# Properties

 Label 3174.2.a.bd.1.3 Level $3174$ Weight $2$ Character 3174.1 Self dual yes Analytic conductor $25.345$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3174,2,Mod(1,3174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3174, 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("3174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3174 = 2 \cdot 3 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3174.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: $$25.3445176016$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.91899$$ of defining polynomial Character $$\chi$$ $$=$$ 3174.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.47889 q^{5} +1.00000 q^{6} +3.20362 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.47889 q^{5} +1.00000 q^{6} +3.20362 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.47889 q^{10} -0.0552927 q^{11} +1.00000 q^{12} -0.805738 q^{13} +3.20362 q^{14} +1.47889 q^{15} +1.00000 q^{16} -2.05954 q^{17} +1.00000 q^{18} +3.67657 q^{19} +1.47889 q^{20} +3.20362 q^{21} -0.0552927 q^{22} +1.00000 q^{24} -2.81288 q^{25} -0.805738 q^{26} +1.00000 q^{27} +3.20362 q^{28} +7.34575 q^{29} +1.47889 q^{30} -7.95546 q^{31} +1.00000 q^{32} -0.0552927 q^{33} -2.05954 q^{34} +4.73780 q^{35} +1.00000 q^{36} +3.08816 q^{37} +3.67657 q^{38} -0.805738 q^{39} +1.47889 q^{40} -1.45973 q^{41} +3.20362 q^{42} +12.5322 q^{43} -0.0552927 q^{44} +1.47889 q^{45} -1.27459 q^{47} +1.00000 q^{48} +3.26315 q^{49} -2.81288 q^{50} -2.05954 q^{51} -0.805738 q^{52} +11.0449 q^{53} +1.00000 q^{54} -0.0817718 q^{55} +3.20362 q^{56} +3.67657 q^{57} +7.34575 q^{58} -5.36077 q^{59} +1.47889 q^{60} -12.2015 q^{61} -7.95546 q^{62} +3.20362 q^{63} +1.00000 q^{64} -1.19160 q^{65} -0.0552927 q^{66} -4.52116 q^{67} -2.05954 q^{68} +4.73780 q^{70} -14.6029 q^{71} +1.00000 q^{72} -9.86160 q^{73} +3.08816 q^{74} -2.81288 q^{75} +3.67657 q^{76} -0.177136 q^{77} -0.805738 q^{78} -3.61983 q^{79} +1.47889 q^{80} +1.00000 q^{81} -1.45973 q^{82} +10.0343 q^{83} +3.20362 q^{84} -3.04583 q^{85} +12.5322 q^{86} +7.34575 q^{87} -0.0552927 q^{88} +8.25248 q^{89} +1.47889 q^{90} -2.58128 q^{91} -7.95546 q^{93} -1.27459 q^{94} +5.43725 q^{95} +1.00000 q^{96} -7.01618 q^{97} +3.26315 q^{98} -0.0552927 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 7 q^{5} + 5 q^{6} + 7 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 7 * q^5 + 5 * q^6 + 7 * q^7 + 5 * q^8 + 5 * q^9 $$5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 7 q^{5} + 5 q^{6} + 7 q^{7} + 5 q^{8} + 5 q^{9} + 7 q^{10} + 13 q^{11} + 5 q^{12} - 4 q^{13} + 7 q^{14} + 7 q^{15} + 5 q^{16} + 9 q^{17} + 5 q^{18} + 11 q^{19} + 7 q^{20} + 7 q^{21} + 13 q^{22} + 5 q^{24} - 2 q^{25} - 4 q^{26} + 5 q^{27} + 7 q^{28} - 7 q^{29} + 7 q^{30} - 8 q^{31} + 5 q^{32} + 13 q^{33} + 9 q^{34} + q^{35} + 5 q^{36} + 12 q^{37} + 11 q^{38} - 4 q^{39} + 7 q^{40} - 10 q^{41} + 7 q^{42} + 4 q^{43} + 13 q^{44} + 7 q^{45} - 24 q^{47} + 5 q^{48} - 12 q^{49} - 2 q^{50} + 9 q^{51} - 4 q^{52} + 9 q^{53} + 5 q^{54} + 16 q^{55} + 7 q^{56} + 11 q^{57} - 7 q^{58} - 14 q^{59} + 7 q^{60} + 5 q^{61} - 8 q^{62} + 7 q^{63} + 5 q^{64} + 12 q^{65} + 13 q^{66} + 13 q^{67} + 9 q^{68} + q^{70} - 19 q^{71} + 5 q^{72} + 4 q^{73} + 12 q^{74} - 2 q^{75} + 11 q^{76} + 5 q^{77} - 4 q^{78} + 4 q^{79} + 7 q^{80} + 5 q^{81} - 10 q^{82} + 24 q^{83} + 7 q^{84} + 17 q^{85} + 4 q^{86} - 7 q^{87} + 13 q^{88} + 4 q^{89} + 7 q^{90} - 21 q^{91} - 8 q^{93} - 24 q^{94} - 11 q^{95} + 5 q^{96} - 9 q^{97} - 12 q^{98} + 13 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 7 * q^5 + 5 * q^6 + 7 * q^7 + 5 * q^8 + 5 * q^9 + 7 * q^10 + 13 * q^11 + 5 * q^12 - 4 * q^13 + 7 * q^14 + 7 * q^15 + 5 * q^16 + 9 * q^17 + 5 * q^18 + 11 * q^19 + 7 * q^20 + 7 * q^21 + 13 * q^22 + 5 * q^24 - 2 * q^25 - 4 * q^26 + 5 * q^27 + 7 * q^28 - 7 * q^29 + 7 * q^30 - 8 * q^31 + 5 * q^32 + 13 * q^33 + 9 * q^34 + q^35 + 5 * q^36 + 12 * q^37 + 11 * q^38 - 4 * q^39 + 7 * q^40 - 10 * q^41 + 7 * q^42 + 4 * q^43 + 13 * q^44 + 7 * q^45 - 24 * q^47 + 5 * q^48 - 12 * q^49 - 2 * q^50 + 9 * q^51 - 4 * q^52 + 9 * q^53 + 5 * q^54 + 16 * q^55 + 7 * q^56 + 11 * q^57 - 7 * q^58 - 14 * q^59 + 7 * q^60 + 5 * q^61 - 8 * q^62 + 7 * q^63 + 5 * q^64 + 12 * q^65 + 13 * q^66 + 13 * q^67 + 9 * q^68 + q^70 - 19 * q^71 + 5 * q^72 + 4 * q^73 + 12 * q^74 - 2 * q^75 + 11 * q^76 + 5 * q^77 - 4 * q^78 + 4 * q^79 + 7 * q^80 + 5 * q^81 - 10 * q^82 + 24 * q^83 + 7 * q^84 + 17 * q^85 + 4 * q^86 - 7 * q^87 + 13 * q^88 + 4 * q^89 + 7 * q^90 - 21 * q^91 - 8 * q^93 - 24 * q^94 - 11 * q^95 + 5 * q^96 - 9 * q^97 - 12 * q^98 + 13 * 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.47889 0.661380 0.330690 0.943739i $$-0.392718\pi$$
0.330690 + 0.943739i $$0.392718\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 3.20362 1.21085 0.605426 0.795901i $$-0.293003\pi$$
0.605426 + 0.795901i $$0.293003\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.47889 0.467667
$$11$$ −0.0552927 −0.0166714 −0.00833568 0.999965i $$-0.502653\pi$$
−0.00833568 + 0.999965i $$0.502653\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −0.805738 −0.223472 −0.111736 0.993738i $$-0.535641\pi$$
−0.111736 + 0.993738i $$0.535641\pi$$
$$14$$ 3.20362 0.856202
$$15$$ 1.47889 0.381848
$$16$$ 1.00000 0.250000
$$17$$ −2.05954 −0.499511 −0.249756 0.968309i $$-0.580350\pi$$
−0.249756 + 0.968309i $$0.580350\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 3.67657 0.843464 0.421732 0.906721i $$-0.361422\pi$$
0.421732 + 0.906721i $$0.361422\pi$$
$$20$$ 1.47889 0.330690
$$21$$ 3.20362 0.699086
$$22$$ −0.0552927 −0.0117884
$$23$$ 0 0
$$24$$ 1.00000 0.204124
$$25$$ −2.81288 −0.562576
$$26$$ −0.805738 −0.158018
$$27$$ 1.00000 0.192450
$$28$$ 3.20362 0.605426
$$29$$ 7.34575 1.36407 0.682036 0.731319i $$-0.261095\pi$$
0.682036 + 0.731319i $$0.261095\pi$$
$$30$$ 1.47889 0.270007
$$31$$ −7.95546 −1.42884 −0.714422 0.699715i $$-0.753310\pi$$
−0.714422 + 0.699715i $$0.753310\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −0.0552927 −0.00962522
$$34$$ −2.05954 −0.353208
$$35$$ 4.73780 0.800834
$$36$$ 1.00000 0.166667
$$37$$ 3.08816 0.507690 0.253845 0.967245i $$-0.418305\pi$$
0.253845 + 0.967245i $$0.418305\pi$$
$$38$$ 3.67657 0.596419
$$39$$ −0.805738 −0.129021
$$40$$ 1.47889 0.233833
$$41$$ −1.45973 −0.227972 −0.113986 0.993482i $$-0.536362\pi$$
−0.113986 + 0.993482i $$0.536362\pi$$
$$42$$ 3.20362 0.494329
$$43$$ 12.5322 1.91115 0.955574 0.294750i $$-0.0952365\pi$$
0.955574 + 0.294750i $$0.0952365\pi$$
$$44$$ −0.0552927 −0.00833568
$$45$$ 1.47889 0.220460
$$46$$ 0 0
$$47$$ −1.27459 −0.185918 −0.0929591 0.995670i $$-0.529633\pi$$
−0.0929591 + 0.995670i $$0.529633\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 3.26315 0.466165
$$50$$ −2.81288 −0.397801
$$51$$ −2.05954 −0.288393
$$52$$ −0.805738 −0.111736
$$53$$ 11.0449 1.51713 0.758564 0.651599i $$-0.225901\pi$$
0.758564 + 0.651599i $$0.225901\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −0.0817718 −0.0110261
$$56$$ 3.20362 0.428101
$$57$$ 3.67657 0.486974
$$58$$ 7.34575 0.964544
$$59$$ −5.36077 −0.697913 −0.348956 0.937139i $$-0.613464\pi$$
−0.348956 + 0.937139i $$0.613464\pi$$
$$60$$ 1.47889 0.190924
$$61$$ −12.2015 −1.56224 −0.781120 0.624380i $$-0.785352\pi$$
−0.781120 + 0.624380i $$0.785352\pi$$
$$62$$ −7.95546 −1.01034
$$63$$ 3.20362 0.403618
$$64$$ 1.00000 0.125000
$$65$$ −1.19160 −0.147800
$$66$$ −0.0552927 −0.00680606
$$67$$ −4.52116 −0.552348 −0.276174 0.961108i $$-0.589067\pi$$
−0.276174 + 0.961108i $$0.589067\pi$$
$$68$$ −2.05954 −0.249756
$$69$$ 0 0
$$70$$ 4.73780 0.566275
$$71$$ −14.6029 −1.73304 −0.866522 0.499138i $$-0.833650\pi$$
−0.866522 + 0.499138i $$0.833650\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −9.86160 −1.15421 −0.577106 0.816669i $$-0.695818\pi$$
−0.577106 + 0.816669i $$0.695818\pi$$
$$74$$ 3.08816 0.358991
$$75$$ −2.81288 −0.324803
$$76$$ 3.67657 0.421732
$$77$$ −0.177136 −0.0201866
$$78$$ −0.805738 −0.0912319
$$79$$ −3.61983 −0.407262 −0.203631 0.979048i $$-0.565274\pi$$
−0.203631 + 0.979048i $$0.565274\pi$$
$$80$$ 1.47889 0.165345
$$81$$ 1.00000 0.111111
$$82$$ −1.45973 −0.161201
$$83$$ 10.0343 1.10140 0.550702 0.834702i $$-0.314360\pi$$
0.550702 + 0.834702i $$0.314360\pi$$
$$84$$ 3.20362 0.349543
$$85$$ −3.04583 −0.330367
$$86$$ 12.5322 1.35139
$$87$$ 7.34575 0.787547
$$88$$ −0.0552927 −0.00589422
$$89$$ 8.25248 0.874762 0.437381 0.899276i $$-0.355906\pi$$
0.437381 + 0.899276i $$0.355906\pi$$
$$90$$ 1.47889 0.155889
$$91$$ −2.58128 −0.270591
$$92$$ 0 0
$$93$$ −7.95546 −0.824943
$$94$$ −1.27459 −0.131464
$$95$$ 5.43725 0.557850
$$96$$ 1.00000 0.102062
$$97$$ −7.01618 −0.712386 −0.356193 0.934412i $$-0.615925\pi$$
−0.356193 + 0.934412i $$0.615925\pi$$
$$98$$ 3.26315 0.329628
$$99$$ −0.0552927 −0.00555712
$$100$$ −2.81288 −0.281288
$$101$$ 15.3507 1.52745 0.763725 0.645542i $$-0.223368\pi$$
0.763725 + 0.645542i $$0.223368\pi$$
$$102$$ −2.05954 −0.203925
$$103$$ −15.3219 −1.50971 −0.754854 0.655893i $$-0.772292\pi$$
−0.754854 + 0.655893i $$0.772292\pi$$
$$104$$ −0.805738 −0.0790091
$$105$$ 4.73780 0.462362
$$106$$ 11.0449 1.07277
$$107$$ 9.67542 0.935358 0.467679 0.883898i $$-0.345090\pi$$
0.467679 + 0.883898i $$0.345090\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 13.4002 1.28350 0.641752 0.766912i $$-0.278208\pi$$
0.641752 + 0.766912i $$0.278208\pi$$
$$110$$ −0.0817718 −0.00779664
$$111$$ 3.08816 0.293115
$$112$$ 3.20362 0.302713
$$113$$ −8.22890 −0.774110 −0.387055 0.922057i $$-0.626508\pi$$
−0.387055 + 0.922057i $$0.626508\pi$$
$$114$$ 3.67657 0.344343
$$115$$ 0 0
$$116$$ 7.34575 0.682036
$$117$$ −0.805738 −0.0744905
$$118$$ −5.36077 −0.493499
$$119$$ −6.59797 −0.604835
$$120$$ 1.47889 0.135004
$$121$$ −10.9969 −0.999722
$$122$$ −12.2015 −1.10467
$$123$$ −1.45973 −0.131620
$$124$$ −7.95546 −0.714422
$$125$$ −11.5544 −1.03346
$$126$$ 3.20362 0.285401
$$127$$ 19.3524 1.71725 0.858624 0.512606i $$-0.171320\pi$$
0.858624 + 0.512606i $$0.171320\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.5322 1.10340
$$130$$ −1.19160 −0.104510
$$131$$ −20.9028 −1.82628 −0.913141 0.407643i $$-0.866351\pi$$
−0.913141 + 0.407643i $$0.866351\pi$$
$$132$$ −0.0552927 −0.00481261
$$133$$ 11.7783 1.02131
$$134$$ −4.52116 −0.390569
$$135$$ 1.47889 0.127283
$$136$$ −2.05954 −0.176604
$$137$$ −5.89909 −0.503993 −0.251997 0.967728i $$-0.581087\pi$$
−0.251997 + 0.967728i $$0.581087\pi$$
$$138$$ 0 0
$$139$$ −4.04356 −0.342971 −0.171485 0.985187i $$-0.554857\pi$$
−0.171485 + 0.985187i $$0.554857\pi$$
$$140$$ 4.73780 0.400417
$$141$$ −1.27459 −0.107340
$$142$$ −14.6029 −1.22545
$$143$$ 0.0445514 0.00372558
$$144$$ 1.00000 0.0833333
$$145$$ 10.8636 0.902170
$$146$$ −9.86160 −0.816152
$$147$$ 3.26315 0.269140
$$148$$ 3.08816 0.253845
$$149$$ 8.21355 0.672880 0.336440 0.941705i $$-0.390777\pi$$
0.336440 + 0.941705i $$0.390777\pi$$
$$150$$ −2.81288 −0.229671
$$151$$ −12.8490 −1.04564 −0.522820 0.852443i $$-0.675120\pi$$
−0.522820 + 0.852443i $$0.675120\pi$$
$$152$$ 3.67657 0.298209
$$153$$ −2.05954 −0.166504
$$154$$ −0.177136 −0.0142741
$$155$$ −11.7653 −0.945009
$$156$$ −0.805738 −0.0645107
$$157$$ −3.91879 −0.312754 −0.156377 0.987697i $$-0.549981\pi$$
−0.156377 + 0.987697i $$0.549981\pi$$
$$158$$ −3.61983 −0.287978
$$159$$ 11.0449 0.875914
$$160$$ 1.47889 0.116917
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 15.7247 1.23165 0.615825 0.787883i $$-0.288823\pi$$
0.615825 + 0.787883i $$0.288823\pi$$
$$164$$ −1.45973 −0.113986
$$165$$ −0.0817718 −0.00636593
$$166$$ 10.0343 0.778810
$$167$$ −4.01965 −0.311050 −0.155525 0.987832i $$-0.549707\pi$$
−0.155525 + 0.987832i $$0.549707\pi$$
$$168$$ 3.20362 0.247164
$$169$$ −12.3508 −0.950060
$$170$$ −3.04583 −0.233605
$$171$$ 3.67657 0.281155
$$172$$ 12.5322 0.955574
$$173$$ 5.25447 0.399490 0.199745 0.979848i $$-0.435989\pi$$
0.199745 + 0.979848i $$0.435989\pi$$
$$174$$ 7.34575 0.556880
$$175$$ −9.01139 −0.681197
$$176$$ −0.0552927 −0.00416784
$$177$$ −5.36077 −0.402940
$$178$$ 8.25248 0.618550
$$179$$ −2.11290 −0.157926 −0.0789629 0.996878i $$-0.525161\pi$$
−0.0789629 + 0.996878i $$0.525161\pi$$
$$180$$ 1.47889 0.110230
$$181$$ 0.230880 0.0171612 0.00858058 0.999963i $$-0.497269\pi$$
0.00858058 + 0.999963i $$0.497269\pi$$
$$182$$ −2.58128 −0.191337
$$183$$ −12.2015 −0.901960
$$184$$ 0 0
$$185$$ 4.56705 0.335776
$$186$$ −7.95546 −0.583323
$$187$$ 0.113877 0.00832753
$$188$$ −1.27459 −0.0929591
$$189$$ 3.20362 0.233029
$$190$$ 5.43725 0.394460
$$191$$ −4.40318 −0.318603 −0.159302 0.987230i $$-0.550924\pi$$
−0.159302 + 0.987230i $$0.550924\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 21.4973 1.54741 0.773704 0.633547i $$-0.218402\pi$$
0.773704 + 0.633547i $$0.218402\pi$$
$$194$$ −7.01618 −0.503733
$$195$$ −1.19160 −0.0853322
$$196$$ 3.26315 0.233082
$$197$$ 8.89209 0.633535 0.316768 0.948503i $$-0.397402\pi$$
0.316768 + 0.948503i $$0.397402\pi$$
$$198$$ −0.0552927 −0.00392948
$$199$$ −18.1550 −1.28698 −0.643488 0.765456i $$-0.722513\pi$$
−0.643488 + 0.765456i $$0.722513\pi$$
$$200$$ −2.81288 −0.198901
$$201$$ −4.52116 −0.318898
$$202$$ 15.3507 1.08007
$$203$$ 23.5330 1.65169
$$204$$ −2.05954 −0.144196
$$205$$ −2.15879 −0.150776
$$206$$ −15.3219 −1.06752
$$207$$ 0 0
$$208$$ −0.805738 −0.0558679
$$209$$ −0.203288 −0.0140617
$$210$$ 4.73780 0.326939
$$211$$ 17.2040 1.18437 0.592187 0.805801i $$-0.298265\pi$$
0.592187 + 0.805801i $$0.298265\pi$$
$$212$$ 11.0449 0.758564
$$213$$ −14.6029 −1.00057
$$214$$ 9.67542 0.661398
$$215$$ 18.5338 1.26400
$$216$$ 1.00000 0.0680414
$$217$$ −25.4862 −1.73012
$$218$$ 13.4002 0.907575
$$219$$ −9.86160 −0.666385
$$220$$ −0.0817718 −0.00551306
$$221$$ 1.65945 0.111627
$$222$$ 3.08816 0.207263
$$223$$ −9.64578 −0.645929 −0.322964 0.946411i $$-0.604679\pi$$
−0.322964 + 0.946411i $$0.604679\pi$$
$$224$$ 3.20362 0.214051
$$225$$ −2.81288 −0.187525
$$226$$ −8.22890 −0.547378
$$227$$ 13.2879 0.881951 0.440975 0.897519i $$-0.354633\pi$$
0.440975 + 0.897519i $$0.354633\pi$$
$$228$$ 3.67657 0.243487
$$229$$ −10.3343 −0.682912 −0.341456 0.939898i $$-0.610920\pi$$
−0.341456 + 0.939898i $$0.610920\pi$$
$$230$$ 0 0
$$231$$ −0.177136 −0.0116547
$$232$$ 7.34575 0.482272
$$233$$ −0.221377 −0.0145029 −0.00725144 0.999974i $$-0.502308\pi$$
−0.00725144 + 0.999974i $$0.502308\pi$$
$$234$$ −0.805738 −0.0526728
$$235$$ −1.88498 −0.122963
$$236$$ −5.36077 −0.348956
$$237$$ −3.61983 −0.235133
$$238$$ −6.59797 −0.427683
$$239$$ 10.0933 0.652880 0.326440 0.945218i $$-0.394151\pi$$
0.326440 + 0.945218i $$0.394151\pi$$
$$240$$ 1.47889 0.0954620
$$241$$ 2.49660 0.160820 0.0804099 0.996762i $$-0.474377\pi$$
0.0804099 + 0.996762i $$0.474377\pi$$
$$242$$ −10.9969 −0.706910
$$243$$ 1.00000 0.0641500
$$244$$ −12.2015 −0.781120
$$245$$ 4.82585 0.308312
$$246$$ −1.45973 −0.0930692
$$247$$ −2.96236 −0.188490
$$248$$ −7.95546 −0.505172
$$249$$ 10.0343 0.635896
$$250$$ −11.5544 −0.730765
$$251$$ −14.3756 −0.907379 −0.453690 0.891160i $$-0.649893\pi$$
−0.453690 + 0.891160i $$0.649893\pi$$
$$252$$ 3.20362 0.201809
$$253$$ 0 0
$$254$$ 19.3524 1.21428
$$255$$ −3.04583 −0.190737
$$256$$ 1.00000 0.0625000
$$257$$ −17.1087 −1.06721 −0.533606 0.845733i $$-0.679164\pi$$
−0.533606 + 0.845733i $$0.679164\pi$$
$$258$$ 12.5322 0.780223
$$259$$ 9.89326 0.614738
$$260$$ −1.19160 −0.0738999
$$261$$ 7.34575 0.454690
$$262$$ −20.9028 −1.29138
$$263$$ 9.82969 0.606125 0.303062 0.952971i $$-0.401991\pi$$
0.303062 + 0.952971i $$0.401991\pi$$
$$264$$ −0.0552927 −0.00340303
$$265$$ 16.3341 1.00340
$$266$$ 11.7783 0.722176
$$267$$ 8.25248 0.505044
$$268$$ −4.52116 −0.276174
$$269$$ 27.0424 1.64881 0.824404 0.566002i $$-0.191511\pi$$
0.824404 + 0.566002i $$0.191511\pi$$
$$270$$ 1.47889 0.0900025
$$271$$ 14.0124 0.851195 0.425597 0.904913i $$-0.360064\pi$$
0.425597 + 0.904913i $$0.360064\pi$$
$$272$$ −2.05954 −0.124878
$$273$$ −2.58128 −0.156226
$$274$$ −5.89909 −0.356377
$$275$$ 0.155532 0.00937891
$$276$$ 0 0
$$277$$ −6.65528 −0.399877 −0.199938 0.979808i $$-0.564074\pi$$
−0.199938 + 0.979808i $$0.564074\pi$$
$$278$$ −4.04356 −0.242517
$$279$$ −7.95546 −0.476281
$$280$$ 4.73780 0.283138
$$281$$ 14.7244 0.878384 0.439192 0.898393i $$-0.355265\pi$$
0.439192 + 0.898393i $$0.355265\pi$$
$$282$$ −1.27459 −0.0759008
$$283$$ 13.6907 0.813827 0.406914 0.913467i $$-0.366605\pi$$
0.406914 + 0.913467i $$0.366605\pi$$
$$284$$ −14.6029 −0.866522
$$285$$ 5.43725 0.322075
$$286$$ 0.0445514 0.00263438
$$287$$ −4.67642 −0.276041
$$288$$ 1.00000 0.0589256
$$289$$ −12.7583 −0.750489
$$290$$ 10.8636 0.637931
$$291$$ −7.01618 −0.411296
$$292$$ −9.86160 −0.577106
$$293$$ −29.6195 −1.73039 −0.865194 0.501437i $$-0.832805\pi$$
−0.865194 + 0.501437i $$0.832805\pi$$
$$294$$ 3.26315 0.190311
$$295$$ −7.92800 −0.461586
$$296$$ 3.08816 0.179495
$$297$$ −0.0552927 −0.00320841
$$298$$ 8.21355 0.475798
$$299$$ 0 0
$$300$$ −2.81288 −0.162402
$$301$$ 40.1485 2.31412
$$302$$ −12.8490 −0.739380
$$303$$ 15.3507 0.881873
$$304$$ 3.67657 0.210866
$$305$$ −18.0447 −1.03324
$$306$$ −2.05954 −0.117736
$$307$$ 9.12670 0.520888 0.260444 0.965489i $$-0.416131\pi$$
0.260444 + 0.965489i $$0.416131\pi$$
$$308$$ −0.177136 −0.0100933
$$309$$ −15.3219 −0.871630
$$310$$ −11.7653 −0.668222
$$311$$ 22.9754 1.30281 0.651407 0.758728i $$-0.274179\pi$$
0.651407 + 0.758728i $$0.274179\pi$$
$$312$$ −0.805738 −0.0456159
$$313$$ −19.7421 −1.11589 −0.557946 0.829877i $$-0.688410\pi$$
−0.557946 + 0.829877i $$0.688410\pi$$
$$314$$ −3.91879 −0.221150
$$315$$ 4.73780 0.266945
$$316$$ −3.61983 −0.203631
$$317$$ −20.8318 −1.17003 −0.585016 0.811022i $$-0.698912\pi$$
−0.585016 + 0.811022i $$0.698912\pi$$
$$318$$ 11.0449 0.619365
$$319$$ −0.406166 −0.0227409
$$320$$ 1.47889 0.0826725
$$321$$ 9.67542 0.540029
$$322$$ 0 0
$$323$$ −7.57204 −0.421320
$$324$$ 1.00000 0.0555556
$$325$$ 2.26645 0.125720
$$326$$ 15.7247 0.870909
$$327$$ 13.4002 0.741032
$$328$$ −1.45973 −0.0806003
$$329$$ −4.08330 −0.225120
$$330$$ −0.0817718 −0.00450139
$$331$$ −15.5468 −0.854527 −0.427264 0.904127i $$-0.640522\pi$$
−0.427264 + 0.904127i $$0.640522\pi$$
$$332$$ 10.0343 0.550702
$$333$$ 3.08816 0.169230
$$334$$ −4.01965 −0.219945
$$335$$ −6.68631 −0.365312
$$336$$ 3.20362 0.174772
$$337$$ −7.07769 −0.385546 −0.192773 0.981243i $$-0.561748\pi$$
−0.192773 + 0.981243i $$0.561748\pi$$
$$338$$ −12.3508 −0.671794
$$339$$ −8.22890 −0.446933
$$340$$ −3.04583 −0.165183
$$341$$ 0.439879 0.0238208
$$342$$ 3.67657 0.198806
$$343$$ −11.9714 −0.646396
$$344$$ 12.5322 0.675693
$$345$$ 0 0
$$346$$ 5.25447 0.282482
$$347$$ −25.4158 −1.36439 −0.682195 0.731170i $$-0.738975\pi$$
−0.682195 + 0.731170i $$0.738975\pi$$
$$348$$ 7.34575 0.393774
$$349$$ −23.1316 −1.23821 −0.619104 0.785309i $$-0.712504\pi$$
−0.619104 + 0.785309i $$0.712504\pi$$
$$350$$ −9.01139 −0.481679
$$351$$ −0.805738 −0.0430071
$$352$$ −0.0552927 −0.00294711
$$353$$ 24.0678 1.28100 0.640500 0.767958i $$-0.278727\pi$$
0.640500 + 0.767958i $$0.278727\pi$$
$$354$$ −5.36077 −0.284922
$$355$$ −21.5961 −1.14620
$$356$$ 8.25248 0.437381
$$357$$ −6.59797 −0.349201
$$358$$ −2.11290 −0.111670
$$359$$ −19.1724 −1.01188 −0.505940 0.862569i $$-0.668854\pi$$
−0.505940 + 0.862569i $$0.668854\pi$$
$$360$$ 1.47889 0.0779444
$$361$$ −5.48281 −0.288569
$$362$$ 0.230880 0.0121348
$$363$$ −10.9969 −0.577190
$$364$$ −2.58128 −0.135296
$$365$$ −14.5842 −0.763374
$$366$$ −12.2015 −0.637782
$$367$$ 26.2686 1.37121 0.685606 0.727973i $$-0.259537\pi$$
0.685606 + 0.727973i $$0.259537\pi$$
$$368$$ 0 0
$$369$$ −1.45973 −0.0759907
$$370$$ 4.56705 0.237429
$$371$$ 35.3835 1.83702
$$372$$ −7.95546 −0.412472
$$373$$ −20.4049 −1.05653 −0.528263 0.849081i $$-0.677156\pi$$
−0.528263 + 0.849081i $$0.677156\pi$$
$$374$$ 0.113877 0.00588846
$$375$$ −11.5544 −0.596667
$$376$$ −1.27459 −0.0657320
$$377$$ −5.91875 −0.304831
$$378$$ 3.20362 0.164776
$$379$$ −3.99029 −0.204968 −0.102484 0.994735i $$-0.532679\pi$$
−0.102484 + 0.994735i $$0.532679\pi$$
$$380$$ 5.43725 0.278925
$$381$$ 19.3524 0.991454
$$382$$ −4.40318 −0.225286
$$383$$ −6.32220 −0.323049 −0.161525 0.986869i $$-0.551641\pi$$
−0.161525 + 0.986869i $$0.551641\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −0.261966 −0.0133510
$$386$$ 21.4973 1.09418
$$387$$ 12.5322 0.637050
$$388$$ −7.01618 −0.356193
$$389$$ 25.4165 1.28867 0.644335 0.764743i $$-0.277134\pi$$
0.644335 + 0.764743i $$0.277134\pi$$
$$390$$ −1.19160 −0.0603390
$$391$$ 0 0
$$392$$ 3.26315 0.164814
$$393$$ −20.9028 −1.05440
$$394$$ 8.89209 0.447977
$$395$$ −5.35333 −0.269355
$$396$$ −0.0552927 −0.00277856
$$397$$ −3.43509 −0.172402 −0.0862011 0.996278i $$-0.527473\pi$$
−0.0862011 + 0.996278i $$0.527473\pi$$
$$398$$ −18.1550 −0.910029
$$399$$ 11.7783 0.589654
$$400$$ −2.81288 −0.140644
$$401$$ 8.35498 0.417228 0.208614 0.977998i $$-0.433105\pi$$
0.208614 + 0.977998i $$0.433105\pi$$
$$402$$ −4.52116 −0.225495
$$403$$ 6.41002 0.319306
$$404$$ 15.3507 0.763725
$$405$$ 1.47889 0.0734867
$$406$$ 23.5330 1.16792
$$407$$ −0.170752 −0.00846388
$$408$$ −2.05954 −0.101962
$$409$$ −4.18657 −0.207013 −0.103506 0.994629i $$-0.533006\pi$$
−0.103506 + 0.994629i $$0.533006\pi$$
$$410$$ −2.15879 −0.106615
$$411$$ −5.89909 −0.290981
$$412$$ −15.3219 −0.754854
$$413$$ −17.1738 −0.845070
$$414$$ 0 0
$$415$$ 14.8396 0.728447
$$416$$ −0.805738 −0.0395046
$$417$$ −4.04356 −0.198014
$$418$$ −0.203288 −0.00994312
$$419$$ −8.98508 −0.438950 −0.219475 0.975618i $$-0.570434\pi$$
−0.219475 + 0.975618i $$0.570434\pi$$
$$420$$ 4.73780 0.231181
$$421$$ 6.75886 0.329407 0.164703 0.986343i $$-0.447333\pi$$
0.164703 + 0.986343i $$0.447333\pi$$
$$422$$ 17.2040 0.837478
$$423$$ −1.27459 −0.0619727
$$424$$ 11.0449 0.536386
$$425$$ 5.79323 0.281013
$$426$$ −14.6029 −0.707513
$$427$$ −39.0889 −1.89164
$$428$$ 9.67542 0.467679
$$429$$ 0.0445514 0.00215096
$$430$$ 18.5338 0.893780
$$431$$ 22.6737 1.09216 0.546078 0.837734i $$-0.316120\pi$$
0.546078 + 0.837734i $$0.316120\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −29.9541 −1.43950 −0.719752 0.694231i $$-0.755744\pi$$
−0.719752 + 0.694231i $$0.755744\pi$$
$$434$$ −25.4862 −1.22338
$$435$$ 10.8636 0.520868
$$436$$ 13.4002 0.641752
$$437$$ 0 0
$$438$$ −9.86160 −0.471205
$$439$$ −1.50141 −0.0716585 −0.0358292 0.999358i $$-0.511407\pi$$
−0.0358292 + 0.999358i $$0.511407\pi$$
$$440$$ −0.0817718 −0.00389832
$$441$$ 3.26315 0.155388
$$442$$ 1.65945 0.0789319
$$443$$ −5.89537 −0.280097 −0.140049 0.990145i $$-0.544726\pi$$
−0.140049 + 0.990145i $$0.544726\pi$$
$$444$$ 3.08816 0.146557
$$445$$ 12.2045 0.578550
$$446$$ −9.64578 −0.456741
$$447$$ 8.21355 0.388487
$$448$$ 3.20362 0.151357
$$449$$ −36.8587 −1.73947 −0.869734 0.493520i $$-0.835710\pi$$
−0.869734 + 0.493520i $$0.835710\pi$$
$$450$$ −2.81288 −0.132600
$$451$$ 0.0807125 0.00380061
$$452$$ −8.22890 −0.387055
$$453$$ −12.8490 −0.603701
$$454$$ 13.2879 0.623633
$$455$$ −3.81743 −0.178964
$$456$$ 3.67657 0.172171
$$457$$ 5.79767 0.271203 0.135602 0.990763i $$-0.456703\pi$$
0.135602 + 0.990763i $$0.456703\pi$$
$$458$$ −10.3343 −0.482892
$$459$$ −2.05954 −0.0961310
$$460$$ 0 0
$$461$$ −19.1277 −0.890864 −0.445432 0.895316i $$-0.646950\pi$$
−0.445432 + 0.895316i $$0.646950\pi$$
$$462$$ −0.177136 −0.00824113
$$463$$ −20.3367 −0.945126 −0.472563 0.881297i $$-0.656671\pi$$
−0.472563 + 0.881297i $$0.656671\pi$$
$$464$$ 7.34575 0.341018
$$465$$ −11.7653 −0.545601
$$466$$ −0.221377 −0.0102551
$$467$$ −27.1467 −1.25620 −0.628101 0.778132i $$-0.716167\pi$$
−0.628101 + 0.778132i $$0.716167\pi$$
$$468$$ −0.805738 −0.0372453
$$469$$ −14.4841 −0.668812
$$470$$ −1.88498 −0.0869477
$$471$$ −3.91879 −0.180568
$$472$$ −5.36077 −0.246749
$$473$$ −0.692941 −0.0318615
$$474$$ −3.61983 −0.166264
$$475$$ −10.3418 −0.474512
$$476$$ −6.59797 −0.302417
$$477$$ 11.0449 0.505709
$$478$$ 10.0933 0.461656
$$479$$ 3.73359 0.170592 0.0852961 0.996356i $$-0.472816\pi$$
0.0852961 + 0.996356i $$0.472816\pi$$
$$480$$ 1.47889 0.0675019
$$481$$ −2.48825 −0.113454
$$482$$ 2.49660 0.113717
$$483$$ 0 0
$$484$$ −10.9969 −0.499861
$$485$$ −10.3762 −0.471158
$$486$$ 1.00000 0.0453609
$$487$$ 23.0573 1.04483 0.522413 0.852693i $$-0.325032\pi$$
0.522413 + 0.852693i $$0.325032\pi$$
$$488$$ −12.2015 −0.552336
$$489$$ 15.7247 0.711094
$$490$$ 4.82585 0.218010
$$491$$ −1.12229 −0.0506483 −0.0253241 0.999679i $$-0.508062\pi$$
−0.0253241 + 0.999679i $$0.508062\pi$$
$$492$$ −1.45973 −0.0658099
$$493$$ −15.1288 −0.681369
$$494$$ −2.96236 −0.133283
$$495$$ −0.0817718 −0.00367537
$$496$$ −7.95546 −0.357211
$$497$$ −46.7821 −2.09846
$$498$$ 10.0343 0.449646
$$499$$ 39.6576 1.77532 0.887659 0.460502i $$-0.152331\pi$$
0.887659 + 0.460502i $$0.152331\pi$$
$$500$$ −11.5544 −0.516729
$$501$$ −4.01965 −0.179585
$$502$$ −14.3756 −0.641614
$$503$$ 34.0385 1.51770 0.758850 0.651266i $$-0.225762\pi$$
0.758850 + 0.651266i $$0.225762\pi$$
$$504$$ 3.20362 0.142700
$$505$$ 22.7020 1.01023
$$506$$ 0 0
$$507$$ −12.3508 −0.548518
$$508$$ 19.3524 0.858624
$$509$$ 0.940632 0.0416928 0.0208464 0.999783i $$-0.493364\pi$$
0.0208464 + 0.999783i $$0.493364\pi$$
$$510$$ −3.04583 −0.134872
$$511$$ −31.5928 −1.39758
$$512$$ 1.00000 0.0441942
$$513$$ 3.67657 0.162325
$$514$$ −17.1087 −0.754633
$$515$$ −22.6594 −0.998491
$$516$$ 12.5322 0.551701
$$517$$ 0.0704755 0.00309951
$$518$$ 9.89326 0.434685
$$519$$ 5.25447 0.230646
$$520$$ −1.19160 −0.0522551
$$521$$ 40.4064 1.77024 0.885119 0.465366i $$-0.154077\pi$$
0.885119 + 0.465366i $$0.154077\pi$$
$$522$$ 7.34575 0.321515
$$523$$ 14.0908 0.616146 0.308073 0.951363i $$-0.400316\pi$$
0.308073 + 0.951363i $$0.400316\pi$$
$$524$$ −20.9028 −0.913141
$$525$$ −9.01139 −0.393289
$$526$$ 9.82969 0.428595
$$527$$ 16.3846 0.713723
$$528$$ −0.0552927 −0.00240630
$$529$$ 0 0
$$530$$ 16.3341 0.709510
$$531$$ −5.36077 −0.232638
$$532$$ 11.7783 0.510655
$$533$$ 1.17616 0.0509453
$$534$$ 8.25248 0.357120
$$535$$ 14.3089 0.618628
$$536$$ −4.52116 −0.195285
$$537$$ −2.11290 −0.0911785
$$538$$ 27.0424 1.16588
$$539$$ −0.180428 −0.00777160
$$540$$ 1.47889 0.0636414
$$541$$ −32.0267 −1.37694 −0.688468 0.725267i $$-0.741716\pi$$
−0.688468 + 0.725267i $$0.741716\pi$$
$$542$$ 14.0124 0.601885
$$543$$ 0.230880 0.00990800
$$544$$ −2.05954 −0.0883019
$$545$$ 19.8174 0.848885
$$546$$ −2.58128 −0.110468
$$547$$ −34.2269 −1.46344 −0.731718 0.681607i $$-0.761281\pi$$
−0.731718 + 0.681607i $$0.761281\pi$$
$$548$$ −5.89909 −0.251997
$$549$$ −12.2015 −0.520747
$$550$$ 0.155532 0.00663189
$$551$$ 27.0072 1.15054
$$552$$ 0 0
$$553$$ −11.5965 −0.493135
$$554$$ −6.65528 −0.282756
$$555$$ 4.56705 0.193860
$$556$$ −4.04356 −0.171485
$$557$$ 11.3076 0.479117 0.239559 0.970882i $$-0.422997\pi$$
0.239559 + 0.970882i $$0.422997\pi$$
$$558$$ −7.95546 −0.336782
$$559$$ −10.0977 −0.427087
$$560$$ 4.73780 0.200209
$$561$$ 0.113877 0.00480790
$$562$$ 14.7244 0.621111
$$563$$ −17.2976 −0.729006 −0.364503 0.931202i $$-0.618761\pi$$
−0.364503 + 0.931202i $$0.618761\pi$$
$$564$$ −1.27459 −0.0536700
$$565$$ −12.1697 −0.511981
$$566$$ 13.6907 0.575463
$$567$$ 3.20362 0.134539
$$568$$ −14.6029 −0.612724
$$569$$ −29.0600 −1.21826 −0.609129 0.793071i $$-0.708481\pi$$
−0.609129 + 0.793071i $$0.708481\pi$$
$$570$$ 5.43725 0.227741
$$571$$ −3.14635 −0.131671 −0.0658354 0.997830i $$-0.520971\pi$$
−0.0658354 + 0.997830i $$0.520971\pi$$
$$572$$ 0.0445514 0.00186279
$$573$$ −4.40318 −0.183946
$$574$$ −4.67642 −0.195190
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 10.8526 0.451799 0.225900 0.974151i $$-0.427468\pi$$
0.225900 + 0.974151i $$0.427468\pi$$
$$578$$ −12.7583 −0.530676
$$579$$ 21.4973 0.893397
$$580$$ 10.8636 0.451085
$$581$$ 32.1459 1.33364
$$582$$ −7.01618 −0.290830
$$583$$ −0.610700 −0.0252926
$$584$$ −9.86160 −0.408076
$$585$$ −1.19160 −0.0492666
$$586$$ −29.6195 −1.22357
$$587$$ −32.2820 −1.33242 −0.666210 0.745764i $$-0.732085\pi$$
−0.666210 + 0.745764i $$0.732085\pi$$
$$588$$ 3.26315 0.134570
$$589$$ −29.2488 −1.20518
$$590$$ −7.92800 −0.326390
$$591$$ 8.89209 0.365772
$$592$$ 3.08816 0.126922
$$593$$ 35.1225 1.44231 0.721154 0.692774i $$-0.243612\pi$$
0.721154 + 0.692774i $$0.243612\pi$$
$$594$$ −0.0552927 −0.00226869
$$595$$ −9.75768 −0.400026
$$596$$ 8.21355 0.336440
$$597$$ −18.1550 −0.743036
$$598$$ 0 0
$$599$$ −33.0831 −1.35174 −0.675870 0.737021i $$-0.736232\pi$$
−0.675870 + 0.737021i $$0.736232\pi$$
$$600$$ −2.81288 −0.114835
$$601$$ −40.9247 −1.66935 −0.834677 0.550740i $$-0.814345\pi$$
−0.834677 + 0.550740i $$0.814345\pi$$
$$602$$ 40.1485 1.63633
$$603$$ −4.52116 −0.184116
$$604$$ −12.8490 −0.522820
$$605$$ −16.2633 −0.661197
$$606$$ 15.3507 0.623579
$$607$$ 0.867250 0.0352006 0.0176003 0.999845i $$-0.494397\pi$$
0.0176003 + 0.999845i $$0.494397\pi$$
$$608$$ 3.67657 0.149105
$$609$$ 23.5330 0.953604
$$610$$ −18.0447 −0.730608
$$611$$ 1.02699 0.0415474
$$612$$ −2.05954 −0.0832519
$$613$$ −13.5982 −0.549227 −0.274614 0.961555i $$-0.588550\pi$$
−0.274614 + 0.961555i $$0.588550\pi$$
$$614$$ 9.12670 0.368324
$$615$$ −2.15879 −0.0870507
$$616$$ −0.177136 −0.00713703
$$617$$ 42.6638 1.71758 0.858790 0.512328i $$-0.171217\pi$$
0.858790 + 0.512328i $$0.171217\pi$$
$$618$$ −15.3219 −0.616335
$$619$$ 31.5251 1.26710 0.633551 0.773701i $$-0.281597\pi$$
0.633551 + 0.773701i $$0.281597\pi$$
$$620$$ −11.7653 −0.472505
$$621$$ 0 0
$$622$$ 22.9754 0.921229
$$623$$ 26.4378 1.05921
$$624$$ −0.805738 −0.0322553
$$625$$ −3.02331 −0.120932
$$626$$ −19.7421 −0.789054
$$627$$ −0.203288 −0.00811852
$$628$$ −3.91879 −0.156377
$$629$$ −6.36017 −0.253597
$$630$$ 4.73780 0.188758
$$631$$ 48.7324 1.94001 0.970003 0.243093i $$-0.0781622\pi$$
0.970003 + 0.243093i $$0.0781622\pi$$
$$632$$ −3.61983 −0.143989
$$633$$ 17.2040 0.683798
$$634$$ −20.8318 −0.827337
$$635$$ 28.6201 1.13575
$$636$$ 11.0449 0.437957
$$637$$ −2.62925 −0.104175
$$638$$ −0.406166 −0.0160803
$$639$$ −14.6029 −0.577682
$$640$$ 1.47889 0.0584583
$$641$$ −30.1834 −1.19217 −0.596086 0.802920i $$-0.703278\pi$$
−0.596086 + 0.802920i $$0.703278\pi$$
$$642$$ 9.67542 0.381858
$$643$$ −34.4723 −1.35946 −0.679728 0.733464i $$-0.737902\pi$$
−0.679728 + 0.733464i $$0.737902\pi$$
$$644$$ 0 0
$$645$$ 18.5338 0.729769
$$646$$ −7.57204 −0.297918
$$647$$ 13.9668 0.549092 0.274546 0.961574i $$-0.411472\pi$$
0.274546 + 0.961574i $$0.411472\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0.296411 0.0116352
$$650$$ 2.26645 0.0888973
$$651$$ −25.4862 −0.998885
$$652$$ 15.7247 0.615825
$$653$$ 36.9272 1.44507 0.722537 0.691332i $$-0.242976\pi$$
0.722537 + 0.691332i $$0.242976\pi$$
$$654$$ 13.4002 0.523988
$$655$$ −30.9129 −1.20787
$$656$$ −1.45973 −0.0569930
$$657$$ −9.86160 −0.384738
$$658$$ −4.08330 −0.159184
$$659$$ 11.0352 0.429872 0.214936 0.976628i $$-0.431046\pi$$
0.214936 + 0.976628i $$0.431046\pi$$
$$660$$ −0.0817718 −0.00318296
$$661$$ −33.9751 −1.32148 −0.660738 0.750616i $$-0.729757\pi$$
−0.660738 + 0.750616i $$0.729757\pi$$
$$662$$ −15.5468 −0.604242
$$663$$ 1.65945 0.0644476
$$664$$ 10.0343 0.389405
$$665$$ 17.4189 0.675475
$$666$$ 3.08816 0.119664
$$667$$ 0 0
$$668$$ −4.01965 −0.155525
$$669$$ −9.64578 −0.372927
$$670$$ −6.68631 −0.258315
$$671$$ 0.674653 0.0260447
$$672$$ 3.20362 0.123582
$$673$$ −31.3928 −1.21010 −0.605052 0.796186i $$-0.706848\pi$$
−0.605052 + 0.796186i $$0.706848\pi$$
$$674$$ −7.07769 −0.272622
$$675$$ −2.81288 −0.108268
$$676$$ −12.3508 −0.475030
$$677$$ 39.3382 1.51189 0.755944 0.654636i $$-0.227178\pi$$
0.755944 + 0.654636i $$0.227178\pi$$
$$678$$ −8.22890 −0.316029
$$679$$ −22.4772 −0.862594
$$680$$ −3.04583 −0.116802
$$681$$ 13.2879 0.509194
$$682$$ 0.439879 0.0168438
$$683$$ 31.6224 1.21000 0.604998 0.796227i $$-0.293174\pi$$
0.604998 + 0.796227i $$0.293174\pi$$
$$684$$ 3.67657 0.140577
$$685$$ −8.72412 −0.333331
$$686$$ −11.9714 −0.457071
$$687$$ −10.3343 −0.394279
$$688$$ 12.5322 0.477787
$$689$$ −8.89927 −0.339035
$$690$$ 0 0
$$691$$ 22.2620 0.846886 0.423443 0.905923i $$-0.360821\pi$$
0.423443 + 0.905923i $$0.360821\pi$$
$$692$$ 5.25447 0.199745
$$693$$ −0.177136 −0.00672886
$$694$$ −25.4158 −0.964770
$$695$$ −5.97999 −0.226834
$$696$$ 7.34575 0.278440
$$697$$ 3.00638 0.113875
$$698$$ −23.1316 −0.875545
$$699$$ −0.221377 −0.00837325
$$700$$ −9.01139 −0.340598
$$701$$ −13.8462 −0.522964 −0.261482 0.965208i $$-0.584211\pi$$
−0.261482 + 0.965208i $$0.584211\pi$$
$$702$$ −0.805738 −0.0304106
$$703$$ 11.3538 0.428218
$$704$$ −0.0552927 −0.00208392
$$705$$ −1.88498 −0.0709925
$$706$$ 24.0678 0.905804
$$707$$ 49.1777 1.84952
$$708$$ −5.36077 −0.201470
$$709$$ −49.1026 −1.84409 −0.922043 0.387087i $$-0.873481\pi$$
−0.922043 + 0.387087i $$0.873481\pi$$
$$710$$ −21.5961 −0.810487
$$711$$ −3.61983 −0.135754
$$712$$ 8.25248 0.309275
$$713$$ 0 0
$$714$$ −6.59797 −0.246923
$$715$$ 0.0658867 0.00246402
$$716$$ −2.11290 −0.0789629
$$717$$ 10.0933 0.376940
$$718$$ −19.1724 −0.715507
$$719$$ 36.5928 1.36468 0.682341 0.731034i $$-0.260962\pi$$
0.682341 + 0.731034i $$0.260962\pi$$
$$720$$ 1.47889 0.0551150
$$721$$ −49.0853 −1.82803
$$722$$ −5.48281 −0.204049
$$723$$ 2.49660 0.0928494
$$724$$ 0.230880 0.00858058
$$725$$ −20.6627 −0.767394
$$726$$ −10.9969 −0.408135
$$727$$ −18.6107 −0.690232 −0.345116 0.938560i $$-0.612160\pi$$
−0.345116 + 0.938560i $$0.612160\pi$$
$$728$$ −2.58128 −0.0956684
$$729$$ 1.00000 0.0370370
$$730$$ −14.5842 −0.539787
$$731$$ −25.8106 −0.954640
$$732$$ −12.2015 −0.450980
$$733$$ 22.6156 0.835327 0.417664 0.908602i $$-0.362849\pi$$
0.417664 + 0.908602i $$0.362849\pi$$
$$734$$ 26.2686 0.969593
$$735$$ 4.82585 0.178004
$$736$$ 0 0
$$737$$ 0.249987 0.00920840
$$738$$ −1.45973 −0.0537335
$$739$$ −17.8869 −0.657982 −0.328991 0.944333i $$-0.606709\pi$$
−0.328991 + 0.944333i $$0.606709\pi$$
$$740$$ 4.56705 0.167888
$$741$$ −2.96236 −0.108825
$$742$$ 35.3835 1.29897
$$743$$ 22.4379 0.823168 0.411584 0.911372i $$-0.364976\pi$$
0.411584 + 0.911372i $$0.364976\pi$$
$$744$$ −7.95546 −0.291661
$$745$$ 12.1469 0.445030
$$746$$ −20.4049 −0.747076
$$747$$ 10.0343 0.367135
$$748$$ 0.113877 0.00416377
$$749$$ 30.9963 1.13258
$$750$$ −11.5544 −0.421907
$$751$$ −46.9846 −1.71449 −0.857246 0.514907i $$-0.827826\pi$$
−0.857246 + 0.514907i $$0.827826\pi$$
$$752$$ −1.27459 −0.0464795
$$753$$ −14.3756 −0.523876
$$754$$ −5.91875 −0.215548
$$755$$ −19.0023 −0.691566
$$756$$ 3.20362 0.116514
$$757$$ −28.7306 −1.04423 −0.522115 0.852875i $$-0.674857\pi$$
−0.522115 + 0.852875i $$0.674857\pi$$
$$758$$ −3.99029 −0.144934
$$759$$ 0 0
$$760$$ 5.43725 0.197230
$$761$$ 15.6869 0.568651 0.284326 0.958728i $$-0.408230\pi$$
0.284326 + 0.958728i $$0.408230\pi$$
$$762$$ 19.3524 0.701064
$$763$$ 42.9290 1.55413
$$764$$ −4.40318 −0.159302
$$765$$ −3.04583 −0.110122
$$766$$ −6.32220 −0.228430
$$767$$ 4.31938 0.155964
$$768$$ 1.00000 0.0360844
$$769$$ 19.8140 0.714509 0.357255 0.934007i $$-0.383713\pi$$
0.357255 + 0.934007i $$0.383713\pi$$
$$770$$ −0.261966 −0.00944058
$$771$$ −17.1087 −0.616155
$$772$$ 21.4973 0.773704
$$773$$ −19.5152 −0.701915 −0.350957 0.936391i $$-0.614144\pi$$
−0.350957 + 0.936391i $$0.614144\pi$$
$$774$$ 12.5322 0.450462
$$775$$ 22.3778 0.803833
$$776$$ −7.01618 −0.251866
$$777$$ 9.89326 0.354919
$$778$$ 25.4165 0.911227
$$779$$ −5.36682 −0.192286
$$780$$ −1.19160 −0.0426661
$$781$$ 0.807433 0.0288922
$$782$$ 0 0
$$783$$ 7.34575 0.262516
$$784$$ 3.26315 0.116541
$$785$$ −5.79547 −0.206849
$$786$$ −20.9028 −0.745577
$$787$$ −20.0112 −0.713321 −0.356661 0.934234i $$-0.616085\pi$$
−0.356661 + 0.934234i $$0.616085\pi$$
$$788$$ 8.89209 0.316768
$$789$$ 9.82969 0.349946
$$790$$ −5.35333 −0.190463
$$791$$ −26.3622 −0.937333
$$792$$ −0.0552927 −0.00196474
$$793$$ 9.83121 0.349116
$$794$$ −3.43509 −0.121907
$$795$$ 16.3341 0.579313
$$796$$ −18.1550 −0.643488
$$797$$ 19.7315 0.698927 0.349464 0.936950i $$-0.386364\pi$$
0.349464 + 0.936950i $$0.386364\pi$$
$$798$$ 11.7783 0.416948
$$799$$ 2.62507 0.0928682
$$800$$ −2.81288 −0.0994503
$$801$$ 8.25248 0.291587
$$802$$ 8.35498 0.295024
$$803$$ 0.545274 0.0192423
$$804$$ −4.52116 −0.159449
$$805$$ 0 0
$$806$$ 6.41002 0.225783
$$807$$ 27.0424 0.951939
$$808$$ 15.3507 0.540035
$$809$$ 27.7606 0.976009 0.488005 0.872841i $$-0.337725\pi$$
0.488005 + 0.872841i $$0.337725\pi$$
$$810$$ 1.47889 0.0519629
$$811$$ 18.4982 0.649560 0.324780 0.945790i $$-0.394710\pi$$
0.324780 + 0.945790i $$0.394710\pi$$
$$812$$ 23.5330 0.825845
$$813$$ 14.0124 0.491437
$$814$$ −0.170752 −0.00598487
$$815$$ 23.2551 0.814590
$$816$$ −2.05954 −0.0720982
$$817$$ 46.0757 1.61198
$$818$$ −4.18657 −0.146380
$$819$$ −2.58128 −0.0901971
$$820$$ −2.15879 −0.0753881
$$821$$ −3.89872 −0.136066 −0.0680332 0.997683i $$-0.521672\pi$$
−0.0680332 + 0.997683i $$0.521672\pi$$
$$822$$ −5.89909 −0.205754
$$823$$ −13.6654 −0.476345 −0.238173 0.971223i $$-0.576548\pi$$
−0.238173 + 0.971223i $$0.576548\pi$$
$$824$$ −15.3219 −0.533762
$$825$$ 0.155532 0.00541492
$$826$$ −17.1738 −0.597554
$$827$$ −1.97288 −0.0686037 −0.0343019 0.999412i $$-0.510921\pi$$
−0.0343019 + 0.999412i $$0.510921\pi$$
$$828$$ 0 0
$$829$$ 6.37133 0.221285 0.110643 0.993860i $$-0.464709\pi$$
0.110643 + 0.993860i $$0.464709\pi$$
$$830$$ 14.8396 0.515090
$$831$$ −6.65528 −0.230869
$$832$$ −0.805738 −0.0279339
$$833$$ −6.72059 −0.232855
$$834$$ −4.04356 −0.140017
$$835$$ −5.94462 −0.205722
$$836$$ −0.203288 −0.00703085
$$837$$ −7.95546 −0.274981
$$838$$ −8.98508 −0.310384
$$839$$ 0.347463 0.0119957 0.00599787 0.999982i $$-0.498091\pi$$
0.00599787 + 0.999982i $$0.498091\pi$$
$$840$$ 4.73780 0.163470
$$841$$ 24.9600 0.860691
$$842$$ 6.75886 0.232926
$$843$$ 14.7244 0.507135
$$844$$ 17.2040 0.592187
$$845$$ −18.2655 −0.628351
$$846$$ −1.27459 −0.0438213
$$847$$ −35.2300 −1.21052
$$848$$ 11.0449 0.379282
$$849$$ 13.6907 0.469863
$$850$$ 5.79323 0.198706
$$851$$ 0 0
$$852$$ −14.6029 −0.500287
$$853$$ −22.0786 −0.755958 −0.377979 0.925814i $$-0.623381\pi$$
−0.377979 + 0.925814i $$0.623381\pi$$
$$854$$ −39.0889 −1.33759
$$855$$ 5.43725 0.185950
$$856$$ 9.67542 0.330699
$$857$$ −19.3638 −0.661454 −0.330727 0.943726i $$-0.607294\pi$$
−0.330727 + 0.943726i $$0.607294\pi$$
$$858$$ 0.0445514 0.00152096
$$859$$ −10.1662 −0.346866 −0.173433 0.984846i $$-0.555486\pi$$
−0.173433 + 0.984846i $$0.555486\pi$$
$$860$$ 18.5338 0.631998
$$861$$ −4.67642 −0.159372
$$862$$ 22.6737 0.772271
$$863$$ −11.7222 −0.399027 −0.199514 0.979895i $$-0.563936\pi$$
−0.199514 + 0.979895i $$0.563936\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 7.77079 0.264215
$$866$$ −29.9541 −1.01788
$$867$$ −12.7583 −0.433295
$$868$$ −25.4862 −0.865060
$$869$$ 0.200150 0.00678962
$$870$$ 10.8636 0.368309
$$871$$ 3.64287 0.123434
$$872$$ 13.4002 0.453787
$$873$$ −7.01618 −0.237462
$$874$$ 0 0
$$875$$ −37.0159 −1.25136
$$876$$ −9.86160 −0.333193
$$877$$ −30.3909 −1.02623 −0.513114 0.858321i $$-0.671508\pi$$
−0.513114 + 0.858321i $$0.671508\pi$$
$$878$$ −1.50141 −0.0506702
$$879$$ −29.6195 −0.999040
$$880$$ −0.0817718 −0.00275653
$$881$$ 33.3403 1.12326 0.561632 0.827387i $$-0.310174\pi$$
0.561632 + 0.827387i $$0.310174\pi$$
$$882$$ 3.26315 0.109876
$$883$$ −5.16297 −0.173748 −0.0868740 0.996219i $$-0.527688\pi$$
−0.0868740 + 0.996219i $$0.527688\pi$$
$$884$$ 1.65945 0.0558133
$$885$$ −7.92800 −0.266497
$$886$$ −5.89537 −0.198059
$$887$$ −13.6271 −0.457555 −0.228777 0.973479i $$-0.573473\pi$$
−0.228777 + 0.973479i $$0.573473\pi$$
$$888$$ 3.08816 0.103632
$$889$$ 61.9977 2.07933
$$890$$ 12.2045 0.409097
$$891$$ −0.0552927 −0.00185237
$$892$$ −9.64578 −0.322964
$$893$$ −4.68613 −0.156815
$$894$$ 8.21355 0.274702
$$895$$ −3.12476 −0.104449
$$896$$ 3.20362 0.107025
$$897$$ 0 0
$$898$$ −36.8587 −1.22999
$$899$$ −58.4388 −1.94904
$$900$$ −2.81288 −0.0937627
$$901$$ −22.7473 −0.757822
$$902$$ 0.0807125 0.00268743
$$903$$ 40.1485 1.33606
$$904$$ −8.22890 −0.273689
$$905$$ 0.341446 0.0113500
$$906$$ −12.8490 −0.426881
$$907$$ 22.2146 0.737623 0.368811 0.929504i $$-0.379765\pi$$
0.368811 + 0.929504i $$0.379765\pi$$
$$908$$ 13.2879 0.440975
$$909$$ 15.3507 0.509150
$$910$$ −3.81743 −0.126546
$$911$$ 18.1822 0.602402 0.301201 0.953561i $$-0.402612\pi$$
0.301201 + 0.953561i $$0.402612\pi$$
$$912$$ 3.67657 0.121744
$$913$$ −0.554821 −0.0183619
$$914$$ 5.79767 0.191770
$$915$$ −18.0447 −0.596539
$$916$$ −10.3343 −0.341456
$$917$$ −66.9644 −2.21136
$$918$$ −2.05954 −0.0679749
$$919$$ 30.2025 0.996289 0.498145 0.867094i $$-0.334015\pi$$
0.498145 + 0.867094i $$0.334015\pi$$
$$920$$ 0 0
$$921$$ 9.12670 0.300735
$$922$$ −19.1277 −0.629936
$$923$$ 11.7661 0.387286
$$924$$ −0.177136 −0.00582736
$$925$$ −8.68661 −0.285614
$$926$$ −20.3367 −0.668305
$$927$$ −15.3219 −0.503236
$$928$$ 7.34575 0.241136
$$929$$ −25.8080 −0.846733 −0.423366 0.905959i $$-0.639152\pi$$
−0.423366 + 0.905959i $$0.639152\pi$$
$$930$$ −11.7653 −0.385798
$$931$$ 11.9972 0.393193
$$932$$ −0.221377 −0.00725144
$$933$$ 22.9754 0.752180
$$934$$ −27.1467 −0.888268
$$935$$ 0.168412 0.00550767
$$936$$ −0.805738 −0.0263364
$$937$$ 15.1514 0.494976 0.247488 0.968891i $$-0.420395\pi$$
0.247488 + 0.968891i $$0.420395\pi$$
$$938$$ −14.4841 −0.472922
$$939$$ −19.7421 −0.644260
$$940$$ −1.88498 −0.0614813
$$941$$ −25.1968 −0.821391 −0.410695 0.911773i $$-0.634714\pi$$
−0.410695 + 0.911773i $$0.634714\pi$$
$$942$$ −3.91879 −0.127681
$$943$$ 0 0
$$944$$ −5.36077 −0.174478
$$945$$ 4.73780 0.154121
$$946$$ −0.692941 −0.0225295
$$947$$ −0.612556 −0.0199054 −0.00995271 0.999950i $$-0.503168\pi$$
−0.00995271 + 0.999950i $$0.503168\pi$$
$$948$$ −3.61983 −0.117567
$$949$$ 7.94587 0.257934
$$950$$ −10.3418 −0.335531
$$951$$ −20.8318 −0.675518
$$952$$ −6.59797 −0.213841
$$953$$ −34.2779 −1.11037 −0.555185 0.831727i $$-0.687353\pi$$
−0.555185 + 0.831727i $$0.687353\pi$$
$$954$$ 11.0449 0.357591
$$955$$ −6.51183 −0.210718
$$956$$ 10.0933 0.326440
$$957$$ −0.406166 −0.0131295
$$958$$ 3.73359 0.120627
$$959$$ −18.8984 −0.610262
$$960$$ 1.47889 0.0477310
$$961$$ 32.2894 1.04159
$$962$$ −2.48825 −0.0802242
$$963$$ 9.67542 0.311786
$$964$$ 2.49660 0.0804099
$$965$$ 31.7921 1.02343
$$966$$ 0 0
$$967$$ 32.9004 1.05801 0.529003 0.848620i $$-0.322566\pi$$
0.529003 + 0.848620i $$0.322566\pi$$
$$968$$ −10.9969 −0.353455
$$969$$ −7.57204 −0.243249
$$970$$ −10.3762 −0.333159
$$971$$ 15.3464 0.492488 0.246244 0.969208i $$-0.420803\pi$$
0.246244 + 0.969208i $$0.420803\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −12.9540 −0.415287
$$974$$ 23.0573 0.738804
$$975$$ 2.26645 0.0725843
$$976$$ −12.2015 −0.390560
$$977$$ 7.59382 0.242948 0.121474 0.992595i $$-0.461238\pi$$
0.121474 + 0.992595i $$0.461238\pi$$
$$978$$ 15.7247 0.502819
$$979$$ −0.456302 −0.0145835
$$980$$ 4.82585 0.154156
$$981$$ 13.4002 0.427835
$$982$$ −1.12229 −0.0358137
$$983$$ −2.79957 −0.0892924 −0.0446462 0.999003i $$-0.514216\pi$$
−0.0446462 + 0.999003i $$0.514216\pi$$
$$984$$ −1.45973 −0.0465346
$$985$$ 13.1504 0.419008
$$986$$ −15.1288 −0.481801
$$987$$ −4.08330 −0.129973
$$988$$ −2.96236 −0.0942451
$$989$$ 0 0
$$990$$ −0.0817718 −0.00259888
$$991$$ 0.506808 0.0160993 0.00804964 0.999968i $$-0.497438\pi$$
0.00804964 + 0.999968i $$0.497438\pi$$
$$992$$ −7.95546 −0.252586
$$993$$ −15.5468 −0.493361
$$994$$ −46.7821 −1.48384
$$995$$ −26.8493 −0.851181
$$996$$ 10.0343 0.317948
$$997$$ 12.1136 0.383640 0.191820 0.981430i $$-0.438561\pi$$
0.191820 + 0.981430i $$0.438561\pi$$
$$998$$ 39.6576 1.25534
$$999$$ 3.08816 0.0977049
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 3174.2.a.bd.1.3 5
3.2 odd 2 9522.2.a.bq.1.3 5
23.17 odd 22 138.2.e.a.13.1 10
23.19 odd 22 138.2.e.a.85.1 yes 10
23.22 odd 2 3174.2.a.bc.1.3 5
69.17 even 22 414.2.i.d.289.1 10
69.65 even 22 414.2.i.d.361.1 10
69.68 even 2 9522.2.a.bt.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.a.13.1 10 23.17 odd 22
138.2.e.a.85.1 yes 10 23.19 odd 22
414.2.i.d.289.1 10 69.17 even 22
414.2.i.d.361.1 10 69.65 even 22
3174.2.a.bc.1.3 5 23.22 odd 2
3174.2.a.bd.1.3 5 1.1 even 1 trivial
9522.2.a.bq.1.3 5 3.2 odd 2
9522.2.a.bt.1.3 5 69.68 even 2