# Properties

 Label 9025.2.a.bn.1.2 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$1$$ 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: no (minimal twist has level 1805) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.25928$$ of defining polynomial Character $$\chi$$ $$=$$ 9025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +2.24264 q^{6} +0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} +O(q^{10})$$ $$q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +2.24264 q^{6} +0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} +2.00000 q^{11} +0.737669 q^{12} -4.29945 q^{13} -1.04322 q^{14} -3.00000 q^{16} -7.65685 q^{17} -0.216058 q^{18} -1.47534 q^{21} -2.51856 q^{22} +0.828427 q^{23} -5.41421 q^{24} +5.41421 q^{26} +5.03712 q^{27} -0.343146 q^{28} +8.59890 q^{29} +3.56178 q^{31} -2.30250 q^{32} -3.56178 q^{33} +9.64212 q^{34} -0.0710678 q^{36} -0.737669 q^{37} +7.65685 q^{39} +1.85786 q^{42} +4.82843 q^{43} -0.828427 q^{44} -1.04322 q^{46} -10.4853 q^{47} +5.34267 q^{48} -6.31371 q^{49} +13.6360 q^{51} +1.78089 q^{52} +7.86123 q^{53} -6.34315 q^{54} +2.51856 q^{56} -10.8284 q^{58} -7.12356 q^{59} -2.82843 q^{61} -4.48528 q^{62} +0.142136 q^{63} +8.89949 q^{64} +4.48528 q^{66} -6.81801 q^{67} +3.17157 q^{68} -1.47534 q^{69} +13.6360 q^{71} +0.521611 q^{72} +11.6569 q^{73} +0.928932 q^{74} +1.65685 q^{77} -9.64212 q^{78} +7.12356 q^{79} -9.48528 q^{81} -8.82843 q^{83} +0.611105 q^{84} -6.08034 q^{86} -15.3137 q^{87} +6.08034 q^{88} -15.7225 q^{89} -3.56178 q^{91} -0.343146 q^{92} -6.34315 q^{93} +13.2039 q^{94} +4.10051 q^{96} -4.29945 q^{97} +7.95073 q^{98} +0.343146 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 8 * q^6 - 8 * q^7 + 12 * q^9 $$4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} - 8 q^{17} - 8 q^{23} - 16 q^{24} + 16 q^{26} - 24 q^{28} + 28 q^{36} + 8 q^{39} + 64 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{47} + 20 q^{49} - 48 q^{54} - 32 q^{58} + 16 q^{62} - 56 q^{63} - 4 q^{64} - 16 q^{66} + 24 q^{68} + 24 q^{73} + 32 q^{74} - 16 q^{77} - 4 q^{81} - 24 q^{83} - 16 q^{87} - 24 q^{92} - 48 q^{93} + 56 q^{96} + 24 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 - 8 * q^6 - 8 * q^7 + 12 * q^9 + 8 * q^11 - 12 * q^16 - 8 * q^17 - 8 * q^23 - 16 * q^24 + 16 * q^26 - 24 * q^28 + 28 * q^36 + 8 * q^39 + 64 * q^42 + 8 * q^43 + 8 * q^44 - 8 * q^47 + 20 * q^49 - 48 * q^54 - 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 - 4 * q^81 - 24 * q^83 - 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
$$5$$ 0 0
$$6$$ 2.24264 0.915554
$$7$$ 0.828427 0.313116 0.156558 0.987669i $$-0.449960\pi$$
0.156558 + 0.987669i $$0.449960\pi$$
$$8$$ 3.04017 1.07486
$$9$$ 0.171573 0.0571910
$$10$$ 0 0
$$11$$ 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
$$15$$ 0 0
$$16$$ −3.00000 −0.750000
$$17$$ −7.65685 −1.85706 −0.928530 0.371257i $$-0.878927\pi$$
−0.928530 + 0.371257i $$0.878927\pi$$
$$18$$ −0.216058 −0.0509254
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −1.47534 −0.321945
$$22$$ −2.51856 −0.536959
$$23$$ 0.828427 0.172739 0.0863695 0.996263i $$-0.472473\pi$$
0.0863695 + 0.996263i $$0.472473\pi$$
$$24$$ −5.41421 −1.10517
$$25$$ 0 0
$$26$$ 5.41421 1.06181
$$27$$ 5.03712 0.969394
$$28$$ −0.343146 −0.0648485
$$29$$ 8.59890 1.59678 0.798388 0.602143i $$-0.205686\pi$$
0.798388 + 0.602143i $$0.205686\pi$$
$$30$$ 0 0
$$31$$ 3.56178 0.639715 0.319857 0.947466i $$-0.396365\pi$$
0.319857 + 0.947466i $$0.396365\pi$$
$$32$$ −2.30250 −0.407029
$$33$$ −3.56178 −0.620027
$$34$$ 9.64212 1.65361
$$35$$ 0 0
$$36$$ −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
$$40$$ 0 0
$$41$$ 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 $$-0.379986\pi$$
0.368164 + 0.929761i $$0.379986\pi$$
$$44$$ −0.828427 −0.124890
$$45$$ 0 0
$$46$$ −1.04322 −0.153815
$$47$$ −10.4853 −1.52944 −0.764718 0.644365i $$-0.777122\pi$$
−0.764718 + 0.644365i $$0.777122\pi$$
$$48$$ 5.34267 0.771148
$$49$$ −6.31371 −0.901958
$$50$$ 0 0
$$51$$ 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
$$55$$ 0 0
$$56$$ 2.51856 0.336557
$$57$$ 0 0
$$58$$ −10.8284 −1.42184
$$59$$ −7.12356 −0.927409 −0.463705 0.885990i $$-0.653480\pi$$
−0.463705 + 0.885990i $$0.653480\pi$$
$$60$$ 0 0
$$61$$ −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
$$65$$ 0 0
$$66$$ 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
$$70$$ 0 0
$$71$$ 13.6360 1.61830 0.809149 0.587603i $$-0.199928\pi$$
0.809149 + 0.587603i $$0.199928\pi$$
$$72$$ 0.521611 0.0614724
$$73$$ 11.6569 1.36433 0.682166 0.731198i $$-0.261038\pi$$
0.682166 + 0.731198i $$0.261038\pi$$
$$74$$ 0.928932 0.107986
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.65685 0.188816
$$78$$ −9.64212 −1.09176
$$79$$ 7.12356 0.801464 0.400732 0.916195i $$-0.368756\pi$$
0.400732 + 0.916195i $$0.368756\pi$$
$$80$$ 0 0
$$81$$ −9.48528 −1.05392
$$82$$ 0 0
$$83$$ −8.82843 −0.969046 −0.484523 0.874779i $$-0.661007\pi$$
−0.484523 + 0.874779i $$0.661007\pi$$
$$84$$ 0.611105 0.0666770
$$85$$ 0 0
$$86$$ −6.08034 −0.655660
$$87$$ −15.3137 −1.64180
$$88$$ 6.08034 0.648167
$$89$$ −15.7225 −1.66658 −0.833289 0.552838i $$-0.813545\pi$$
−0.833289 + 0.552838i $$0.813545\pi$$
$$90$$ 0 0
$$91$$ −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 0
$$101$$ 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
$$105$$ 0 0
$$106$$ −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 $$-0.852313\pi$$
−0.894281 + 0.447505i $$0.852313\pi$$
$$110$$ 0 0
$$111$$ 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 0
$$116$$ −3.56178 −0.330703
$$117$$ −0.737669 −0.0681975
$$118$$ 8.97056 0.825807
$$119$$ −6.34315 −0.581475
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 3.56178 0.322469
$$123$$ 0 0
$$124$$ −1.47534 −0.132489
$$125$$ 0 0
$$126$$ −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
$$130$$ 0 0
$$131$$ 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
$$135$$ 0 0
$$136$$ −23.2781 −1.99608
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 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 0
$$141$$ 18.6731 1.57256
$$142$$ −17.1716 −1.44101
$$143$$ −8.59890 −0.719076
$$144$$ −0.514719 −0.0428932
$$145$$ 0 0
$$146$$ −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 $$0.707786\pi$$
$$150$$ 0 0
$$151$$ −10.6853 −0.869561 −0.434781 0.900536i $$-0.643174\pi$$
−0.434781 + 0.900536i $$0.643174\pi$$
$$152$$ 0 0
$$153$$ −1.31371 −0.106207
$$154$$ −2.08644 −0.168130
$$155$$ 0 0
$$156$$ −3.17157 −0.253929
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −8.97056 −0.713660
$$159$$ −14.0000 −1.11027
$$160$$ 0 0
$$161$$ 0.686292 0.0540873
$$162$$ 11.9446 0.938458
$$163$$ −7.17157 −0.561721 −0.280860 0.959749i $$-0.590620\pi$$
−0.280860 + 0.959749i $$0.590620\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 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
$$170$$ 0 0
$$171$$ 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 0
$$176$$ −6.00000 −0.452267
$$177$$ 12.6863 0.953560
$$178$$ 19.7990 1.48400
$$179$$ 17.1978 1.28542 0.642712 0.766108i $$-0.277809\pi$$
0.642712 + 0.766108i $$0.277809\pi$$
$$180$$ 0 0
$$181$$ −10.0742 −0.748812 −0.374406 0.927265i $$-0.622153\pi$$
−0.374406 + 0.927265i $$0.622153\pi$$
$$182$$ 4.48528 0.332471
$$183$$ 5.03712 0.372355
$$184$$ 2.51856 0.185671
$$185$$ 0 0
$$186$$ 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
$$195$$ 0 0
$$196$$ 2.61522 0.186802
$$197$$ −9.31371 −0.663574 −0.331787 0.943354i $$-0.607652\pi$$
−0.331787 + 0.943354i $$0.607652\pi$$
$$198$$ −0.432117 −0.0307092
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 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
$$210$$ 0 0
$$211$$ −13.6360 −0.938743 −0.469371 0.883001i $$-0.655519\pi$$
−0.469371 + 0.883001i $$0.655519\pi$$
$$212$$ −3.25623 −0.223639
$$213$$ −24.2843 −1.66393
$$214$$ −17.5563 −1.20013
$$215$$ 0 0
$$216$$ 15.3137 1.04197
$$217$$ 2.95068 0.200305
$$218$$ 23.5147 1.59262
$$219$$ −20.7596 −1.40280
$$220$$ 0 0
$$221$$ 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 0
$$226$$ −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 $$0.547347\pi$$
$$230$$ 0 0
$$231$$ −2.95068 −0.194140
$$232$$ 26.1421 1.71632
$$233$$ 9.31371 0.610161 0.305081 0.952327i $$-0.401317\pi$$
0.305081 + 0.952327i $$0.401317\pi$$
$$234$$ 0.928932 0.0607262
$$235$$ 0 0
$$236$$ 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 $$0.398892\pi$$
$$240$$ 0 0
$$241$$ 24.3214 1.56668 0.783339 0.621595i $$-0.213515\pi$$
0.783339 + 0.621595i $$0.213515\pi$$
$$242$$ 8.81496 0.566647
$$243$$ 1.78089 0.114244
$$244$$ 1.17157 0.0750023
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 10.8284 0.687606
$$249$$ 15.7225 0.996371
$$250$$ 0 0
$$251$$ 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
$$255$$ 0 0
$$256$$ −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
$$260$$ 0 0
$$261$$ 1.47534 0.0913212
$$262$$ −19.2842 −1.19138
$$263$$ 16.1421 0.995367 0.497683 0.867359i $$-0.334184\pi$$
0.497683 + 0.867359i $$0.334184\pi$$
$$264$$ −10.8284 −0.666444
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 28.0000 1.71357
$$268$$ 2.82411 0.172510
$$269$$ 8.59890 0.524284 0.262142 0.965029i $$-0.415571\pi$$
0.262142 + 0.965029i $$0.415571\pi$$
$$270$$ 0 0
$$271$$ 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
$$275$$ 0 0
$$276$$ 0.611105 0.0367842
$$277$$ 17.3137 1.04028 0.520140 0.854081i $$-0.325880\pi$$
0.520140 + 0.854081i $$0.325880\pi$$
$$278$$ −24.7535 −1.48462
$$279$$ 0.611105 0.0365859
$$280$$ 0 0
$$281$$ 14.2471 0.849912 0.424956 0.905214i $$-0.360289\pi$$
0.424956 + 0.905214i $$0.360289\pi$$
$$282$$ −23.5147 −1.40028
$$283$$ 9.79899 0.582489 0.291245 0.956649i $$-0.405931\pi$$
0.291245 + 0.956649i $$0.405931\pi$$
$$284$$ −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
$$290$$ 0 0
$$291$$ 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
$$295$$ 0 0
$$296$$ −2.24264 −0.130351
$$297$$ 10.0742 0.584567
$$298$$ 18.6731 1.08171
$$299$$ −3.56178 −0.205983
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 13.4558 0.774297
$$303$$ −0.864233 −0.0496489
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 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
$$310$$ 0 0
$$311$$ −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 $$-0.745044\pi$$
−0.696012 + 0.718030i $$0.745044\pi$$
$$314$$ 22.6670 1.27918
$$315$$ 0 0
$$316$$ −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
$$320$$ 0 0
$$321$$ −24.8284 −1.38579
$$322$$ −0.864233 −0.0481618
$$323$$ 0 0
$$324$$ 3.92893 0.218274
$$325$$ 0 0
$$326$$ 9.03102 0.500182
$$327$$ 33.2548 1.83900
$$328$$ 0 0
$$329$$ −8.68629 −0.478891
$$330$$ 0 0
$$331$$ −17.8089 −0.978866 −0.489433 0.872041i $$-0.662796\pi$$
−0.489433 + 0.872041i $$0.662796\pi$$
$$332$$ 3.65685 0.200696
$$333$$ −0.126564 −0.00693567
$$334$$ −21.2721 −1.16396
$$335$$ 0 0
$$336$$ 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
$$340$$ 0 0
$$341$$ 7.12356 0.385763
$$342$$ 0 0
$$343$$ −11.0294 −0.595534
$$344$$ 14.6792 0.791452
$$345$$ 0 0
$$346$$ 3.55635 0.191191
$$347$$ −19.1716 −1.02918 −0.514592 0.857435i $$-0.672057\pi$$
−0.514592 + 0.857435i $$0.672057\pi$$
$$348$$ 6.34315 0.340028
$$349$$ −29.3137 −1.56913 −0.784563 0.620049i $$-0.787113\pi$$
−0.784563 + 0.620049i $$0.787113\pi$$
$$350$$ 0 0
$$351$$ −21.6569 −1.15596
$$352$$ −4.60500 −0.245448
$$353$$ −7.65685 −0.407533 −0.203767 0.979019i $$-0.565318\pi$$
−0.203767 + 0.979019i $$0.565318\pi$$
$$354$$ −15.9756 −0.849093
$$355$$ 0 0
$$356$$ 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 0
$$361$$ 0 0
$$362$$ 12.6863 0.666777
$$363$$ 12.4662 0.654308
$$364$$ 1.47534 0.0773287
$$365$$ 0 0
$$366$$ −6.34315 −0.331562
$$367$$ −3.17157 −0.165555 −0.0827774 0.996568i $$-0.526379\pi$$
−0.0827774 + 0.996568i $$0.526379\pi$$
$$368$$ −2.48528 −0.129554
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 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
$$375$$ 0 0
$$376$$ −31.8771 −1.64393
$$377$$ −36.9706 −1.90408
$$378$$ −5.25483 −0.270279
$$379$$ −17.1978 −0.883392 −0.441696 0.897165i $$-0.645623\pi$$
−0.441696 + 0.897165i $$0.645623\pi$$
$$380$$ 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
$$385$$ 0 0
$$386$$ −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 $$0.675160\pi$$
$$390$$ 0 0
$$391$$ −6.34315 −0.320787
$$392$$ −19.1948 −0.969482
$$393$$ −27.2720 −1.37569
$$394$$ 11.7286 0.590877
$$395$$ 0 0
$$396$$ −0.142136 −0.00714258
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 5.03712 0.252488
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −10.0742 −0.503084 −0.251542 0.967846i $$-0.580938\pi$$
−0.251542 + 0.967846i $$0.580938\pi$$
$$402$$ −15.2904 −0.762613
$$403$$ −15.3137 −0.762830
$$404$$ −0.201010 −0.0100006
$$405$$ 0 0
$$406$$ −8.97056 −0.445202
$$407$$ −1.47534 −0.0731298
$$408$$ 41.4558 2.05237
$$409$$ 12.7718 0.631524 0.315762 0.948838i $$-0.397740\pi$$
0.315762 + 0.948838i $$0.397740\pi$$
$$410$$ 0 0
$$411$$ −3.56178 −0.175690
$$412$$ −2.21301 −0.109027
$$413$$ −5.90135 −0.290387
$$414$$ −0.178989 −0.00879681
$$415$$ 0 0
$$416$$ 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 0
$$421$$ −10.0742 −0.490988 −0.245494 0.969398i $$-0.578950\pi$$
−0.245494 + 0.969398i $$0.578950\pi$$
$$422$$ 17.1716 0.835899
$$423$$ −1.79899 −0.0874699
$$424$$ 23.8995 1.16066
$$425$$ 0 0
$$426$$ 30.5807 1.48164
$$427$$ −2.34315 −0.113393
$$428$$ −5.77479 −0.279135
$$429$$ 15.3137 0.739353
$$430$$ 0 0
$$431$$ 3.56178 0.171565 0.0857825 0.996314i $$-0.472661\pi$$
0.0857825 + 0.996314i $$0.472661\pi$$
$$432$$ −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
$$435$$ 0 0
$$436$$ 7.73467 0.370423
$$437$$ 0 0
$$438$$ 26.1421 1.24912
$$439$$ 2.95068 0.140828 0.0704141 0.997518i $$-0.477568\pi$$
0.0704141 + 0.997518i $$0.477568\pi$$
$$440$$ 0 0
$$441$$ −1.08326 −0.0515839
$$442$$ −41.4558 −1.97185
$$443$$ −20.1421 −0.956982 −0.478491 0.878093i $$-0.658816\pi$$
−0.478491 + 0.878093i $$0.658816\pi$$
$$444$$ −0.544156 −0.0258245
$$445$$ 0 0
$$446$$ 0.384776 0.0182197
$$447$$ 26.4078 1.24905
$$448$$ 7.37258 0.348322
$$449$$ −8.59890 −0.405807 −0.202904 0.979199i $$-0.565038\pi$$
−0.202904 + 0.979199i $$0.565038\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −3.86733 −0.181904
$$453$$ 19.0294 0.894081
$$454$$ −21.2721 −0.998348
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5.31371 −0.248565 −0.124282 0.992247i $$-0.539663\pi$$
−0.124282 + 0.992247i $$0.539663\pi$$
$$458$$ 5.64823 0.263924
$$459$$ −38.5685 −1.80022
$$460$$ 0 0
$$461$$ −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 $$-0.223808\pi$$
0.762833 + 0.646595i $$0.223808\pi$$
$$464$$ −25.7967 −1.19758
$$465$$ 0 0
$$466$$ −11.7286 −0.543315
$$467$$ −38.4853 −1.78089 −0.890443 0.455094i $$-0.849606\pi$$
−0.890443 + 0.455094i $$0.849606\pi$$
$$468$$ 0.305553 0.0141242
$$469$$ −5.64823 −0.260811
$$470$$ 0 0
$$471$$ 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 $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 3.17157 0.144611
$$482$$ −30.6274 −1.39504
$$483$$ −1.22221 −0.0556125
$$484$$ 2.89949 0.131795
$$485$$ 0 0
$$486$$ −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
$$490$$ 0 0
$$491$$ 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 0
$$496$$ −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 0
$$501$$ −30.0833 −1.34402
$$502$$ −26.4078 −1.17864
$$503$$ −2.48528 −0.110813 −0.0554066 0.998464i $$-0.517646\pi$$
−0.0554066 + 0.998464i $$0.517646\pi$$
$$504$$ 0.432117 0.0192480
$$505$$ 0 0
$$506$$ −2.08644 −0.0927537
$$507$$ −9.76869 −0.433843
$$508$$ −7.25013 −0.321672
$$509$$ −32.9203 −1.45917 −0.729583 0.683893i $$-0.760286\pi$$
−0.729583 + 0.683893i $$0.760286\pi$$
$$510$$ 0 0
$$511$$ 9.65685 0.427194
$$512$$ 25.1485 1.11142
$$513$$ 0 0
$$514$$ 25.2132 1.11211
$$515$$ 0 0
$$516$$ 3.56178 0.156799
$$517$$ −20.9706 −0.922284
$$518$$ 0.769553 0.0338122
$$519$$ 5.02944 0.220768
$$520$$ 0 0
$$521$$ −17.1978 −0.753450 −0.376725 0.926325i $$-0.622950\pi$$
−0.376725 + 0.926325i $$0.622950\pi$$
$$522$$ −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
$$525$$ 0 0
$$526$$ −20.3275 −0.886320
$$527$$ −27.2720 −1.18799
$$528$$ 10.6853 0.465020
$$529$$ −22.3137 −0.970161
$$530$$ 0 0
$$531$$ −1.22221 −0.0530394
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −35.2598 −1.52584
$$535$$ 0 0
$$536$$ −20.7279 −0.895310
$$537$$ −30.6274 −1.32167
$$538$$ −10.8284 −0.466847
$$539$$ −12.6274 −0.543901
$$540$$ 0 0
$$541$$ 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
$$545$$ 0 0
$$546$$ −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
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −4.48528 −0.190906
$$553$$ 5.90135 0.250951
$$554$$ −21.8028 −0.926313
$$555$$ 0 0
$$556$$ −8.14214 −0.345303
$$557$$ 28.6274 1.21298 0.606491 0.795090i $$-0.292576\pi$$
0.606491 + 0.795090i $$0.292576\pi$$
$$558$$ −0.769553 −0.0325778
$$559$$ −20.7596 −0.878037
$$560$$ 0 0
$$561$$ 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
$$565$$ 0 0
$$566$$ −12.3397 −0.518675
$$567$$ −7.85786 −0.329999
$$568$$ 41.4558 1.73945
$$569$$ −15.7225 −0.659120 −0.329560 0.944135i $$-0.606900\pi$$
−0.329560 + 0.944135i $$0.606900\pi$$
$$570$$ 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 0
$$576$$ 1.52691 0.0636213
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ −52.4206 −2.18041
$$579$$ −35.6569 −1.48185
$$580$$ 0 0
$$581$$ −7.31371 −0.303424
$$582$$ −9.64212 −0.399679
$$583$$ 15.7225 0.651158
$$584$$ 35.4388 1.46647
$$585$$ 0 0
$$586$$ −22.5858 −0.933010
$$587$$ −18.4853 −0.762969 −0.381485 0.924375i $$-0.624587\pi$$
−0.381485 + 0.924375i $$0.624587\pi$$
$$588$$ −4.65743 −0.192069
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 16.5867 0.682286
$$592$$ 2.21301 0.0909541
$$593$$ 9.31371 0.382468 0.191234 0.981544i $$-0.438751\pi$$
0.191234 + 0.981544i $$0.438751\pi$$
$$594$$ −12.6863 −0.520525
$$595$$ 0 0
$$596$$ 6.14214 0.251592
$$597$$ 7.12356 0.291548
$$598$$ 4.48528 0.183417
$$599$$ −34.3956 −1.40537 −0.702683 0.711503i $$-0.748015\pi$$
−0.702683 + 0.711503i $$0.748015\pi$$
$$600$$ 0 0
$$601$$ −17.1978 −0.701513 −0.350757 0.936467i $$-0.614076\pi$$
−0.350757 + 0.936467i $$0.614076\pi$$
$$602$$ −5.03712 −0.205298
$$603$$ −1.16979 −0.0476374
$$604$$ 4.42602 0.180092
$$605$$ 0 0
$$606$$ 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
$$610$$ 0 0
$$611$$ 45.0810 1.82378
$$612$$ 0.544156 0.0219962
$$613$$ 14.2843 0.576936 0.288468 0.957489i $$-0.406854\pi$$
0.288468 + 0.957489i $$0.406854\pi$$
$$614$$ 4.10051 0.165483
$$615$$ 0 0
$$616$$ 5.03712 0.202951
$$617$$ −30.2843 −1.21920 −0.609599 0.792710i $$-0.708670\pi$$
−0.609599 + 0.792710i $$0.708670\pi$$
$$618$$ 11.9817 0.481975
$$619$$ −18.9706 −0.762491 −0.381246 0.924474i $$-0.624505\pi$$
−0.381246 + 0.924474i $$0.624505\pi$$
$$620$$ 0 0
$$621$$ 4.17289 0.167452
$$622$$ 25.6177 1.02718
$$623$$ −13.0249 −0.521832
$$624$$ −22.9706 −0.919558
$$625$$ 0 0
$$626$$ 31.0128 1.23952
$$627$$ 0 0
$$628$$ 7.45584 0.297521
$$629$$ 5.64823 0.225210
$$630$$ 0 0
$$631$$ 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
$$635$$ 0 0
$$636$$ 5.79899 0.229945
$$637$$ 27.1455 1.07554
$$638$$ −21.6569 −0.857403
$$639$$ 2.33957 0.0925520
$$640$$ 0 0
$$641$$ −21.3707 −0.844092 −0.422046 0.906575i $$-0.638688\pi$$
−0.422046 + 0.906575i $$0.638688\pi$$
$$642$$ 31.2659 1.23397
$$643$$ −9.51472 −0.375224 −0.187612 0.982243i $$-0.560075\pi$$
−0.187612 + 0.982243i $$0.560075\pi$$
$$644$$ −0.284271 −0.0112019
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27.1716 1.06822 0.534112 0.845413i $$-0.320646\pi$$
0.534112 + 0.845413i $$0.320646\pi$$
$$648$$ −28.8369 −1.13282
$$649$$ −14.2471 −0.559249
$$650$$ 0 0
$$651$$ −5.25483 −0.205953
$$652$$ 2.97056 0.116336
$$653$$ −27.6569 −1.08230 −0.541148 0.840927i $$-0.682010\pi$$
−0.541148 + 0.840927i $$0.682010\pi$$
$$654$$ −41.8772 −1.63753
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 10.9385 0.426426
$$659$$ −20.1485 −0.784873 −0.392437 0.919779i $$-0.628368\pi$$
−0.392437 + 0.919779i $$0.628368\pi$$
$$660$$ 0 0
$$661$$ 2.95068 0.114768 0.0573840 0.998352i $$-0.481724\pi$$
0.0573840 + 0.998352i $$0.481724\pi$$
$$662$$ 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
$$670$$ 0 0
$$671$$ −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
$$675$$ 0 0
$$676$$ −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
$$680$$ 0 0
$$681$$ −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
$$685$$ 0 0
$$686$$ 13.8892 0.530290
$$687$$ 7.98780 0.304753
$$688$$ −14.4853 −0.552246
$$689$$ −33.7990 −1.28764
$$690$$ 0 0
$$691$$ 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
$$695$$ 0 0
$$696$$ −46.5563 −1.76471
$$697$$ 0 0
$$698$$ 36.9142 1.39722
$$699$$ −16.5867 −0.627367
$$700$$ 0 0
$$701$$ −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
$$705$$ 0 0
$$706$$ 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 $$0.531814\pi$$
$$710$$ 0 0
$$711$$ 1.22221 0.0458365
$$712$$ −47.7990 −1.79134
$$713$$ 2.95068 0.110504
$$714$$ −14.2254 −0.532372
$$715$$ 0 0
$$716$$ −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 0
$$721$$ 4.42602 0.164833
$$722$$ 0 0
$$723$$ −43.3137 −1.61085
$$724$$ 4.17289 0.155084
$$725$$ 0 0
$$726$$ −15.6985 −0.582625
$$727$$ −6.48528 −0.240526 −0.120263 0.992742i $$-0.538374\pi$$
−0.120263 + 0.992742i $$0.538374\pi$$
$$728$$ −10.8284 −0.401328
$$729$$ 25.2843 0.936454
$$730$$ 0 0
$$731$$ −36.9706 −1.36741
$$732$$ −2.08644 −0.0771172
$$733$$ 35.6569 1.31702 0.658508 0.752574i $$-0.271188\pi$$
0.658508 + 0.752574i $$0.271188\pi$$
$$734$$ 3.99390 0.147417
$$735$$ 0 0
$$736$$ −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 0
$$741$$ 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
$$745$$ 0 0
$$746$$ −13.6152 −0.498489
$$747$$ −1.51472 −0.0554207
$$748$$ 6.34315 0.231928
$$749$$ 11.5496 0.422012
$$750$$ 0 0
$$751$$ −6.51246 −0.237643 −0.118822 0.992916i $$-0.537912\pi$$
−0.118822 + 0.992916i $$0.537912\pi$$
$$752$$ 31.4558 1.14708
$$753$$ −37.3463 −1.36097
$$754$$ 46.5563 1.69548
$$755$$ 0 0
$$756$$ −1.72847 −0.0628637
$$757$$ 6.68629 0.243017 0.121509 0.992590i $$-0.461227\pi$$
0.121509 + 0.992590i $$0.461227\pi$$
$$758$$ 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
$$765$$ 0 0
$$766$$ 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 $$0.417928\pi$$
$$770$$ 0 0
$$771$$ 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
$$775$$ 0 0
$$776$$ −13.0711 −0.469224
$$777$$ 1.08831 0.0390430
$$778$$ 25.9757 0.931274
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 27.2720 0.975871
$$782$$ 7.98780 0.285643
$$783$$ 43.3137 1.54791
$$784$$ 18.9411 0.676469
$$785$$ 0 0
$$786$$ 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
$$790$$ 0 0
$$791$$ 7.73467 0.275013
$$792$$ 1.04322 0.0370693
$$793$$ 12.1607 0.431839
$$794$$ −17.6299 −0.625663
$$795$$ 0 0
$$796$$ 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
$$800$$ 0 0
$$801$$ −2.69755 −0.0953132
$$802$$ 12.6863 0.447969
$$803$$ 23.3137 0.822723
$$804$$ −5.02944 −0.177375
$$805$$ 0 0
$$806$$ 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 $$0.575196\pi$$
$$810$$ 0 0
$$811$$ −33.7845 −1.18633 −0.593167 0.805079i $$-0.702123\pi$$
−0.593167 + 0.805079i $$0.702123\pi$$
$$812$$ −2.95068 −0.103548
$$813$$ −55.1552 −1.93438
$$814$$ 1.85786 0.0651181
$$815$$ 0 0
$$816$$ −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 $$-0.762098\pi$$
−0.733465 + 0.679727i $$0.762098\pi$$
$$824$$ 16.2426 0.565839
$$825$$ 0 0
$$826$$ 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 $$-0.231670\pi$$
0.746631 + 0.665239i $$0.231670\pi$$
$$830$$ 0 0
$$831$$ −30.8338 −1.06961
$$832$$ −38.2629 −1.32653
$$833$$ 48.3431 1.67499
$$834$$ 44.0833 1.52648
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 17.9411 0.620136
$$838$$ 31.4449 1.08625
$$839$$ −31.4449 −1.08560 −0.542800 0.839862i $$-0.682636\pi$$
−0.542800 + 0.839862i $$0.682636\pi$$
$$840$$ 0 0
$$841$$ 44.9411 1.54969
$$842$$ 12.6863 0.437198
$$843$$ −25.3726 −0.873878
$$844$$ 5.64823 0.194420
$$845$$ 0 0
$$846$$ 2.26543 0.0778872
$$847$$ −5.79899 −0.199256
$$848$$ −23.5837 −0.809868
$$849$$ −17.4509 −0.598914
$$850$$ 0 0
$$851$$ −0.611105 −0.0209484
$$852$$ 10.0589 0.344611
$$853$$ −36.3431 −1.24437 −0.622183 0.782872i $$-0.713754\pi$$
−0.622183 + 0.782872i $$0.713754\pi$$
$$854$$ 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 $$0.772596\pi$$
$$860$$ 0 0
$$861$$ 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
$$865$$ 0 0
$$866$$ 6.18377 0.210133
$$867$$ −74.1339 −2.51772
$$868$$ −1.22221 −0.0414845
$$869$$ 14.2471 0.483301
$$870$$ 0 0
$$871$$ 29.3137 0.993257
$$872$$ −56.7696 −1.92246
$$873$$ −0.737669 −0.0249663
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 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
$$880$$ 0 0
$$881$$ 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 $$-0.520677\pi$$
−0.0649137 + 0.997891i $$0.520677\pi$$
$$884$$ −13.6360 −0.458629
$$885$$ 0 0
$$886$$ 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
$$890$$ 0 0
$$891$$ −18.9706 −0.635538
$$892$$ 0.126564 0.00423768
$$893$$ 0 0
$$894$$ −33.2548 −1.11221
$$895$$ 0 0
$$896$$ −5.46924 −0.182714
$$897$$ 6.34315 0.211791
$$898$$ 10.8284 0.361349
$$899$$ 30.6274 1.02148
$$900$$ 0 0
$$901$$ −60.1923 −2.00530
$$902$$ 0 0
$$903$$ −7.12356 −0.237057
$$904$$ 28.3848 0.944064
$$905$$ 0 0
$$906$$ −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
$$910$$ 0 0
$$911$$ −26.6609 −0.883316 −0.441658 0.897183i $$-0.645610\pi$$
−0.441658 + 0.897183i $$0.645610\pi$$
$$912$$ 0 0
$$913$$ −17.6569 −0.584357
$$914$$ 6.69145 0.221333
$$915$$ 0 0
$$916$$ 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 $$0.498508\pi$$
$$920$$ 0 0
$$921$$ 5.79899 0.191083
$$922$$ 13.4570 0.443184
$$923$$ −58.6274 −1.92974
$$924$$ 1.22221 0.0402078
$$925$$ 0 0
$$926$$ −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 $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −3.85786 −0.126369
$$933$$ 36.2289 1.18608
$$934$$ 48.4638 1.58578
$$935$$ 0 0
$$936$$ −2.24264 −0.0733030
$$937$$ −10.9706 −0.358393 −0.179196 0.983813i $$-0.557350\pi$$
−0.179196 + 0.983813i $$0.557350\pi$$
$$938$$ 7.11270 0.232238
$$939$$ 43.8587 1.43128
$$940$$ 0 0
$$941$$ 2.95068 0.0961893 0.0480947 0.998843i $$-0.484685\pi$$
0.0480947 + 0.998843i $$0.484685\pi$$
$$942$$ −40.3675 −1.31525
$$943$$ 0 0
$$944$$ 21.3707 0.695557
$$945$$ 0 0
$$946$$ −12.1607 −0.395378
$$947$$ 4.14214 0.134601 0.0673007 0.997733i $$-0.478561\pi$$
0.0673007 + 0.997733i $$0.478561\pi$$
$$948$$ 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
$$955$$ 0 0
$$956$$ −4.00000 −0.129369
$$957$$ −30.6274 −0.990044
$$958$$ −12.5928 −0.406855
$$959$$ 1.65685 0.0535026
$$960$$ 0 0
$$961$$ −18.3137 −0.590765
$$962$$ −3.99390 −0.128768
$$963$$ 2.39200 0.0770810
$$964$$ −10.0742 −0.324469
$$965$$ 0 0
$$966$$ 1.53911 0.0495199
$$967$$ −21.5147 −0.691867 −0.345933 0.938259i $$-0.612438\pi$$
−0.345933 + 0.938259i $$0.612438\pi$$
$$968$$ −21.2812 −0.684004
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 7.73467 0.248217 0.124109 0.992269i $$-0.460393\pi$$
0.124109 + 0.992269i $$0.460393\pi$$
$$972$$ −0.737669 −0.0236608
$$973$$ 16.2843 0.522050
$$974$$ −37.3553 −1.19694
$$975$$ 0 0
$$976$$ 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
$$980$$ 0 0
$$981$$ −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
$$985$$ 0 0
$$986$$ 82.9117 2.64045
$$987$$ 15.4693 0.492394
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 27.8832 0.885737 0.442869 0.896586i $$-0.353961\pi$$
0.442869 + 0.896586i $$0.353961\pi$$
$$992$$ −8.20101 −0.260382
$$993$$ 31.7157 1.00647
$$994$$ −14.2254 −0.451202
$$995$$ 0 0
$$996$$ −6.51246 −0.206355
$$997$$ 7.65685 0.242495 0.121248 0.992622i $$-0.461311\pi$$
0.121248 + 0.992622i $$0.461311\pi$$
$$998$$ −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 9025.2.a.bn.1.2 4
5.4 even 2 1805.2.a.m.1.3 yes 4
19.18 odd 2 inner 9025.2.a.bn.1.3 4
95.94 odd 2 1805.2.a.m.1.2 4

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