.120263 0.992742i $$-0.461626\pi$$
0.120263 + 0.992742i $$0.461626\pi$$
$$728$$ 10.8284 0.401328
$$729$$ 25.2843 0.936454
$$730$$ 14.6792 0.543303
$$731$$ −36.9706 −1.36741
$$732$$ −2.08644 −0.0771172
$$733$$ −35.6569 −1.31702 −0.658508 0.752574i $$-0.728812\pi$$
−0.658508 + 0.752574i $$0.728812\pi$$
$$734$$ −3.99390 −0.147417
$$735$$ 11.2440 0.414742
$$736$$ 1.90746 0.0703097
$$737$$ −13.6360 −0.502289
$$738$$ 0 0
$$739$$ 1.37258 0.0504913 0.0252456 0.999681i $$-0.491963\pi$$
0.0252456 + 0.999681i $$0.491963\pi$$
$$740$$ 0.305553 0.0112323
$$741$$ 0 0
$$742$$ 8.20101 0.301069
$$743$$ −6.81801 −0.250129 −0.125064 0.992149i $$-0.539914\pi$$
−0.125064 + 0.992149i $$0.539914\pi$$
$$744$$ 19.2842 0.706995
$$745$$ −14.8284 −0.543272
$$746$$ −13.6152 −0.498489
$$747$$ 1.51472 0.0554207
$$748$$ −6.34315 −0.231928
$$749$$ −11.5496 −0.422012
$$750$$ 2.24264 0.0818897
$$751$$ 6.51246 0.237643 0.118822 0.992916i $$-0.462088\pi$$
0.118822 + 0.992916i $$0.462088\pi$$
$$752$$ −31.4558 −1.14708
$$753$$ −37.3463 −1.36097
$$754$$ −46.5563 −1.69548
$$755$$ 10.6853 0.388880
$$756$$ 1.72847 0.0628637
$$757$$ −6.68629 −0.243017 −0.121509 0.992590i $$-0.538773\pi$$
−0.121509 + 0.992590i $$0.538773\pi$$
$$758$$ −21.6569 −0.786612
$$759$$ 2.95068 0.107103
$$760$$ 0 0
$$761$$ −9.17157 −0.332469 −0.166235 0.986086i $$-0.553161\pi$$
−0.166235 + 0.986086i $$0.553161\pi$$
$$762$$ 39.2537 1.42201
$$763$$ −15.4693 −0.560028
$$764$$ 0.970563 0.0351137
$$765$$ 1.31371 0.0474972
$$766$$ 41.0711 1.48396
$$767$$ −30.6274 −1.10589
$$768$$ 16.8923 0.609547
$$769$$ 14.1421 0.509978 0.254989 0.966944i $$-0.417928\pi$$
0.254989 + 0.966944i $$0.417928\pi$$
$$770$$ 2.08644 0.0751902
$$771$$ 35.6569 1.28415
$$772$$ −8.29335 −0.298484
$$773$$ −18.5466 −0.667074 −0.333537 0.942737i $$-0.608242\pi$$
−0.333537 + 0.942737i $$0.608242\pi$$
$$774$$ 1.04322 0.0374978
$$775$$ −3.56178 −0.127943
$$776$$ −13.0711 −0.469224
$$777$$ −1.08831 −0.0390430
$$778$$ 25.9757 0.931274
$$779$$ 0 0
$$780$$ −3.17157 −0.113561
$$781$$ −27.2720 −0.975871
$$782$$ 7.98780 0.285643
$$783$$ −43.3137 −1.54791
$$784$$ 18.9411 0.676469
$$785$$ 18.0000 0.642448
$$786$$ 34.3431 1.22498
$$787$$ −46.2507 −1.64866 −0.824330 0.566109i $$-0.808448\pi$$
−0.824330 + 0.566109i $$0.808448\pi$$
$$788$$ −3.85786 −0.137431
$$789$$ 28.7474 1.02343
$$790$$ 8.97056 0.319158
$$791$$ −7.73467 −0.275013
$$792$$ 1.04322 0.0370693
$$793$$ 12.1607 0.431839
$$794$$ 17.6299 0.625663
$$795$$ −14.0000 −0.496529
$$796$$ 1.65685 0.0587256
$$797$$ 0.737669 0.0261296 0.0130648 0.999915i $$-0.495841\pi$$
0.0130648 + 0.999915i $$0.495841\pi$$
$$798$$ 0 0
$$799$$ 80.2843 2.84025
$$800$$ −2.30250 −0.0814057
$$801$$ 2.69755 0.0953132
$$802$$ −12.6863 −0.447969
$$803$$ −23.3137 −0.822723
$$804$$ −5.02944 −0.177375
$$805$$ 0.686292 0.0241886
$$806$$ −19.2842 −0.679259
$$807$$ 15.3137 0.539068
$$808$$ 1.47534 0.0519022
$$809$$ −13.3137 −0.468085 −0.234043 0.972226i $$-0.575196\pi$$
−0.234043 + 0.972226i $$0.575196\pi$$
$$810$$ 11.9446 0.419691
$$811$$ 33.7845 1.18633 0.593167 0.805079i $$-0.297877\pi$$
0.593167 + 0.805079i $$0.297877\pi$$
$$812$$ −2.95068 −0.103548
$$813$$ −55.1552 −1.93438
$$814$$ 1.85786 0.0651181
$$815$$ 7.17157 0.251209
$$816$$ 40.9081 1.43207
$$817$$ 0 0
$$818$$ 16.0833 0.562338
$$819$$ 0.611105 0.0213537
$$820$$ 0 0
$$821$$ 2.68629 0.0937522 0.0468761 0.998901i $$-0.485073\pi$$
0.0468761 + 0.998901i $$0.485073\pi$$
$$822$$ −4.48528 −0.156442
$$823$$ 42.0833 1.46693 0.733465 0.679727i $$-0.237902\pi$$
0.733465 + 0.679727i $$0.237902\pi$$
$$824$$ 16.2426 0.565839
$$825$$ −3.56178 −0.124005
$$826$$ 7.43146 0.258573
$$827$$ 18.9787 0.659954 0.329977 0.943989i $$-0.392959\pi$$
0.329977 + 0.943989i $$0.392959\pi$$
$$828$$ 0.0588745 0.00204603
$$829$$ −42.9945 −1.49326 −0.746631 0.665239i $$-0.768330\pi$$
−0.746631 + 0.665239i $$0.768330\pi$$
$$830$$ −11.1175 −0.385893
$$831$$ 30.8338 1.06961
$$832$$ −38.2629 −1.32653
$$833$$ −48.3431 −1.67499
$$834$$ 44.0833 1.52648
$$835$$ 16.8923 0.584581
$$836$$ 0 0
$$837$$ −17.9411 −0.620136
$$838$$ 31.4449 1.08625
$$839$$ 31.4449 1.08560 0.542800 0.839862i $$-0.317364\pi$$
0.542800 + 0.839862i $$0.317364\pi$$
$$840$$ 4.48528 0.154757
$$841$$ 44.9411 1.54969
$$842$$ −12.6863 −0.437198
$$843$$ 25.3726 0.873878
$$844$$ −5.64823 −0.194420
$$845$$ 5.48528 0.188699
$$846$$ −2.26543 −0.0778872
$$847$$ 5.79899 0.199256
$$848$$ −23.5837 −0.809868
$$849$$ 17.4509 0.598914
$$850$$ −9.64212 −0.330722
$$851$$ 0.611105 0.0209484
$$852$$ −10.0589 −0.344611
$$853$$ 36.3431 1.24437 0.622183 0.782872i $$-0.286246\pi$$
0.622183 + 0.782872i $$0.286246\pi$$
$$854$$ −2.95068 −0.100970
$$855$$ 0 0
$$856$$ 42.3848 1.44868
$$857$$ −17.0712 −0.583142 −0.291571 0.956549i $$-0.594178\pi$$
−0.291571 + 0.956549i $$0.594178\pi$$
$$858$$ −19.2842 −0.658353
$$859$$ −44.2843 −1.51096 −0.755480 0.655172i $$-0.772596\pi$$
−0.755480 + 0.655172i $$0.772596\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ 4.48528 0.152769
$$863$$ 24.6269 0.838310 0.419155 0.907915i $$-0.362326\pi$$
0.419155 + 0.907915i $$0.362326\pi$$
$$864$$ −11.5980 −0.394571
$$865$$ −2.82411 −0.0960227
$$866$$ 6.18377 0.210133
$$867$$ −74.1339 −2.51772
$$868$$ −1.22221 −0.0414845
$$869$$ −14.2471 −0.483301
$$870$$ −19.2842 −0.653797
$$871$$ 29.3137 0.993257
$$872$$ 56.7696 1.92246
$$873$$ −0.737669 −0.0249663
$$874$$ 0 0
$$875$$ −0.828427 −0.0280059
$$876$$ −8.59890 −0.290530
$$877$$ 10.8119 0.365092 0.182546 0.983197i $$-0.441566\pi$$
0.182546 + 0.983197i $$0.441566\pi$$
$$878$$ 3.71573 0.125400
$$879$$ −31.9411 −1.07735
$$880$$ −6.00000 −0.202260
$$881$$ 4.48528 0.151113 0.0755565 0.997142i $$-0.475927\pi$$
0.0755565 + 0.997142i $$0.475927\pi$$
$$882$$ 1.36413 0.0459326
$$883$$ 3.85786 0.129827 0.0649137 0.997891i $$-0.479323\pi$$
0.0649137 + 0.997891i $$0.479323\pi$$
$$884$$ 13.6360 0.458629
$$885$$ −12.6863 −0.426445
$$886$$ −25.3646 −0.852140
$$887$$ −1.16979 −0.0392776 −0.0196388 0.999807i $$-0.506252\pi$$
−0.0196388 + 0.999807i $$0.506252\pi$$
$$888$$ 3.99390 0.134026
$$889$$ −14.5003 −0.486323
$$890$$ −19.7990 −0.663664
$$891$$ −18.9706 −0.635538
$$892$$ 0.126564 0.00423768
$$893$$ 0 0
$$894$$ −33.2548 −1.11221
$$895$$ −17.1978 −0.574859
$$896$$ 5.46924 0.182714
$$897$$ −6.34315 −0.211791
$$898$$ −10.8284 −0.361349
$$899$$ 30.6274 1.02148
$$900$$ −0.0710678 −0.00236893
$$901$$ 60.1923 2.00530
$$902$$ 0 0
$$903$$ −7.12356 −0.237057
$$904$$ 28.3848 0.944064
$$905$$ 10.0742 0.334879
$$906$$ 23.9634 0.796130
$$907$$ −30.5283 −1.01367 −0.506837 0.862042i $$-0.669186\pi$$
−0.506837 + 0.862042i $$0.669186\pi$$
$$908$$ −6.99700 −0.232204
$$909$$ 0.0832611 0.00276160
$$910$$ −4.48528 −0.148686
$$911$$ 26.6609 0.883316 0.441658 0.897183i $$-0.354390\pi$$
0.441658 + 0.897183i $$0.354390\pi$$
$$912$$ 0 0
$$913$$ 17.6569 0.584357
$$914$$ −6.69145 −0.221333
$$915$$ 5.03712 0.166522
$$916$$ 1.85786 0.0613856
$$917$$ −12.6863 −0.418938
$$918$$ −48.5685 −1.60300
$$919$$ 0.284271 0.00937724 0.00468862 0.999989i $$-0.498508\pi$$
0.00468862 + 0.999989i $$0.498508\pi$$
$$920$$ −2.51856 −0.0830345
$$921$$ 5.79899 0.191083
$$922$$ 13.4570 0.443184
$$923$$ 58.6274 1.92974
$$924$$ −1.22221 −0.0402078
$$925$$ −0.737669 −0.0242544
$$926$$ 41.3402 1.35852
$$927$$ 0.916658 0.0301070
$$928$$ 19.7990 0.649934
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ −7.98780 −0.261930
$$931$$ 0 0
$$932$$ 3.85786 0.126369
$$933$$ 36.2289 1.18608
$$934$$ −48.4638 −1.58578
$$935$$ 15.3137 0.500812
$$936$$ −2.24264 −0.0733030
$$937$$ 10.9706 0.358393 0.179196 0.983813i $$-0.442650\pi$$
0.179196 + 0.983813i $$0.442650\pi$$
$$938$$ −7.11270 −0.232238
$$939$$ −43.8587 −1.43128
$$940$$ −4.34315 −0.141658
$$941$$ −2.95068 −0.0961893 −0.0480947 0.998843i $$-0.515315\pi$$
−0.0480947 + 0.998843i $$0.515315\pi$$
$$942$$ 40.3675 1.31525
$$943$$ 0 0
$$944$$ −21.3707 −0.695557
$$945$$ −4.17289 −0.135744
$$946$$ 12.1607 0.395378
$$947$$ −4.14214 −0.134601 −0.0673007 0.997733i $$-0.521439\pi$$
−0.0673007 + 0.997733i $$0.521439\pi$$
$$948$$ −5.25483 −0.170669
$$949$$ 50.1181 1.62690
$$950$$ 0 0
$$951$$ −10.2843 −0.333490
$$952$$ −19.2842 −0.625006
$$953$$ 1.34877 0.0436911 0.0218455 0.999761i $$-0.493046\pi$$
0.0218455 + 0.999761i $$0.493046\pi$$
$$954$$ −1.69848 −0.0549905
$$955$$ −2.34315 −0.0758224
$$956$$ −4.00000 −0.129369
$$957$$ 30.6274 0.990044
$$958$$ −12.5928 −0.406855
$$959$$ 1.65685 0.0535026
$$960$$ −15.8490 −0.511525
$$961$$ −18.3137 −0.590765
$$962$$ −3.99390 −0.128768
$$963$$ 2.39200 0.0770810
$$964$$ 10.0742 0.324469
$$965$$ 20.0219 0.644528
$$966$$ 1.53911 0.0495199
$$967$$ 21.5147 0.691867 0.345933 0.938259i $$-0.387562\pi$$
0.345933 + 0.938259i $$0.387562\pi$$
$$968$$ −21.2812 −0.684004
$$969$$ 0 0
$$970$$ 5.41421 0.173840
$$971$$ −7.73467 −0.248217 −0.124109 0.992269i $$-0.539607\pi$$
−0.124109 + 0.992269i $$0.539607\pi$$
$$972$$ −0.737669 −0.0236608
$$973$$ −16.2843 −0.522050
$$974$$ −37.3553 −1.19694
$$975$$ 7.65685 0.245216
$$976$$ 8.48528 0.271607
$$977$$ −47.9051 −1.53262 −0.766309 0.642472i $$-0.777909\pi$$
−0.766309 + 0.642472i $$0.777909\pi$$
$$978$$ 16.0833 0.514286
$$979$$ 31.4449 1.00498
$$980$$ 2.61522 0.0835403
$$981$$ 3.20380 0.102290
$$982$$ −40.6549 −1.29735
$$983$$ −33.8369 −1.07923 −0.539615 0.841912i $$-0.681430\pi$$
−0.539615 + 0.841912i $$0.681430\pi$$
$$984$$ 0 0
$$985$$ 9.31371 0.296759
$$986$$ 82.9117 2.64045
$$987$$ 15.4693 0.492394
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ −0.432117 −0.0137336
$$991$$ −27.8832 −0.885737 −0.442869 0.896586i $$-0.646039\pi$$
−0.442869 + 0.896586i $$0.646039\pi$$
$$992$$ 8.20101 0.260382
$$993$$ −31.7157 −1.00647
$$994$$ −14.2254 −0.451202
$$995$$ −4.00000 −0.126809
$$996$$ 6.51246 0.206355
$$997$$ −7.65685 −0.242495 −0.121248 0.992622i $$-0.538689\pi$$
−0.121248 + 0.992622i $$0.538689\pi$$
$$998$$ −1.29635 −0.0410352
$$999$$ −3.71573 −0.117560
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 1805.2.a.m.1.2 4
5.4 even 2 9025.2.a.bn.1.3 4
19.18 odd 2 inner 1805.2.a.m.1.3 yes 4
95.94 odd 2 9025.2.a.bn.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.m.1.2 4 1.1 even 1 trivial
1805.2.a.m.1.3 yes 4 19.18 odd 2 inner
9025.2.a.bn.1.2 4 95.94 odd 2
9025.2.a.bn.1.3 4 5.4 even 2