Properties

 Label 74.2.a.a.1.2 Level $74$ Weight $2$ Character 74.1 Self dual yes Analytic conductor $0.591$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(1,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("74.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.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: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 74.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} -2.30278 q^{5} -3.30278 q^{6} -2.60555 q^{7} -1.00000 q^{8} +7.90833 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} -2.30278 q^{5} -3.30278 q^{6} -2.60555 q^{7} -1.00000 q^{8} +7.90833 q^{9} +2.30278 q^{10} -2.30278 q^{11} +3.30278 q^{12} +1.30278 q^{13} +2.60555 q^{14} -7.60555 q^{15} +1.00000 q^{16} -6.00000 q^{17} -7.90833 q^{18} +2.00000 q^{19} -2.30278 q^{20} -8.60555 q^{21} +2.30278 q^{22} +3.90833 q^{23} -3.30278 q^{24} +0.302776 q^{25} -1.30278 q^{26} +16.2111 q^{27} -2.60555 q^{28} -3.90833 q^{29} +7.60555 q^{30} -0.302776 q^{31} -1.00000 q^{32} -7.60555 q^{33} +6.00000 q^{34} +6.00000 q^{35} +7.90833 q^{36} +1.00000 q^{37} -2.00000 q^{38} +4.30278 q^{39} +2.30278 q^{40} +9.90833 q^{41} +8.60555 q^{42} +0.605551 q^{43} -2.30278 q^{44} -18.2111 q^{45} -3.90833 q^{46} +4.60555 q^{47} +3.30278 q^{48} -0.211103 q^{49} -0.302776 q^{50} -19.8167 q^{51} +1.30278 q^{52} -6.00000 q^{53} -16.2111 q^{54} +5.30278 q^{55} +2.60555 q^{56} +6.60555 q^{57} +3.90833 q^{58} +10.6056 q^{59} -7.60555 q^{60} +7.51388 q^{61} +0.302776 q^{62} -20.6056 q^{63} +1.00000 q^{64} -3.00000 q^{65} +7.60555 q^{66} -3.51388 q^{67} -6.00000 q^{68} +12.9083 q^{69} -6.00000 q^{70} +6.00000 q^{71} -7.90833 q^{72} -12.3028 q^{73} -1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +6.00000 q^{77} -4.30278 q^{78} +9.11943 q^{79} -2.30278 q^{80} +29.8167 q^{81} -9.90833 q^{82} +2.78890 q^{83} -8.60555 q^{84} +13.8167 q^{85} -0.605551 q^{86} -12.9083 q^{87} +2.30278 q^{88} -9.21110 q^{89} +18.2111 q^{90} -3.39445 q^{91} +3.90833 q^{92} -1.00000 q^{93} -4.60555 q^{94} -4.60555 q^{95} -3.30278 q^{96} -16.4222 q^{97} +0.211103 q^{98} -18.2111 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - q^5 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} - q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 12 q^{17} - 5 q^{18} + 4 q^{19} - q^{20} - 10 q^{21} + q^{22} - 3 q^{23} - 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} + 2 q^{28} + 3 q^{29} + 8 q^{30} + 3 q^{31} - 2 q^{32} - 8 q^{33} + 12 q^{34} + 12 q^{35} + 5 q^{36} + 2 q^{37} - 4 q^{38} + 5 q^{39} + q^{40} + 9 q^{41} + 10 q^{42} - 6 q^{43} - q^{44} - 22 q^{45} + 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} + 3 q^{50} - 18 q^{51} - q^{52} - 12 q^{53} - 18 q^{54} + 7 q^{55} - 2 q^{56} + 6 q^{57} - 3 q^{58} + 14 q^{59} - 8 q^{60} - 3 q^{61} - 3 q^{62} - 34 q^{63} + 2 q^{64} - 6 q^{65} + 8 q^{66} + 11 q^{67} - 12 q^{68} + 15 q^{69} - 12 q^{70} + 12 q^{71} - 5 q^{72} - 21 q^{73} - 2 q^{74} + 2 q^{75} + 4 q^{76} + 12 q^{77} - 5 q^{78} - 7 q^{79} - q^{80} + 38 q^{81} - 9 q^{82} + 20 q^{83} - 10 q^{84} + 6 q^{85} + 6 q^{86} - 15 q^{87} + q^{88} - 4 q^{89} + 22 q^{90} - 14 q^{91} - 3 q^{92} - 2 q^{93} - 2 q^{94} - 2 q^{95} - 3 q^{96} - 4 q^{97} - 14 q^{98} - 22 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - q^5 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 5 * q^9 + q^10 - q^11 + 3 * q^12 - q^13 - 2 * q^14 - 8 * q^15 + 2 * q^16 - 12 * q^17 - 5 * q^18 + 4 * q^19 - q^20 - 10 * q^21 + q^22 - 3 * q^23 - 3 * q^24 - 3 * q^25 + q^26 + 18 * q^27 + 2 * q^28 + 3 * q^29 + 8 * q^30 + 3 * q^31 - 2 * q^32 - 8 * q^33 + 12 * q^34 + 12 * q^35 + 5 * q^36 + 2 * q^37 - 4 * q^38 + 5 * q^39 + q^40 + 9 * q^41 + 10 * q^42 - 6 * q^43 - q^44 - 22 * q^45 + 3 * q^46 + 2 * q^47 + 3 * q^48 + 14 * q^49 + 3 * q^50 - 18 * q^51 - q^52 - 12 * q^53 - 18 * q^54 + 7 * q^55 - 2 * q^56 + 6 * q^57 - 3 * q^58 + 14 * q^59 - 8 * q^60 - 3 * q^61 - 3 * q^62 - 34 * q^63 + 2 * q^64 - 6 * q^65 + 8 * q^66 + 11 * q^67 - 12 * q^68 + 15 * q^69 - 12 * q^70 + 12 * q^71 - 5 * q^72 - 21 * q^73 - 2 * q^74 + 2 * q^75 + 4 * q^76 + 12 * q^77 - 5 * q^78 - 7 * q^79 - q^80 + 38 * q^81 - 9 * q^82 + 20 * q^83 - 10 * q^84 + 6 * q^85 + 6 * q^86 - 15 * q^87 + q^88 - 4 * q^89 + 22 * q^90 - 14 * q^91 - 3 * q^92 - 2 * q^93 - 2 * q^94 - 2 * q^95 - 3 * q^96 - 4 * q^97 - 14 * q^98 - 22 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 3.30278 1.90686 0.953429 0.301617i $$-0.0975264\pi$$
0.953429 + 0.301617i $$0.0975264\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −2.30278 −1.02983 −0.514916 0.857240i $$-0.672177\pi$$
−0.514916 + 0.857240i $$0.672177\pi$$
$$6$$ −3.30278 −1.34835
$$7$$ −2.60555 −0.984806 −0.492403 0.870367i $$-0.663881\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 7.90833 2.63611
$$10$$ 2.30278 0.728202
$$11$$ −2.30278 −0.694313 −0.347156 0.937807i $$-0.612853\pi$$
−0.347156 + 0.937807i $$0.612853\pi$$
$$12$$ 3.30278 0.953429
$$13$$ 1.30278 0.361325 0.180662 0.983545i $$-0.442176\pi$$
0.180662 + 0.983545i $$0.442176\pi$$
$$14$$ 2.60555 0.696363
$$15$$ −7.60555 −1.96374
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −7.90833 −1.86401
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ −2.30278 −0.514916
$$21$$ −8.60555 −1.87789
$$22$$ 2.30278 0.490953
$$23$$ 3.90833 0.814942 0.407471 0.913218i $$-0.366411\pi$$
0.407471 + 0.913218i $$0.366411\pi$$
$$24$$ −3.30278 −0.674176
$$25$$ 0.302776 0.0605551
$$26$$ −1.30278 −0.255495
$$27$$ 16.2111 3.11983
$$28$$ −2.60555 −0.492403
$$29$$ −3.90833 −0.725758 −0.362879 0.931836i $$-0.618206\pi$$
−0.362879 + 0.931836i $$0.618206\pi$$
$$30$$ 7.60555 1.38858
$$31$$ −0.302776 −0.0543801 −0.0271901 0.999630i $$-0.508656\pi$$
−0.0271901 + 0.999630i $$0.508656\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −7.60555 −1.32396
$$34$$ 6.00000 1.02899
$$35$$ 6.00000 1.01419
$$36$$ 7.90833 1.31805
$$37$$ 1.00000 0.164399
$$38$$ −2.00000 −0.324443
$$39$$ 4.30278 0.688996
$$40$$ 2.30278 0.364101
$$41$$ 9.90833 1.54742 0.773710 0.633540i $$-0.218399\pi$$
0.773710 + 0.633540i $$0.218399\pi$$
$$42$$ 8.60555 1.32787
$$43$$ 0.605551 0.0923457 0.0461729 0.998933i $$-0.485297\pi$$
0.0461729 + 0.998933i $$0.485297\pi$$
$$44$$ −2.30278 −0.347156
$$45$$ −18.2111 −2.71475
$$46$$ −3.90833 −0.576251
$$47$$ 4.60555 0.671789 0.335894 0.941900i $$-0.390961\pi$$
0.335894 + 0.941900i $$0.390961\pi$$
$$48$$ 3.30278 0.476715
$$49$$ −0.211103 −0.0301575
$$50$$ −0.302776 −0.0428189
$$51$$ −19.8167 −2.77489
$$52$$ 1.30278 0.180662
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −16.2111 −2.20605
$$55$$ 5.30278 0.715026
$$56$$ 2.60555 0.348181
$$57$$ 6.60555 0.874927
$$58$$ 3.90833 0.513188
$$59$$ 10.6056 1.38073 0.690363 0.723464i $$-0.257451\pi$$
0.690363 + 0.723464i $$0.257451\pi$$
$$60$$ −7.60555 −0.981872
$$61$$ 7.51388 0.962054 0.481027 0.876706i $$-0.340264\pi$$
0.481027 + 0.876706i $$0.340264\pi$$
$$62$$ 0.302776 0.0384525
$$63$$ −20.6056 −2.59606
$$64$$ 1.00000 0.125000
$$65$$ −3.00000 −0.372104
$$66$$ 7.60555 0.936179
$$67$$ −3.51388 −0.429289 −0.214644 0.976692i $$-0.568859\pi$$
−0.214644 + 0.976692i $$0.568859\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 12.9083 1.55398
$$70$$ −6.00000 −0.717137
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −7.90833 −0.932005
$$73$$ −12.3028 −1.43993 −0.719965 0.694010i $$-0.755842\pi$$
−0.719965 + 0.694010i $$0.755842\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 1.00000 0.115470
$$76$$ 2.00000 0.229416
$$77$$ 6.00000 0.683763
$$78$$ −4.30278 −0.487193
$$79$$ 9.11943 1.02602 0.513008 0.858384i $$-0.328531\pi$$
0.513008 + 0.858384i $$0.328531\pi$$
$$80$$ −2.30278 −0.257458
$$81$$ 29.8167 3.31296
$$82$$ −9.90833 −1.09419
$$83$$ 2.78890 0.306121 0.153061 0.988217i $$-0.451087\pi$$
0.153061 + 0.988217i $$0.451087\pi$$
$$84$$ −8.60555 −0.938943
$$85$$ 13.8167 1.49863
$$86$$ −0.605551 −0.0652983
$$87$$ −12.9083 −1.38392
$$88$$ 2.30278 0.245477
$$89$$ −9.21110 −0.976375 −0.488187 0.872739i $$-0.662342\pi$$
−0.488187 + 0.872739i $$0.662342\pi$$
$$90$$ 18.2111 1.91962
$$91$$ −3.39445 −0.355835
$$92$$ 3.90833 0.407471
$$93$$ −1.00000 −0.103695
$$94$$ −4.60555 −0.475026
$$95$$ −4.60555 −0.472520
$$96$$ −3.30278 −0.337088
$$97$$ −16.4222 −1.66742 −0.833711 0.552201i $$-0.813788\pi$$
−0.833711 + 0.552201i $$0.813788\pi$$
$$98$$ 0.211103 0.0213246
$$99$$ −18.2111 −1.83028
$$100$$ 0.302776 0.0302776
$$101$$ −12.4222 −1.23606 −0.618028 0.786156i $$-0.712068\pi$$
−0.618028 + 0.786156i $$0.712068\pi$$
$$102$$ 19.8167 1.96214
$$103$$ −0.302776 −0.0298334 −0.0149167 0.999889i $$-0.504748\pi$$
−0.0149167 + 0.999889i $$0.504748\pi$$
$$104$$ −1.30278 −0.127748
$$105$$ 19.8167 1.93391
$$106$$ 6.00000 0.582772
$$107$$ 0.697224 0.0674032 0.0337016 0.999432i $$-0.489270\pi$$
0.0337016 + 0.999432i $$0.489270\pi$$
$$108$$ 16.2111 1.55991
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ −5.30278 −0.505600
$$111$$ 3.30278 0.313486
$$112$$ −2.60555 −0.246201
$$113$$ −3.21110 −0.302075 −0.151038 0.988528i $$-0.548261\pi$$
−0.151038 + 0.988528i $$0.548261\pi$$
$$114$$ −6.60555 −0.618667
$$115$$ −9.00000 −0.839254
$$116$$ −3.90833 −0.362879
$$117$$ 10.3028 0.952492
$$118$$ −10.6056 −0.976320
$$119$$ 15.6333 1.43310
$$120$$ 7.60555 0.694289
$$121$$ −5.69722 −0.517929
$$122$$ −7.51388 −0.680275
$$123$$ 32.7250 2.95071
$$124$$ −0.302776 −0.0271901
$$125$$ 10.8167 0.967471
$$126$$ 20.6056 1.83569
$$127$$ −19.2111 −1.70471 −0.852355 0.522964i $$-0.824826\pi$$
−0.852355 + 0.522964i $$0.824826\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 2.00000 0.176090
$$130$$ 3.00000 0.263117
$$131$$ 10.6056 0.926611 0.463306 0.886199i $$-0.346663\pi$$
0.463306 + 0.886199i $$0.346663\pi$$
$$132$$ −7.60555 −0.661978
$$133$$ −5.21110 −0.451860
$$134$$ 3.51388 0.303553
$$135$$ −37.3305 −3.21290
$$136$$ 6.00000 0.514496
$$137$$ 0.908327 0.0776036 0.0388018 0.999247i $$-0.487646\pi$$
0.0388018 + 0.999247i $$0.487646\pi$$
$$138$$ −12.9083 −1.09883
$$139$$ −1.90833 −0.161862 −0.0809311 0.996720i $$-0.525789\pi$$
−0.0809311 + 0.996720i $$0.525789\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 15.2111 1.28101
$$142$$ −6.00000 −0.503509
$$143$$ −3.00000 −0.250873
$$144$$ 7.90833 0.659027
$$145$$ 9.00000 0.747409
$$146$$ 12.3028 1.01818
$$147$$ −0.697224 −0.0575061
$$148$$ 1.00000 0.0821995
$$149$$ 19.8167 1.62344 0.811722 0.584044i $$-0.198531\pi$$
0.811722 + 0.584044i $$0.198531\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −20.6056 −1.67686 −0.838428 0.545012i $$-0.816525\pi$$
−0.838428 + 0.545012i $$0.816525\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −47.4500 −3.83610
$$154$$ −6.00000 −0.483494
$$155$$ 0.697224 0.0560024
$$156$$ 4.30278 0.344498
$$157$$ −7.21110 −0.575509 −0.287754 0.957704i $$-0.592909\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ −9.11943 −0.725503
$$159$$ −19.8167 −1.57156
$$160$$ 2.30278 0.182050
$$161$$ −10.1833 −0.802560
$$162$$ −29.8167 −2.34262
$$163$$ 8.42221 0.659678 0.329839 0.944037i $$-0.393006\pi$$
0.329839 + 0.944037i $$0.393006\pi$$
$$164$$ 9.90833 0.773710
$$165$$ 17.5139 1.36345
$$166$$ −2.78890 −0.216460
$$167$$ −5.51388 −0.426677 −0.213338 0.976978i $$-0.568434\pi$$
−0.213338 + 0.976978i $$0.568434\pi$$
$$168$$ 8.60555 0.663933
$$169$$ −11.3028 −0.869444
$$170$$ −13.8167 −1.05969
$$171$$ 15.8167 1.20953
$$172$$ 0.605551 0.0461729
$$173$$ −8.78890 −0.668207 −0.334104 0.942536i $$-0.608434\pi$$
−0.334104 + 0.942536i $$0.608434\pi$$
$$174$$ 12.9083 0.978578
$$175$$ −0.788897 −0.0596350
$$176$$ −2.30278 −0.173578
$$177$$ 35.0278 2.63285
$$178$$ 9.21110 0.690401
$$179$$ −13.8167 −1.03271 −0.516353 0.856376i $$-0.672711\pi$$
−0.516353 + 0.856376i $$0.672711\pi$$
$$180$$ −18.2111 −1.35738
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 3.39445 0.251613
$$183$$ 24.8167 1.83450
$$184$$ −3.90833 −0.288126
$$185$$ −2.30278 −0.169303
$$186$$ 1.00000 0.0733236
$$187$$ 13.8167 1.01037
$$188$$ 4.60555 0.335894
$$189$$ −42.2389 −3.07242
$$190$$ 4.60555 0.334122
$$191$$ −5.51388 −0.398970 −0.199485 0.979901i $$-0.563927\pi$$
−0.199485 + 0.979901i $$0.563927\pi$$
$$192$$ 3.30278 0.238357
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 16.4222 1.17905
$$195$$ −9.90833 −0.709550
$$196$$ −0.211103 −0.0150788
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 18.2111 1.29421
$$199$$ 26.4222 1.87302 0.936510 0.350640i $$-0.114036\pi$$
0.936510 + 0.350640i $$0.114036\pi$$
$$200$$ −0.302776 −0.0214095
$$201$$ −11.6056 −0.818592
$$202$$ 12.4222 0.874023
$$203$$ 10.1833 0.714731
$$204$$ −19.8167 −1.38744
$$205$$ −22.8167 −1.59358
$$206$$ 0.302776 0.0210954
$$207$$ 30.9083 2.14828
$$208$$ 1.30278 0.0903312
$$209$$ −4.60555 −0.318573
$$210$$ −19.8167 −1.36748
$$211$$ 10.3028 0.709272 0.354636 0.935004i $$-0.384605\pi$$
0.354636 + 0.935004i $$0.384605\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 19.8167 1.35781
$$214$$ −0.697224 −0.0476613
$$215$$ −1.39445 −0.0951006
$$216$$ −16.2111 −1.10303
$$217$$ 0.788897 0.0535538
$$218$$ −2.00000 −0.135457
$$219$$ −40.6333 −2.74574
$$220$$ 5.30278 0.357513
$$221$$ −7.81665 −0.525805
$$222$$ −3.30278 −0.221668
$$223$$ −5.81665 −0.389512 −0.194756 0.980852i $$-0.562391\pi$$
−0.194756 + 0.980852i $$0.562391\pi$$
$$224$$ 2.60555 0.174091
$$225$$ 2.39445 0.159630
$$226$$ 3.21110 0.213599
$$227$$ 13.8167 0.917044 0.458522 0.888683i $$-0.348379\pi$$
0.458522 + 0.888683i $$0.348379\pi$$
$$228$$ 6.60555 0.437463
$$229$$ 24.6056 1.62598 0.812990 0.582277i $$-0.197838\pi$$
0.812990 + 0.582277i $$0.197838\pi$$
$$230$$ 9.00000 0.593442
$$231$$ 19.8167 1.30384
$$232$$ 3.90833 0.256594
$$233$$ 8.51388 0.557763 0.278881 0.960326i $$-0.410036\pi$$
0.278881 + 0.960326i $$0.410036\pi$$
$$234$$ −10.3028 −0.673514
$$235$$ −10.6056 −0.691830
$$236$$ 10.6056 0.690363
$$237$$ 30.1194 1.95647
$$238$$ −15.6333 −1.01336
$$239$$ −17.5139 −1.13288 −0.566439 0.824103i $$-0.691679\pi$$
−0.566439 + 0.824103i $$0.691679\pi$$
$$240$$ −7.60555 −0.490936
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 5.69722 0.366231
$$243$$ 49.8444 3.19752
$$244$$ 7.51388 0.481027
$$245$$ 0.486122 0.0310572
$$246$$ −32.7250 −2.08647
$$247$$ 2.60555 0.165787
$$248$$ 0.302776 0.0192263
$$249$$ 9.21110 0.583730
$$250$$ −10.8167 −0.684105
$$251$$ −21.2111 −1.33883 −0.669416 0.742887i $$-0.733456\pi$$
−0.669416 + 0.742887i $$0.733456\pi$$
$$252$$ −20.6056 −1.29803
$$253$$ −9.00000 −0.565825
$$254$$ 19.2111 1.20541
$$255$$ 45.6333 2.85767
$$256$$ 1.00000 0.0625000
$$257$$ 3.21110 0.200303 0.100152 0.994972i $$-0.468067\pi$$
0.100152 + 0.994972i $$0.468067\pi$$
$$258$$ −2.00000 −0.124515
$$259$$ −2.60555 −0.161901
$$260$$ −3.00000 −0.186052
$$261$$ −30.9083 −1.91318
$$262$$ −10.6056 −0.655213
$$263$$ 13.8167 0.851971 0.425986 0.904730i $$-0.359927\pi$$
0.425986 + 0.904730i $$0.359927\pi$$
$$264$$ 7.60555 0.468089
$$265$$ 13.8167 0.848750
$$266$$ 5.21110 0.319513
$$267$$ −30.4222 −1.86181
$$268$$ −3.51388 −0.214644
$$269$$ −21.2111 −1.29326 −0.646632 0.762802i $$-0.723823\pi$$
−0.646632 + 0.762802i $$0.723823\pi$$
$$270$$ 37.3305 2.27186
$$271$$ −22.4222 −1.36205 −0.681026 0.732259i $$-0.738466\pi$$
−0.681026 + 0.732259i $$0.738466\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ −11.2111 −0.678527
$$274$$ −0.908327 −0.0548740
$$275$$ −0.697224 −0.0420442
$$276$$ 12.9083 0.776990
$$277$$ 0.119429 0.00717582 0.00358791 0.999994i $$-0.498858\pi$$
0.00358791 + 0.999994i $$0.498858\pi$$
$$278$$ 1.90833 0.114454
$$279$$ −2.39445 −0.143352
$$280$$ −6.00000 −0.358569
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ −15.2111 −0.905808
$$283$$ 24.6056 1.46265 0.731324 0.682030i $$-0.238903\pi$$
0.731324 + 0.682030i $$0.238903\pi$$
$$284$$ 6.00000 0.356034
$$285$$ −15.2111 −0.901028
$$286$$ 3.00000 0.177394
$$287$$ −25.8167 −1.52391
$$288$$ −7.90833 −0.466003
$$289$$ 19.0000 1.11765
$$290$$ −9.00000 −0.528498
$$291$$ −54.2389 −3.17954
$$292$$ −12.3028 −0.719965
$$293$$ 11.0278 0.644248 0.322124 0.946697i $$-0.395603\pi$$
0.322124 + 0.946697i $$0.395603\pi$$
$$294$$ 0.697224 0.0406630
$$295$$ −24.4222 −1.42192
$$296$$ −1.00000 −0.0581238
$$297$$ −37.3305 −2.16614
$$298$$ −19.8167 −1.14795
$$299$$ 5.09167 0.294459
$$300$$ 1.00000 0.0577350
$$301$$ −1.57779 −0.0909426
$$302$$ 20.6056 1.18572
$$303$$ −41.0278 −2.35698
$$304$$ 2.00000 0.114708
$$305$$ −17.3028 −0.990754
$$306$$ 47.4500 2.71253
$$307$$ 17.9083 1.02208 0.511041 0.859556i $$-0.329260\pi$$
0.511041 + 0.859556i $$0.329260\pi$$
$$308$$ 6.00000 0.341882
$$309$$ −1.00000 −0.0568880
$$310$$ −0.697224 −0.0395997
$$311$$ 15.9083 0.902078 0.451039 0.892504i $$-0.351053\pi$$
0.451039 + 0.892504i $$0.351053\pi$$
$$312$$ −4.30278 −0.243597
$$313$$ −9.02776 −0.510279 −0.255139 0.966904i $$-0.582121\pi$$
−0.255139 + 0.966904i $$0.582121\pi$$
$$314$$ 7.21110 0.406946
$$315$$ 47.4500 2.67350
$$316$$ 9.11943 0.513008
$$317$$ 9.21110 0.517347 0.258674 0.965965i $$-0.416715\pi$$
0.258674 + 0.965965i $$0.416715\pi$$
$$318$$ 19.8167 1.11126
$$319$$ 9.00000 0.503903
$$320$$ −2.30278 −0.128729
$$321$$ 2.30278 0.128528
$$322$$ 10.1833 0.567496
$$323$$ −12.0000 −0.667698
$$324$$ 29.8167 1.65648
$$325$$ 0.394449 0.0218801
$$326$$ −8.42221 −0.466463
$$327$$ 6.60555 0.365288
$$328$$ −9.90833 −0.547096
$$329$$ −12.0000 −0.661581
$$330$$ −17.5139 −0.964107
$$331$$ −13.2111 −0.726148 −0.363074 0.931760i $$-0.618273\pi$$
−0.363074 + 0.931760i $$0.618273\pi$$
$$332$$ 2.78890 0.153061
$$333$$ 7.90833 0.433374
$$334$$ 5.51388 0.301706
$$335$$ 8.09167 0.442095
$$336$$ −8.60555 −0.469471
$$337$$ 6.11943 0.333347 0.166673 0.986012i $$-0.446697\pi$$
0.166673 + 0.986012i $$0.446697\pi$$
$$338$$ 11.3028 0.614790
$$339$$ −10.6056 −0.576014
$$340$$ 13.8167 0.749313
$$341$$ 0.697224 0.0377568
$$342$$ −15.8167 −0.855267
$$343$$ 18.7889 1.01451
$$344$$ −0.605551 −0.0326491
$$345$$ −29.7250 −1.60034
$$346$$ 8.78890 0.472494
$$347$$ 10.1833 0.546671 0.273335 0.961919i $$-0.411873\pi$$
0.273335 + 0.961919i $$0.411873\pi$$
$$348$$ −12.9083 −0.691959
$$349$$ 28.2389 1.51159 0.755796 0.654807i $$-0.227250\pi$$
0.755796 + 0.654807i $$0.227250\pi$$
$$350$$ 0.788897 0.0421683
$$351$$ 21.1194 1.12727
$$352$$ 2.30278 0.122738
$$353$$ 10.1833 0.542005 0.271002 0.962579i $$-0.412645\pi$$
0.271002 + 0.962579i $$0.412645\pi$$
$$354$$ −35.0278 −1.86170
$$355$$ −13.8167 −0.733312
$$356$$ −9.21110 −0.488187
$$357$$ 51.6333 2.73272
$$358$$ 13.8167 0.730233
$$359$$ 3.21110 0.169476 0.0847378 0.996403i $$-0.472995\pi$$
0.0847378 + 0.996403i $$0.472995\pi$$
$$360$$ 18.2111 0.959809
$$361$$ −15.0000 −0.789474
$$362$$ −20.0000 −1.05118
$$363$$ −18.8167 −0.987618
$$364$$ −3.39445 −0.177917
$$365$$ 28.3305 1.48289
$$366$$ −24.8167 −1.29719
$$367$$ 3.81665 0.199228 0.0996139 0.995026i $$-0.468239\pi$$
0.0996139 + 0.995026i $$0.468239\pi$$
$$368$$ 3.90833 0.203736
$$369$$ 78.3583 4.07917
$$370$$ 2.30278 0.119716
$$371$$ 15.6333 0.811641
$$372$$ −1.00000 −0.0518476
$$373$$ −17.8167 −0.922511 −0.461256 0.887267i $$-0.652601\pi$$
−0.461256 + 0.887267i $$0.652601\pi$$
$$374$$ −13.8167 −0.714442
$$375$$ 35.7250 1.84483
$$376$$ −4.60555 −0.237513
$$377$$ −5.09167 −0.262235
$$378$$ 42.2389 2.17253
$$379$$ 24.3305 1.24978 0.624888 0.780715i $$-0.285145\pi$$
0.624888 + 0.780715i $$0.285145\pi$$
$$380$$ −4.60555 −0.236260
$$381$$ −63.4500 −3.25064
$$382$$ 5.51388 0.282115
$$383$$ −36.8444 −1.88266 −0.941331 0.337486i $$-0.890424\pi$$
−0.941331 + 0.337486i $$0.890424\pi$$
$$384$$ −3.30278 −0.168544
$$385$$ −13.8167 −0.704162
$$386$$ 4.00000 0.203595
$$387$$ 4.78890 0.243433
$$388$$ −16.4222 −0.833711
$$389$$ −37.1194 −1.88203 −0.941015 0.338365i $$-0.890126\pi$$
−0.941015 + 0.338365i $$0.890126\pi$$
$$390$$ 9.90833 0.501728
$$391$$ −23.4500 −1.18592
$$392$$ 0.211103 0.0106623
$$393$$ 35.0278 1.76692
$$394$$ 6.00000 0.302276
$$395$$ −21.0000 −1.05662
$$396$$ −18.2111 −0.915142
$$397$$ 6.18335 0.310333 0.155167 0.987888i $$-0.450409\pi$$
0.155167 + 0.987888i $$0.450409\pi$$
$$398$$ −26.4222 −1.32443
$$399$$ −17.2111 −0.861633
$$400$$ 0.302776 0.0151388
$$401$$ −7.81665 −0.390345 −0.195173 0.980769i $$-0.562527\pi$$
−0.195173 + 0.980769i $$0.562527\pi$$
$$402$$ 11.6056 0.578832
$$403$$ −0.394449 −0.0196489
$$404$$ −12.4222 −0.618028
$$405$$ −68.6611 −3.41180
$$406$$ −10.1833 −0.505391
$$407$$ −2.30278 −0.114144
$$408$$ 19.8167 0.981071
$$409$$ 31.0278 1.53422 0.767112 0.641513i $$-0.221693\pi$$
0.767112 + 0.641513i $$0.221693\pi$$
$$410$$ 22.8167 1.12683
$$411$$ 3.00000 0.147979
$$412$$ −0.302776 −0.0149167
$$413$$ −27.6333 −1.35975
$$414$$ −30.9083 −1.51906
$$415$$ −6.42221 −0.315254
$$416$$ −1.30278 −0.0638738
$$417$$ −6.30278 −0.308648
$$418$$ 4.60555 0.225265
$$419$$ 36.1472 1.76591 0.882953 0.469462i $$-0.155552\pi$$
0.882953 + 0.469462i $$0.155552\pi$$
$$420$$ 19.8167 0.966954
$$421$$ −3.72498 −0.181544 −0.0907722 0.995872i $$-0.528934\pi$$
−0.0907722 + 0.995872i $$0.528934\pi$$
$$422$$ −10.3028 −0.501531
$$423$$ 36.4222 1.77091
$$424$$ 6.00000 0.291386
$$425$$ −1.81665 −0.0881207
$$426$$ −19.8167 −0.960120
$$427$$ −19.5778 −0.947436
$$428$$ 0.697224 0.0337016
$$429$$ −9.90833 −0.478379
$$430$$ 1.39445 0.0672463
$$431$$ 9.21110 0.443683 0.221842 0.975083i $$-0.428793\pi$$
0.221842 + 0.975083i $$0.428793\pi$$
$$432$$ 16.2111 0.779957
$$433$$ 34.9361 1.67892 0.839461 0.543421i $$-0.182871\pi$$
0.839461 + 0.543421i $$0.182871\pi$$
$$434$$ −0.788897 −0.0378683
$$435$$ 29.7250 1.42520
$$436$$ 2.00000 0.0957826
$$437$$ 7.81665 0.373921
$$438$$ 40.6333 1.94153
$$439$$ 30.3305 1.44760 0.723799 0.690011i $$-0.242394\pi$$
0.723799 + 0.690011i $$0.242394\pi$$
$$440$$ −5.30278 −0.252800
$$441$$ −1.66947 −0.0794985
$$442$$ 7.81665 0.371800
$$443$$ 32.7250 1.55481 0.777405 0.629000i $$-0.216535\pi$$
0.777405 + 0.629000i $$0.216535\pi$$
$$444$$ 3.30278 0.156743
$$445$$ 21.2111 1.00550
$$446$$ 5.81665 0.275427
$$447$$ 65.4500 3.09568
$$448$$ −2.60555 −0.123101
$$449$$ −15.2111 −0.717856 −0.358928 0.933365i $$-0.616858\pi$$
−0.358928 + 0.933365i $$0.616858\pi$$
$$450$$ −2.39445 −0.112875
$$451$$ −22.8167 −1.07439
$$452$$ −3.21110 −0.151038
$$453$$ −68.0555 −3.19753
$$454$$ −13.8167 −0.648448
$$455$$ 7.81665 0.366450
$$456$$ −6.60555 −0.309333
$$457$$ −2.60555 −0.121883 −0.0609413 0.998141i $$-0.519410\pi$$
−0.0609413 + 0.998141i $$0.519410\pi$$
$$458$$ −24.6056 −1.14974
$$459$$ −97.2666 −4.54002
$$460$$ −9.00000 −0.419627
$$461$$ 12.4222 0.578560 0.289280 0.957245i $$-0.406584\pi$$
0.289280 + 0.957245i $$0.406584\pi$$
$$462$$ −19.8167 −0.921954
$$463$$ 26.6972 1.24073 0.620363 0.784315i $$-0.286985\pi$$
0.620363 + 0.784315i $$0.286985\pi$$
$$464$$ −3.90833 −0.181440
$$465$$ 2.30278 0.106789
$$466$$ −8.51388 −0.394398
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 10.3028 0.476246
$$469$$ 9.15559 0.422766
$$470$$ 10.6056 0.489198
$$471$$ −23.8167 −1.09741
$$472$$ −10.6056 −0.488160
$$473$$ −1.39445 −0.0641168
$$474$$ −30.1194 −1.38343
$$475$$ 0.605551 0.0277846
$$476$$ 15.6333 0.716551
$$477$$ −47.4500 −2.17258
$$478$$ 17.5139 0.801066
$$479$$ −13.1194 −0.599442 −0.299721 0.954027i $$-0.596894\pi$$
−0.299721 + 0.954027i $$0.596894\pi$$
$$480$$ 7.60555 0.347144
$$481$$ 1.30278 0.0594015
$$482$$ −8.00000 −0.364390
$$483$$ −33.6333 −1.53037
$$484$$ −5.69722 −0.258965
$$485$$ 37.8167 1.71717
$$486$$ −49.8444 −2.26099
$$487$$ −37.2111 −1.68620 −0.843098 0.537760i $$-0.819271\pi$$
−0.843098 + 0.537760i $$0.819271\pi$$
$$488$$ −7.51388 −0.340137
$$489$$ 27.8167 1.25791
$$490$$ −0.486122 −0.0219607
$$491$$ 17.7250 0.799917 0.399959 0.916533i $$-0.369025\pi$$
0.399959 + 0.916533i $$0.369025\pi$$
$$492$$ 32.7250 1.47536
$$493$$ 23.4500 1.05613
$$494$$ −2.60555 −0.117229
$$495$$ 41.9361 1.88489
$$496$$ −0.302776 −0.0135950
$$497$$ −15.6333 −0.701250
$$498$$ −9.21110 −0.412759
$$499$$ −42.2389 −1.89087 −0.945436 0.325809i $$-0.894363\pi$$
−0.945436 + 0.325809i $$0.894363\pi$$
$$500$$ 10.8167 0.483735
$$501$$ −18.2111 −0.813612
$$502$$ 21.2111 0.946698
$$503$$ −6.48612 −0.289202 −0.144601 0.989490i $$-0.546190\pi$$
−0.144601 + 0.989490i $$0.546190\pi$$
$$504$$ 20.6056 0.917844
$$505$$ 28.6056 1.27293
$$506$$ 9.00000 0.400099
$$507$$ −37.3305 −1.65791
$$508$$ −19.2111 −0.852355
$$509$$ −4.18335 −0.185424 −0.0927118 0.995693i $$-0.529554\pi$$
−0.0927118 + 0.995693i $$0.529554\pi$$
$$510$$ −45.6333 −2.02068
$$511$$ 32.0555 1.41805
$$512$$ −1.00000 −0.0441942
$$513$$ 32.4222 1.43148
$$514$$ −3.21110 −0.141636
$$515$$ 0.697224 0.0307234
$$516$$ 2.00000 0.0880451
$$517$$ −10.6056 −0.466432
$$518$$ 2.60555 0.114481
$$519$$ −29.0278 −1.27418
$$520$$ 3.00000 0.131559
$$521$$ 33.6333 1.47350 0.736751 0.676164i $$-0.236359\pi$$
0.736751 + 0.676164i $$0.236359\pi$$
$$522$$ 30.9083 1.35282
$$523$$ −18.2389 −0.797530 −0.398765 0.917053i $$-0.630561\pi$$
−0.398765 + 0.917053i $$0.630561\pi$$
$$524$$ 10.6056 0.463306
$$525$$ −2.60555 −0.113716
$$526$$ −13.8167 −0.602435
$$527$$ 1.81665 0.0791347
$$528$$ −7.60555 −0.330989
$$529$$ −7.72498 −0.335869
$$530$$ −13.8167 −0.600157
$$531$$ 83.8722 3.63974
$$532$$ −5.21110 −0.225930
$$533$$ 12.9083 0.559122
$$534$$ 30.4222 1.31650
$$535$$ −1.60555 −0.0694140
$$536$$ 3.51388 0.151776
$$537$$ −45.6333 −1.96922
$$538$$ 21.2111 0.914476
$$539$$ 0.486122 0.0209387
$$540$$ −37.3305 −1.60645
$$541$$ 25.9361 1.11508 0.557540 0.830150i $$-0.311745\pi$$
0.557540 + 0.830150i $$0.311745\pi$$
$$542$$ 22.4222 0.963116
$$543$$ 66.0555 2.83471
$$544$$ 6.00000 0.257248
$$545$$ −4.60555 −0.197280
$$546$$ 11.2111 0.479791
$$547$$ −20.6056 −0.881030 −0.440515 0.897745i $$-0.645204\pi$$
−0.440515 + 0.897745i $$0.645204\pi$$
$$548$$ 0.908327 0.0388018
$$549$$ 59.4222 2.53608
$$550$$ 0.697224 0.0297297
$$551$$ −7.81665 −0.333001
$$552$$ −12.9083 −0.549415
$$553$$ −23.7611 −1.01043
$$554$$ −0.119429 −0.00507407
$$555$$ −7.60555 −0.322838
$$556$$ −1.90833 −0.0809311
$$557$$ 11.5139 0.487859 0.243929 0.969793i $$-0.421564\pi$$
0.243929 + 0.969793i $$0.421564\pi$$
$$558$$ 2.39445 0.101365
$$559$$ 0.788897 0.0333668
$$560$$ 6.00000 0.253546
$$561$$ 45.6333 1.92664
$$562$$ 12.0000 0.506189
$$563$$ −28.0555 −1.18240 −0.591199 0.806525i $$-0.701345\pi$$
−0.591199 + 0.806525i $$0.701345\pi$$
$$564$$ 15.2111 0.640503
$$565$$ 7.39445 0.311087
$$566$$ −24.6056 −1.03425
$$567$$ −77.6888 −3.26262
$$568$$ −6.00000 −0.251754
$$569$$ 18.4222 0.772299 0.386150 0.922436i $$-0.373805\pi$$
0.386150 + 0.922436i $$0.373805\pi$$
$$570$$ 15.2111 0.637123
$$571$$ −16.6972 −0.698757 −0.349379 0.936982i $$-0.613607\pi$$
−0.349379 + 0.936982i $$0.613607\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ −18.2111 −0.760780
$$574$$ 25.8167 1.07757
$$575$$ 1.18335 0.0493489
$$576$$ 7.90833 0.329514
$$577$$ 22.2389 0.925816 0.462908 0.886406i $$-0.346806\pi$$
0.462908 + 0.886406i $$0.346806\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −13.2111 −0.549035
$$580$$ 9.00000 0.373705
$$581$$ −7.26662 −0.301470
$$582$$ 54.2389 2.24827
$$583$$ 13.8167 0.572227
$$584$$ 12.3028 0.509092
$$585$$ −23.7250 −0.980907
$$586$$ −11.0278 −0.455552
$$587$$ −45.6333 −1.88349 −0.941744 0.336330i $$-0.890814\pi$$
−0.941744 + 0.336330i $$0.890814\pi$$
$$588$$ −0.697224 −0.0287530
$$589$$ −0.605551 −0.0249513
$$590$$ 24.4222 1.00545
$$591$$ −19.8167 −0.815148
$$592$$ 1.00000 0.0410997
$$593$$ 18.4861 0.759134 0.379567 0.925164i $$-0.376073\pi$$
0.379567 + 0.925164i $$0.376073\pi$$
$$594$$ 37.3305 1.53169
$$595$$ −36.0000 −1.47586
$$596$$ 19.8167 0.811722
$$597$$ 87.2666 3.57158
$$598$$ −5.09167 −0.208214
$$599$$ −20.7889 −0.849411 −0.424706 0.905331i $$-0.639622\pi$$
−0.424706 + 0.905331i $$0.639622\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −24.3028 −0.991331 −0.495665 0.868514i $$-0.665076\pi$$
−0.495665 + 0.868514i $$0.665076\pi$$
$$602$$ 1.57779 0.0643061
$$603$$ −27.7889 −1.13165
$$604$$ −20.6056 −0.838428
$$605$$ 13.1194 0.533381
$$606$$ 41.0278 1.66664
$$607$$ −13.4861 −0.547385 −0.273692 0.961817i $$-0.588245\pi$$
−0.273692 + 0.961817i $$0.588245\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 33.6333 1.36289
$$610$$ 17.3028 0.700569
$$611$$ 6.00000 0.242734
$$612$$ −47.4500 −1.91805
$$613$$ −29.8167 −1.20428 −0.602142 0.798389i $$-0.705686\pi$$
−0.602142 + 0.798389i $$0.705686\pi$$
$$614$$ −17.9083 −0.722721
$$615$$ −75.3583 −3.03874
$$616$$ −6.00000 −0.241747
$$617$$ −42.5694 −1.71378 −0.856890 0.515500i $$-0.827606\pi$$
−0.856890 + 0.515500i $$0.827606\pi$$
$$618$$ 1.00000 0.0402259
$$619$$ −6.30278 −0.253330 −0.126665 0.991946i $$-0.540427\pi$$
−0.126665 + 0.991946i $$0.540427\pi$$
$$620$$ 0.697224 0.0280012
$$621$$ 63.3583 2.54248
$$622$$ −15.9083 −0.637866
$$623$$ 24.0000 0.961540
$$624$$ 4.30278 0.172249
$$625$$ −26.4222 −1.05689
$$626$$ 9.02776 0.360822
$$627$$ −15.2111 −0.607473
$$628$$ −7.21110 −0.287754
$$629$$ −6.00000 −0.239236
$$630$$ −47.4500 −1.89045
$$631$$ 14.6972 0.585087 0.292544 0.956252i $$-0.405498\pi$$
0.292544 + 0.956252i $$0.405498\pi$$
$$632$$ −9.11943 −0.362751
$$633$$ 34.0278 1.35248
$$634$$ −9.21110 −0.365820
$$635$$ 44.2389 1.75557
$$636$$ −19.8167 −0.785781
$$637$$ −0.275019 −0.0108967
$$638$$ −9.00000 −0.356313
$$639$$ 47.4500 1.87709
$$640$$ 2.30278 0.0910252
$$641$$ −20.5139 −0.810249 −0.405125 0.914261i $$-0.632772\pi$$
−0.405125 + 0.914261i $$0.632772\pi$$
$$642$$ −2.30278 −0.0908833
$$643$$ −8.18335 −0.322720 −0.161360 0.986896i $$-0.551588\pi$$
−0.161360 + 0.986896i $$0.551588\pi$$
$$644$$ −10.1833 −0.401280
$$645$$ −4.60555 −0.181343
$$646$$ 12.0000 0.472134
$$647$$ 20.9361 0.823082 0.411541 0.911391i $$-0.364991\pi$$
0.411541 + 0.911391i $$0.364991\pi$$
$$648$$ −29.8167 −1.17131
$$649$$ −24.4222 −0.958655
$$650$$ −0.394449 −0.0154716
$$651$$ 2.60555 0.102120
$$652$$ 8.42221 0.329839
$$653$$ −3.90833 −0.152945 −0.0764723 0.997072i $$-0.524366\pi$$
−0.0764723 + 0.997072i $$0.524366\pi$$
$$654$$ −6.60555 −0.258297
$$655$$ −24.4222 −0.954255
$$656$$ 9.90833 0.386855
$$657$$ −97.2944 −3.79581
$$658$$ 12.0000 0.467809
$$659$$ −16.8806 −0.657574 −0.328787 0.944404i $$-0.606640\pi$$
−0.328787 + 0.944404i $$0.606640\pi$$
$$660$$ 17.5139 0.681727
$$661$$ −30.5139 −1.18685 −0.593426 0.804888i $$-0.702225\pi$$
−0.593426 + 0.804888i $$0.702225\pi$$
$$662$$ 13.2111 0.513464
$$663$$ −25.8167 −1.00264
$$664$$ −2.78890 −0.108230
$$665$$ 12.0000 0.465340
$$666$$ −7.90833 −0.306441
$$667$$ −15.2750 −0.591451
$$668$$ −5.51388 −0.213338
$$669$$ −19.2111 −0.742744
$$670$$ −8.09167 −0.312609
$$671$$ −17.3028 −0.667966
$$672$$ 8.60555 0.331966
$$673$$ 20.6972 0.797819 0.398910 0.916990i $$-0.369389\pi$$
0.398910 + 0.916990i $$0.369389\pi$$
$$674$$ −6.11943 −0.235712
$$675$$ 4.90833 0.188922
$$676$$ −11.3028 −0.434722
$$677$$ 14.2389 0.547244 0.273622 0.961837i $$-0.411778\pi$$
0.273622 + 0.961837i $$0.411778\pi$$
$$678$$ 10.6056 0.407304
$$679$$ 42.7889 1.64209
$$680$$ −13.8167 −0.529844
$$681$$ 45.6333 1.74867
$$682$$ −0.697224 −0.0266981
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 15.8167 0.604765
$$685$$ −2.09167 −0.0799187
$$686$$ −18.7889 −0.717363
$$687$$ 81.2666 3.10051
$$688$$ 0.605551 0.0230864
$$689$$ −7.81665 −0.297791
$$690$$ 29.7250 1.13161
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ −8.78890 −0.334104
$$693$$ 47.4500 1.80247
$$694$$ −10.1833 −0.386555
$$695$$ 4.39445 0.166691
$$696$$ 12.9083 0.489289
$$697$$ −59.4500 −2.25183
$$698$$ −28.2389 −1.06886
$$699$$ 28.1194 1.06357
$$700$$ −0.788897 −0.0298175
$$701$$ 40.1194 1.51529 0.757645 0.652667i $$-0.226350\pi$$
0.757645 + 0.652667i $$0.226350\pi$$
$$702$$ −21.1194 −0.797101
$$703$$ 2.00000 0.0754314
$$704$$ −2.30278 −0.0867891
$$705$$ −35.0278 −1.31922
$$706$$ −10.1833 −0.383255
$$707$$ 32.3667 1.21727
$$708$$ 35.0278 1.31642
$$709$$ −41.3305 −1.55220 −0.776100 0.630609i $$-0.782805\pi$$
−0.776100 + 0.630609i $$0.782805\pi$$
$$710$$ 13.8167 0.518530
$$711$$ 72.1194 2.70469
$$712$$ 9.21110 0.345201
$$713$$ −1.18335 −0.0443167
$$714$$ −51.6333 −1.93233
$$715$$ 6.90833 0.258357
$$716$$ −13.8167 −0.516353
$$717$$ −57.8444 −2.16024
$$718$$ −3.21110 −0.119837
$$719$$ −51.6333 −1.92560 −0.962799 0.270220i $$-0.912904\pi$$
−0.962799 + 0.270220i $$0.912904\pi$$
$$720$$ −18.2111 −0.678688
$$721$$ 0.788897 0.0293801
$$722$$ 15.0000 0.558242
$$723$$ 26.4222 0.982652
$$724$$ 20.0000 0.743294
$$725$$ −1.18335 −0.0439484
$$726$$ 18.8167 0.698352
$$727$$ 19.0917 0.708071 0.354035 0.935232i $$-0.384809\pi$$
0.354035 + 0.935232i $$0.384809\pi$$
$$728$$ 3.39445 0.125807
$$729$$ 75.1749 2.78426
$$730$$ −28.3305 −1.04856
$$731$$ −3.63331 −0.134383
$$732$$ 24.8167 0.917250
$$733$$ −13.6333 −0.503558 −0.251779 0.967785i $$-0.581016\pi$$
−0.251779 + 0.967785i $$0.581016\pi$$
$$734$$ −3.81665 −0.140875
$$735$$ 1.60555 0.0592217
$$736$$ −3.90833 −0.144063
$$737$$ 8.09167 0.298061
$$738$$ −78.3583 −2.88441
$$739$$ −2.66947 −0.0981980 −0.0490990 0.998794i $$-0.515635\pi$$
−0.0490990 + 0.998794i $$0.515635\pi$$
$$740$$ −2.30278 −0.0846517
$$741$$ 8.60555 0.316133
$$742$$ −15.6333 −0.573917
$$743$$ −29.4500 −1.08041 −0.540207 0.841532i $$-0.681654\pi$$
−0.540207 + 0.841532i $$0.681654\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ −45.6333 −1.67188
$$746$$ 17.8167 0.652314
$$747$$ 22.0555 0.806969
$$748$$ 13.8167 0.505187
$$749$$ −1.81665 −0.0663791
$$750$$ −35.7250 −1.30449
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 4.60555 0.167947
$$753$$ −70.0555 −2.55296
$$754$$ 5.09167 0.185428
$$755$$ 47.4500 1.72688
$$756$$ −42.2389 −1.53621
$$757$$ 5.69722 0.207069 0.103535 0.994626i $$-0.466985\pi$$
0.103535 + 0.994626i $$0.466985\pi$$
$$758$$ −24.3305 −0.883725
$$759$$ −29.7250 −1.07895
$$760$$ 4.60555 0.167061
$$761$$ 16.8806 0.611920 0.305960 0.952044i $$-0.401023\pi$$
0.305960 + 0.952044i $$0.401023\pi$$
$$762$$ 63.4500 2.29855
$$763$$ −5.21110 −0.188655
$$764$$ −5.51388 −0.199485
$$765$$ 109.267 3.95054
$$766$$ 36.8444 1.33124
$$767$$ 13.8167 0.498890
$$768$$ 3.30278 0.119179
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 13.8167 0.497918
$$771$$ 10.6056 0.381950
$$772$$ −4.00000 −0.143963
$$773$$ 22.0555 0.793282 0.396641 0.917974i $$-0.370176\pi$$
0.396641 + 0.917974i $$0.370176\pi$$
$$774$$ −4.78890 −0.172133
$$775$$ −0.0916731 −0.00329299
$$776$$ 16.4222 0.589523
$$777$$ −8.60555 −0.308722
$$778$$ 37.1194 1.33080
$$779$$ 19.8167 0.710005
$$780$$ −9.90833 −0.354775
$$781$$ −13.8167 −0.494399
$$782$$ 23.4500 0.838569
$$783$$ −63.3583 −2.26424
$$784$$ −0.211103 −0.00753938
$$785$$ 16.6056 0.592678
$$786$$ −35.0278 −1.24940
$$787$$ 10.7889 0.384583 0.192291 0.981338i $$-0.438408\pi$$
0.192291 + 0.981338i $$0.438408\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 45.6333 1.62459
$$790$$ 21.0000 0.747146
$$791$$ 8.36669 0.297485
$$792$$ 18.2111 0.647103
$$793$$ 9.78890 0.347614
$$794$$ −6.18335 −0.219439
$$795$$ 45.6333 1.61845
$$796$$ 26.4222 0.936510
$$797$$ 22.3305 0.790988 0.395494 0.918469i $$-0.370573\pi$$
0.395494 + 0.918469i $$0.370573\pi$$
$$798$$ 17.2111 0.609266
$$799$$ −27.6333 −0.977596
$$800$$ −0.302776 −0.0107047
$$801$$ −72.8444 −2.57383
$$802$$ 7.81665 0.276016
$$803$$ 28.3305 0.999763
$$804$$ −11.6056 −0.409296
$$805$$ 23.4500 0.826503
$$806$$ 0.394449 0.0138939
$$807$$ −70.0555 −2.46607
$$808$$ 12.4222 0.437012
$$809$$ −35.4500 −1.24635 −0.623177 0.782081i $$-0.714158\pi$$
−0.623177 + 0.782081i $$0.714158\pi$$
$$810$$ 68.6611 2.41250
$$811$$ −7.14719 −0.250972 −0.125486 0.992095i $$-0.540049\pi$$
−0.125486 + 0.992095i $$0.540049\pi$$
$$812$$ 10.1833 0.357365
$$813$$ −74.0555 −2.59724
$$814$$ 2.30278 0.0807122
$$815$$ −19.3944 −0.679358
$$816$$ −19.8167 −0.693722
$$817$$ 1.21110 0.0423711
$$818$$ −31.0278 −1.08486
$$819$$ −26.8444 −0.938020
$$820$$ −22.8167 −0.796792
$$821$$ −3.21110 −0.112068 −0.0560341 0.998429i $$-0.517846\pi$$
−0.0560341 + 0.998429i $$0.517846\pi$$
$$822$$ −3.00000 −0.104637
$$823$$ 44.8444 1.56318 0.781589 0.623794i $$-0.214410\pi$$
0.781589 + 0.623794i $$0.214410\pi$$
$$824$$ 0.302776 0.0105477
$$825$$ −2.30278 −0.0801724
$$826$$ 27.6333 0.961486
$$827$$ 34.6056 1.20335 0.601676 0.798740i $$-0.294500\pi$$
0.601676 + 0.798740i $$0.294500\pi$$
$$828$$ 30.9083 1.07414
$$829$$ −27.7250 −0.962928 −0.481464 0.876466i $$-0.659895\pi$$
−0.481464 + 0.876466i $$0.659895\pi$$
$$830$$ 6.42221 0.222918
$$831$$ 0.394449 0.0136833
$$832$$ 1.30278 0.0451656
$$833$$ 1.26662 0.0438856
$$834$$ 6.30278 0.218247
$$835$$ 12.6972 0.439406
$$836$$ −4.60555 −0.159286
$$837$$ −4.90833 −0.169657
$$838$$ −36.1472 −1.24868
$$839$$ −12.9722 −0.447852 −0.223926 0.974606i $$-0.571887\pi$$
−0.223926 + 0.974606i $$0.571887\pi$$
$$840$$ −19.8167 −0.683740
$$841$$ −13.7250 −0.473275
$$842$$ 3.72498 0.128371
$$843$$ −39.6333 −1.36504
$$844$$ 10.3028 0.354636
$$845$$ 26.0278 0.895382
$$846$$ −36.4222 −1.25222
$$847$$ 14.8444 0.510060
$$848$$ −6.00000 −0.206041
$$849$$ 81.2666 2.78906
$$850$$ 1.81665 0.0623107
$$851$$ 3.90833 0.133976
$$852$$ 19.8167 0.678907
$$853$$ 42.5416 1.45660 0.728299 0.685260i $$-0.240311\pi$$
0.728299 + 0.685260i $$0.240311\pi$$
$$854$$ 19.5778 0.669938
$$855$$ −36.4222 −1.24561
$$856$$ −0.697224 −0.0238306
$$857$$ 42.8444 1.46354 0.731769 0.681553i $$-0.238695\pi$$
0.731769 + 0.681553i $$0.238695\pi$$
$$858$$ 9.90833 0.338265
$$859$$ 48.0555 1.63963 0.819816 0.572626i $$-0.194075\pi$$
0.819816 + 0.572626i $$0.194075\pi$$
$$860$$ −1.39445 −0.0475503
$$861$$ −85.2666 −2.90588
$$862$$ −9.21110 −0.313731
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ −16.2111 −0.551513
$$865$$ 20.2389 0.688142
$$866$$ −34.9361 −1.18718
$$867$$ 62.7527 2.13119
$$868$$ 0.788897 0.0267769
$$869$$ −21.0000 −0.712376
$$870$$ −29.7250 −1.00777
$$871$$ −4.57779 −0.155113
$$872$$ −2.00000 −0.0677285
$$873$$ −129.872 −4.39551
$$874$$ −7.81665 −0.264402
$$875$$ −28.1833 −0.952771
$$876$$ −40.6333 −1.37287
$$877$$ −7.21110 −0.243502 −0.121751 0.992561i $$-0.538851\pi$$
−0.121751 + 0.992561i $$0.538851\pi$$
$$878$$ −30.3305 −1.02361
$$879$$ 36.4222 1.22849
$$880$$ 5.30278 0.178757
$$881$$ −28.5416 −0.961592 −0.480796 0.876832i $$-0.659652\pi$$
−0.480796 + 0.876832i $$0.659652\pi$$
$$882$$ 1.66947 0.0562139
$$883$$ 26.4222 0.889178 0.444589 0.895735i $$-0.353350\pi$$
0.444589 + 0.895735i $$0.353350\pi$$
$$884$$ −7.81665 −0.262903
$$885$$ −80.6611 −2.71139
$$886$$ −32.7250 −1.09942
$$887$$ −0.422205 −0.0141763 −0.00708813 0.999975i $$-0.502256\pi$$
−0.00708813 + 0.999975i $$0.502256\pi$$
$$888$$ −3.30278 −0.110834
$$889$$ 50.0555 1.67881
$$890$$ −21.2111 −0.710998
$$891$$ −68.6611 −2.30023
$$892$$ −5.81665 −0.194756
$$893$$ 9.21110 0.308238
$$894$$ −65.4500 −2.18897
$$895$$ 31.8167 1.06351
$$896$$ 2.60555 0.0870454
$$897$$ 16.8167 0.561492
$$898$$ 15.2111 0.507601
$$899$$ 1.18335 0.0394668
$$900$$ 2.39445 0.0798150
$$901$$ 36.0000 1.19933
$$902$$ 22.8167 0.759711
$$903$$ −5.21110 −0.173415
$$904$$ 3.21110 0.106800
$$905$$ −46.0555 −1.53094
$$906$$ 68.0555 2.26099
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ 13.8167 0.458522
$$909$$ −98.2389 −3.25838
$$910$$ −7.81665 −0.259120
$$911$$ −17.5778 −0.582378 −0.291189 0.956665i $$-0.594051\pi$$
−0.291189 + 0.956665i $$0.594051\pi$$
$$912$$ 6.60555 0.218732
$$913$$ −6.42221 −0.212544
$$914$$ 2.60555 0.0861840
$$915$$ −57.1472 −1.88923
$$916$$ 24.6056 0.812990
$$917$$ −27.6333 −0.912532
$$918$$ 97.2666 3.21028
$$919$$ −9.57779 −0.315942 −0.157971 0.987444i $$-0.550495\pi$$
−0.157971 + 0.987444i $$0.550495\pi$$
$$920$$ 9.00000 0.296721
$$921$$ 59.1472 1.94897
$$922$$ −12.4222 −0.409104
$$923$$ 7.81665 0.257288
$$924$$ 19.8167 0.651920
$$925$$ 0.302776 0.00995520
$$926$$ −26.6972 −0.877325
$$927$$ −2.39445 −0.0786440
$$928$$ 3.90833 0.128297
$$929$$ −18.4861 −0.606510 −0.303255 0.952909i $$-0.598073\pi$$
−0.303255 + 0.952909i $$0.598073\pi$$
$$930$$ −2.30278 −0.0755110
$$931$$ −0.422205 −0.0138372
$$932$$ 8.51388 0.278881
$$933$$ 52.5416 1.72014
$$934$$ 0 0
$$935$$ −31.8167 −1.04052
$$936$$ −10.3028 −0.336757
$$937$$ −18.0917 −0.591029 −0.295515 0.955338i $$-0.595491\pi$$
−0.295515 + 0.955338i $$0.595491\pi$$
$$938$$ −9.15559 −0.298941
$$939$$ −29.8167 −0.973030
$$940$$ −10.6056 −0.345915
$$941$$ 13.8167 0.450410 0.225205 0.974311i $$-0.427695\pi$$
0.225205 + 0.974311i $$0.427695\pi$$
$$942$$ 23.8167 0.775989
$$943$$ 38.7250 1.26106
$$944$$ 10.6056 0.345181
$$945$$ 97.2666 3.16408
$$946$$ 1.39445 0.0453374
$$947$$ −3.63331 −0.118067 −0.0590333 0.998256i $$-0.518802\pi$$
−0.0590333 + 0.998256i $$0.518802\pi$$
$$948$$ 30.1194 0.978234
$$949$$ −16.0278 −0.520283
$$950$$ −0.605551 −0.0196467
$$951$$ 30.4222 0.986508
$$952$$ −15.6333 −0.506678
$$953$$ 49.7527 1.61165 0.805825 0.592154i $$-0.201722\pi$$
0.805825 + 0.592154i $$0.201722\pi$$
$$954$$ 47.4500 1.53625
$$955$$ 12.6972 0.410873
$$956$$ −17.5139 −0.566439
$$957$$ 29.7250 0.960872
$$958$$ 13.1194 0.423870
$$959$$ −2.36669 −0.0764245
$$960$$ −7.60555 −0.245468
$$961$$ −30.9083 −0.997043
$$962$$ −1.30278 −0.0420032
$$963$$ 5.51388 0.177682
$$964$$ 8.00000 0.257663
$$965$$ 9.21110 0.296516
$$966$$ 33.6333 1.08213
$$967$$ −6.72498 −0.216261 −0.108130 0.994137i $$-0.534486\pi$$
−0.108130 + 0.994137i $$0.534486\pi$$
$$968$$ 5.69722 0.183116
$$969$$ −39.6333 −1.27321
$$970$$ −37.8167 −1.21422
$$971$$ −22.5416 −0.723395 −0.361698 0.932295i $$-0.617803\pi$$
−0.361698 + 0.932295i $$0.617803\pi$$
$$972$$ 49.8444 1.59876
$$973$$ 4.97224 0.159403
$$974$$ 37.2111 1.19232
$$975$$ 1.30278 0.0417222
$$976$$ 7.51388 0.240513
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ −27.8167 −0.889479
$$979$$ 21.2111 0.677910
$$980$$ 0.486122 0.0155286
$$981$$ 15.8167 0.504987
$$982$$ −17.7250 −0.565627
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ −32.7250 −1.04323
$$985$$ 13.8167 0.440235
$$986$$ −23.4500 −0.746799
$$987$$ −39.6333 −1.26154
$$988$$ 2.60555 0.0828936
$$989$$ 2.36669 0.0752564
$$990$$ −41.9361 −1.33282
$$991$$ 50.6972 1.61045 0.805225 0.592969i $$-0.202044\pi$$
0.805225 + 0.592969i $$0.202044\pi$$
$$992$$ 0.302776 0.00961314
$$993$$ −43.6333 −1.38466
$$994$$ 15.6333 0.495858
$$995$$ −60.8444 −1.92890
$$996$$ 9.21110 0.291865
$$997$$ −52.4222 −1.66023 −0.830114 0.557594i $$-0.811725\pi$$
−0.830114 + 0.557594i $$0.811725\pi$$
$$998$$ 42.2389 1.33705
$$999$$ 16.2111 0.512897
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 74.2.a.a.1.2 2
3.2 odd 2 666.2.a.j.1.2 2
4.3 odd 2 592.2.a.f.1.1 2
5.2 odd 4 1850.2.b.i.149.1 4
5.3 odd 4 1850.2.b.i.149.4 4
5.4 even 2 1850.2.a.u.1.1 2
7.6 odd 2 3626.2.a.a.1.1 2
8.3 odd 2 2368.2.a.ba.1.2 2
8.5 even 2 2368.2.a.s.1.1 2
11.10 odd 2 8954.2.a.p.1.2 2
12.11 even 2 5328.2.a.bf.1.2 2
37.36 even 2 2738.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 1.1 even 1 trivial
592.2.a.f.1.1 2 4.3 odd 2
666.2.a.j.1.2 2 3.2 odd 2
1850.2.a.u.1.1 2 5.4 even 2
1850.2.b.i.149.1 4 5.2 odd 4
1850.2.b.i.149.4 4 5.3 odd 4
2368.2.a.s.1.1 2 8.5 even 2
2368.2.a.ba.1.2 2 8.3 odd 2
2738.2.a.l.1.2 2 37.36 even 2
3626.2.a.a.1.1 2 7.6 odd 2
5328.2.a.bf.1.2 2 12.11 even 2
8954.2.a.p.1.2 2 11.10 odd 2