# Properties

 Label 145.2.a.c.1.2 Level $145$ Weight $2$ Character 145.1 Self dual yes Analytic conductor $1.158$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [145,2,Mod(1,145)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(145, 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("145.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 145.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: $$1.15783082931$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 145.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.311108 q^{2} +2.90321 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.903212 q^{6} -0.903212 q^{7} -1.21432 q^{8} +5.42864 q^{9} +O(q^{10})$$ $$q+0.311108 q^{2} +2.90321 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.903212 q^{6} -0.903212 q^{7} -1.21432 q^{8} +5.42864 q^{9} +0.311108 q^{10} -1.52543 q^{11} -5.52543 q^{12} -0.622216 q^{13} -0.280996 q^{14} +2.90321 q^{15} +3.42864 q^{16} -7.95407 q^{17} +1.68889 q^{18} -1.09679 q^{19} -1.90321 q^{20} -2.62222 q^{21} -0.474572 q^{22} +7.52543 q^{23} -3.52543 q^{24} +1.00000 q^{25} -0.193576 q^{26} +7.05086 q^{27} +1.71900 q^{28} -1.00000 q^{29} +0.903212 q^{30} -6.90321 q^{31} +3.49532 q^{32} -4.42864 q^{33} -2.47457 q^{34} -0.903212 q^{35} -10.3319 q^{36} +3.95407 q^{37} -0.341219 q^{38} -1.80642 q^{39} -1.21432 q^{40} +3.67307 q^{41} -0.815792 q^{42} -10.5161 q^{43} +2.90321 q^{44} +5.42864 q^{45} +2.34122 q^{46} +6.90321 q^{47} +9.95407 q^{48} -6.18421 q^{49} +0.311108 q^{50} -23.0923 q^{51} +1.18421 q^{52} +6.42864 q^{53} +2.19358 q^{54} -1.52543 q^{55} +1.09679 q^{56} -3.18421 q^{57} -0.311108 q^{58} -1.67307 q^{59} -5.52543 q^{60} -1.86665 q^{61} -2.14764 q^{62} -4.90321 q^{63} -5.76986 q^{64} -0.622216 q^{65} -1.37778 q^{66} +11.5254 q^{67} +15.1383 q^{68} +21.8479 q^{69} -0.280996 q^{70} +13.6731 q^{71} -6.59210 q^{72} +10.1891 q^{73} +1.23014 q^{74} +2.90321 q^{75} +2.08742 q^{76} +1.37778 q^{77} -0.561993 q^{78} +9.13828 q^{79} +3.42864 q^{80} +4.18421 q^{81} +1.14272 q^{82} +10.7096 q^{83} +4.99063 q^{84} -7.95407 q^{85} -3.27163 q^{86} -2.90321 q^{87} +1.85236 q^{88} -7.80642 q^{89} +1.68889 q^{90} +0.561993 q^{91} -14.3225 q^{92} -20.0415 q^{93} +2.14764 q^{94} -1.09679 q^{95} +10.1476 q^{96} -4.08742 q^{97} -1.92396 q^{98} -8.28100 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 + 2 * q^3 + q^4 + 3 * q^5 - 4 * q^6 + 4 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + q^{2} + 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 2 q^{11} - 10 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{15} - 3 q^{16} - 4 q^{17} + 5 q^{18} - 10 q^{19} + q^{20} - 8 q^{21} - 8 q^{22} + 16 q^{23} - 4 q^{24} + 3 q^{25} - 14 q^{26} + 8 q^{27} + 12 q^{28} - 3 q^{29} - 4 q^{30} - 14 q^{31} - 3 q^{32} - 14 q^{34} + 4 q^{35} - 11 q^{36} - 8 q^{37} - 8 q^{38} + 8 q^{39} + 3 q^{40} - 2 q^{41} - 16 q^{42} + 2 q^{43} + 2 q^{44} + 3 q^{45} + 14 q^{46} + 14 q^{47} + 10 q^{48} - 5 q^{49} + q^{50} - 16 q^{51} - 10 q^{52} + 6 q^{53} + 20 q^{54} + 2 q^{55} + 10 q^{56} + 4 q^{57} - q^{58} + 8 q^{59} - 10 q^{60} - 6 q^{61} - 8 q^{63} - 11 q^{64} - 2 q^{65} - 4 q^{66} + 28 q^{67} + 12 q^{68} + 12 q^{69} + 6 q^{70} + 28 q^{71} - 13 q^{72} - 16 q^{73} + 10 q^{74} + 2 q^{75} - 14 q^{76} + 4 q^{77} + 12 q^{78} - 6 q^{79} - 3 q^{80} - q^{81} + 30 q^{82} + 12 q^{83} - 12 q^{84} - 4 q^{85} + 24 q^{86} - 2 q^{87} + 12 q^{88} - 10 q^{89} + 5 q^{90} - 12 q^{91} + 4 q^{92} - 20 q^{93} - 10 q^{95} + 24 q^{96} + 8 q^{97} + 21 q^{98} - 18 q^{99}+O(q^{100})$$ 3 * q + q^2 + 2 * q^3 + q^4 + 3 * q^5 - 4 * q^6 + 4 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 2 * q^11 - 10 * q^12 - 2 * q^13 + 6 * q^14 + 2 * q^15 - 3 * q^16 - 4 * q^17 + 5 * q^18 - 10 * q^19 + q^20 - 8 * q^21 - 8 * q^22 + 16 * q^23 - 4 * q^24 + 3 * q^25 - 14 * q^26 + 8 * q^27 + 12 * q^28 - 3 * q^29 - 4 * q^30 - 14 * q^31 - 3 * q^32 - 14 * q^34 + 4 * q^35 - 11 * q^36 - 8 * q^37 - 8 * q^38 + 8 * q^39 + 3 * q^40 - 2 * q^41 - 16 * q^42 + 2 * q^43 + 2 * q^44 + 3 * q^45 + 14 * q^46 + 14 * q^47 + 10 * q^48 - 5 * q^49 + q^50 - 16 * q^51 - 10 * q^52 + 6 * q^53 + 20 * q^54 + 2 * q^55 + 10 * q^56 + 4 * q^57 - q^58 + 8 * q^59 - 10 * q^60 - 6 * q^61 - 8 * q^63 - 11 * q^64 - 2 * q^65 - 4 * q^66 + 28 * q^67 + 12 * q^68 + 12 * q^69 + 6 * q^70 + 28 * q^71 - 13 * q^72 - 16 * q^73 + 10 * q^74 + 2 * q^75 - 14 * q^76 + 4 * q^77 + 12 * q^78 - 6 * q^79 - 3 * q^80 - q^81 + 30 * q^82 + 12 * q^83 - 12 * q^84 - 4 * q^85 + 24 * q^86 - 2 * q^87 + 12 * q^88 - 10 * q^89 + 5 * q^90 - 12 * q^91 + 4 * q^92 - 20 * q^93 - 10 * q^95 + 24 * q^96 + 8 * q^97 + 21 * q^98 - 18 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.311108 0.219986 0.109993 0.993932i $$-0.464917\pi$$
0.109993 + 0.993932i $$0.464917\pi$$
$$3$$ 2.90321 1.67617 0.838085 0.545540i $$-0.183675\pi$$
0.838085 + 0.545540i $$0.183675\pi$$
$$4$$ −1.90321 −0.951606
$$5$$ 1.00000 0.447214
$$6$$ 0.903212 0.368735
$$7$$ −0.903212 −0.341382 −0.170691 0.985325i $$-0.554600\pi$$
−0.170691 + 0.985325i $$0.554600\pi$$
$$8$$ −1.21432 −0.429327
$$9$$ 5.42864 1.80955
$$10$$ 0.311108 0.0983809
$$11$$ −1.52543 −0.459934 −0.229967 0.973198i $$-0.573862\pi$$
−0.229967 + 0.973198i $$0.573862\pi$$
$$12$$ −5.52543 −1.59505
$$13$$ −0.622216 −0.172572 −0.0862858 0.996270i $$-0.527500\pi$$
−0.0862858 + 0.996270i $$0.527500\pi$$
$$14$$ −0.280996 −0.0750994
$$15$$ 2.90321 0.749606
$$16$$ 3.42864 0.857160
$$17$$ −7.95407 −1.92914 −0.964572 0.263819i $$-0.915018\pi$$
−0.964572 + 0.263819i $$0.915018\pi$$
$$18$$ 1.68889 0.398076
$$19$$ −1.09679 −0.251620 −0.125810 0.992054i $$-0.540153\pi$$
−0.125810 + 0.992054i $$0.540153\pi$$
$$20$$ −1.90321 −0.425571
$$21$$ −2.62222 −0.572214
$$22$$ −0.474572 −0.101179
$$23$$ 7.52543 1.56916 0.784580 0.620028i $$-0.212879\pi$$
0.784580 + 0.620028i $$0.212879\pi$$
$$24$$ −3.52543 −0.719625
$$25$$ 1.00000 0.200000
$$26$$ −0.193576 −0.0379634
$$27$$ 7.05086 1.35694
$$28$$ 1.71900 0.324861
$$29$$ −1.00000 −0.185695
$$30$$ 0.903212 0.164903
$$31$$ −6.90321 −1.23985 −0.619927 0.784660i $$-0.712838\pi$$
−0.619927 + 0.784660i $$0.712838\pi$$
$$32$$ 3.49532 0.617890
$$33$$ −4.42864 −0.770927
$$34$$ −2.47457 −0.424386
$$35$$ −0.903212 −0.152671
$$36$$ −10.3319 −1.72198
$$37$$ 3.95407 0.650045 0.325022 0.945706i $$-0.394628\pi$$
0.325022 + 0.945706i $$0.394628\pi$$
$$38$$ −0.341219 −0.0553531
$$39$$ −1.80642 −0.289259
$$40$$ −1.21432 −0.192001
$$41$$ 3.67307 0.573637 0.286819 0.957985i $$-0.407402\pi$$
0.286819 + 0.957985i $$0.407402\pi$$
$$42$$ −0.815792 −0.125879
$$43$$ −10.5161 −1.60368 −0.801842 0.597536i $$-0.796146\pi$$
−0.801842 + 0.597536i $$0.796146\pi$$
$$44$$ 2.90321 0.437676
$$45$$ 5.42864 0.809254
$$46$$ 2.34122 0.345194
$$47$$ 6.90321 1.00694 0.503468 0.864014i $$-0.332057\pi$$
0.503468 + 0.864014i $$0.332057\pi$$
$$48$$ 9.95407 1.43675
$$49$$ −6.18421 −0.883458
$$50$$ 0.311108 0.0439973
$$51$$ −23.0923 −3.23357
$$52$$ 1.18421 0.164220
$$53$$ 6.42864 0.883042 0.441521 0.897251i $$-0.354439\pi$$
0.441521 + 0.897251i $$0.354439\pi$$
$$54$$ 2.19358 0.298508
$$55$$ −1.52543 −0.205689
$$56$$ 1.09679 0.146564
$$57$$ −3.18421 −0.421759
$$58$$ −0.311108 −0.0408505
$$59$$ −1.67307 −0.217815 −0.108908 0.994052i $$-0.534735\pi$$
−0.108908 + 0.994052i $$0.534735\pi$$
$$60$$ −5.52543 −0.713330
$$61$$ −1.86665 −0.239000 −0.119500 0.992834i $$-0.538129\pi$$
−0.119500 + 0.992834i $$0.538129\pi$$
$$62$$ −2.14764 −0.272751
$$63$$ −4.90321 −0.617747
$$64$$ −5.76986 −0.721232
$$65$$ −0.622216 −0.0771764
$$66$$ −1.37778 −0.169594
$$67$$ 11.5254 1.40806 0.704028 0.710173i $$-0.251383\pi$$
0.704028 + 0.710173i $$0.251383\pi$$
$$68$$ 15.1383 1.83579
$$69$$ 21.8479 2.63018
$$70$$ −0.280996 −0.0335855
$$71$$ 13.6731 1.62269 0.811347 0.584564i $$-0.198734\pi$$
0.811347 + 0.584564i $$0.198734\pi$$
$$72$$ −6.59210 −0.776887
$$73$$ 10.1891 1.19255 0.596274 0.802781i $$-0.296647\pi$$
0.596274 + 0.802781i $$0.296647\pi$$
$$74$$ 1.23014 0.143001
$$75$$ 2.90321 0.335234
$$76$$ 2.08742 0.239444
$$77$$ 1.37778 0.157013
$$78$$ −0.561993 −0.0636331
$$79$$ 9.13828 1.02814 0.514068 0.857749i $$-0.328138\pi$$
0.514068 + 0.857749i $$0.328138\pi$$
$$80$$ 3.42864 0.383334
$$81$$ 4.18421 0.464912
$$82$$ 1.14272 0.126192
$$83$$ 10.7096 1.17554 0.587768 0.809030i $$-0.300007\pi$$
0.587768 + 0.809030i $$0.300007\pi$$
$$84$$ 4.99063 0.544523
$$85$$ −7.95407 −0.862740
$$86$$ −3.27163 −0.352789
$$87$$ −2.90321 −0.311257
$$88$$ 1.85236 0.197462
$$89$$ −7.80642 −0.827479 −0.413740 0.910395i $$-0.635778\pi$$
−0.413740 + 0.910395i $$0.635778\pi$$
$$90$$ 1.68889 0.178025
$$91$$ 0.561993 0.0589128
$$92$$ −14.3225 −1.49322
$$93$$ −20.0415 −2.07821
$$94$$ 2.14764 0.221512
$$95$$ −1.09679 −0.112528
$$96$$ 10.1476 1.03569
$$97$$ −4.08742 −0.415015 −0.207507 0.978233i $$-0.566535\pi$$
−0.207507 + 0.978233i $$0.566535\pi$$
$$98$$ −1.92396 −0.194349
$$99$$ −8.28100 −0.832271
$$100$$ −1.90321 −0.190321
$$101$$ 13.9081 1.38391 0.691956 0.721940i $$-0.256749\pi$$
0.691956 + 0.721940i $$0.256749\pi$$
$$102$$ −7.18421 −0.711343
$$103$$ −12.9447 −1.27548 −0.637740 0.770252i $$-0.720130\pi$$
−0.637740 + 0.770252i $$0.720130\pi$$
$$104$$ 0.755569 0.0740896
$$105$$ −2.62222 −0.255902
$$106$$ 2.00000 0.194257
$$107$$ 11.0049 1.06389 0.531943 0.846780i $$-0.321462\pi$$
0.531943 + 0.846780i $$0.321462\pi$$
$$108$$ −13.4193 −1.29127
$$109$$ −18.0415 −1.72806 −0.864031 0.503439i $$-0.832068\pi$$
−0.864031 + 0.503439i $$0.832068\pi$$
$$110$$ −0.474572 −0.0452487
$$111$$ 11.4795 1.08959
$$112$$ −3.09679 −0.292619
$$113$$ −10.2810 −0.967155 −0.483577 0.875302i $$-0.660663\pi$$
−0.483577 + 0.875302i $$0.660663\pi$$
$$114$$ −0.990632 −0.0927812
$$115$$ 7.52543 0.701750
$$116$$ 1.90321 0.176709
$$117$$ −3.37778 −0.312276
$$118$$ −0.520505 −0.0479164
$$119$$ 7.18421 0.658575
$$120$$ −3.52543 −0.321826
$$121$$ −8.67307 −0.788461
$$122$$ −0.580728 −0.0525767
$$123$$ 10.6637 0.961514
$$124$$ 13.1383 1.17985
$$125$$ 1.00000 0.0894427
$$126$$ −1.52543 −0.135896
$$127$$ 6.22077 0.552004 0.276002 0.961157i $$-0.410990\pi$$
0.276002 + 0.961157i $$0.410990\pi$$
$$128$$ −8.78568 −0.776552
$$129$$ −30.5303 −2.68805
$$130$$ −0.193576 −0.0169778
$$131$$ −11.7605 −1.02752 −0.513759 0.857934i $$-0.671748\pi$$
−0.513759 + 0.857934i $$0.671748\pi$$
$$132$$ 8.42864 0.733619
$$133$$ 0.990632 0.0858987
$$134$$ 3.58565 0.309753
$$135$$ 7.05086 0.606841
$$136$$ 9.65878 0.828234
$$137$$ −3.56691 −0.304742 −0.152371 0.988323i $$-0.548691\pi$$
−0.152371 + 0.988323i $$0.548691\pi$$
$$138$$ 6.79706 0.578604
$$139$$ −8.56199 −0.726219 −0.363109 0.931747i $$-0.618285\pi$$
−0.363109 + 0.931747i $$0.618285\pi$$
$$140$$ 1.71900 0.145282
$$141$$ 20.0415 1.68780
$$142$$ 4.25380 0.356971
$$143$$ 0.949145 0.0793715
$$144$$ 18.6128 1.55107
$$145$$ −1.00000 −0.0830455
$$146$$ 3.16992 0.262344
$$147$$ −17.9541 −1.48083
$$148$$ −7.52543 −0.618586
$$149$$ −5.61285 −0.459822 −0.229911 0.973212i $$-0.573844\pi$$
−0.229911 + 0.973212i $$0.573844\pi$$
$$150$$ 0.903212 0.0737469
$$151$$ 10.7971 0.878652 0.439326 0.898328i $$-0.355217\pi$$
0.439326 + 0.898328i $$0.355217\pi$$
$$152$$ 1.33185 0.108027
$$153$$ −43.1798 −3.49088
$$154$$ 0.428639 0.0345408
$$155$$ −6.90321 −0.554479
$$156$$ 3.43801 0.275261
$$157$$ 2.28100 0.182043 0.0910217 0.995849i $$-0.470987\pi$$
0.0910217 + 0.995849i $$0.470987\pi$$
$$158$$ 2.84299 0.226176
$$159$$ 18.6637 1.48013
$$160$$ 3.49532 0.276329
$$161$$ −6.79706 −0.535683
$$162$$ 1.30174 0.102274
$$163$$ 16.3225 1.27848 0.639238 0.769009i $$-0.279250\pi$$
0.639238 + 0.769009i $$0.279250\pi$$
$$164$$ −6.99063 −0.545877
$$165$$ −4.42864 −0.344769
$$166$$ 3.33185 0.258602
$$167$$ −4.76986 −0.369103 −0.184551 0.982823i $$-0.559083\pi$$
−0.184551 + 0.982823i $$0.559083\pi$$
$$168$$ 3.18421 0.245667
$$169$$ −12.6128 −0.970219
$$170$$ −2.47457 −0.189791
$$171$$ −5.95407 −0.455319
$$172$$ 20.0143 1.52608
$$173$$ 4.23506 0.321986 0.160993 0.986956i $$-0.448530\pi$$
0.160993 + 0.986956i $$0.448530\pi$$
$$174$$ −0.903212 −0.0684723
$$175$$ −0.903212 −0.0682764
$$176$$ −5.23014 −0.394237
$$177$$ −4.85728 −0.365095
$$178$$ −2.42864 −0.182034
$$179$$ 9.71456 0.726100 0.363050 0.931770i $$-0.381735\pi$$
0.363050 + 0.931770i $$0.381735\pi$$
$$180$$ −10.3319 −0.770091
$$181$$ 0.326929 0.0243005 0.0121502 0.999926i $$-0.496132\pi$$
0.0121502 + 0.999926i $$0.496132\pi$$
$$182$$ 0.174840 0.0129600
$$183$$ −5.41927 −0.400604
$$184$$ −9.13828 −0.673683
$$185$$ 3.95407 0.290709
$$186$$ −6.23506 −0.457177
$$187$$ 12.1334 0.887279
$$188$$ −13.1383 −0.958207
$$189$$ −6.36842 −0.463234
$$190$$ −0.341219 −0.0247547
$$191$$ −14.9447 −1.08136 −0.540680 0.841228i $$-0.681833\pi$$
−0.540680 + 0.841228i $$0.681833\pi$$
$$192$$ −16.7511 −1.20891
$$193$$ −14.1476 −1.01837 −0.509185 0.860657i $$-0.670053\pi$$
−0.509185 + 0.860657i $$0.670053\pi$$
$$194$$ −1.27163 −0.0912976
$$195$$ −1.80642 −0.129361
$$196$$ 11.7699 0.840704
$$197$$ −5.70471 −0.406444 −0.203222 0.979133i $$-0.565141\pi$$
−0.203222 + 0.979133i $$0.565141\pi$$
$$198$$ −2.57628 −0.183088
$$199$$ −22.1432 −1.56969 −0.784845 0.619692i $$-0.787257\pi$$
−0.784845 + 0.619692i $$0.787257\pi$$
$$200$$ −1.21432 −0.0858654
$$201$$ 33.4608 2.36014
$$202$$ 4.32693 0.304442
$$203$$ 0.903212 0.0633930
$$204$$ 43.9496 3.07709
$$205$$ 3.67307 0.256538
$$206$$ −4.02720 −0.280588
$$207$$ 40.8528 2.83947
$$208$$ −2.13335 −0.147921
$$209$$ 1.67307 0.115729
$$210$$ −0.815792 −0.0562950
$$211$$ −20.8430 −1.43489 −0.717445 0.696615i $$-0.754689\pi$$
−0.717445 + 0.696615i $$0.754689\pi$$
$$212$$ −12.2351 −0.840308
$$213$$ 39.6958 2.71991
$$214$$ 3.42372 0.234040
$$215$$ −10.5161 −0.717189
$$216$$ −8.56199 −0.582570
$$217$$ 6.23506 0.423264
$$218$$ −5.61285 −0.380150
$$219$$ 29.5812 1.99891
$$220$$ 2.90321 0.195735
$$221$$ 4.94914 0.332916
$$222$$ 3.57136 0.239694
$$223$$ 9.03657 0.605133 0.302567 0.953128i $$-0.402157\pi$$
0.302567 + 0.953128i $$0.402157\pi$$
$$224$$ −3.15701 −0.210937
$$225$$ 5.42864 0.361909
$$226$$ −3.19850 −0.212761
$$227$$ 19.4050 1.28795 0.643977 0.765045i $$-0.277283\pi$$
0.643977 + 0.765045i $$0.277283\pi$$
$$228$$ 6.06022 0.401348
$$229$$ −25.6128 −1.69254 −0.846272 0.532751i $$-0.821158\pi$$
−0.846272 + 0.532751i $$0.821158\pi$$
$$230$$ 2.34122 0.154375
$$231$$ 4.00000 0.263181
$$232$$ 1.21432 0.0797240
$$233$$ −3.12399 −0.204659 −0.102330 0.994751i $$-0.532630\pi$$
−0.102330 + 0.994751i $$0.532630\pi$$
$$234$$ −1.05086 −0.0686965
$$235$$ 6.90321 0.450316
$$236$$ 3.18421 0.207274
$$237$$ 26.5303 1.72333
$$238$$ 2.23506 0.144878
$$239$$ 13.9398 0.901689 0.450845 0.892602i $$-0.351123\pi$$
0.450845 + 0.892602i $$0.351123\pi$$
$$240$$ 9.95407 0.642532
$$241$$ −18.4701 −1.18977 −0.594883 0.803813i $$-0.702802\pi$$
−0.594883 + 0.803813i $$0.702802\pi$$
$$242$$ −2.69826 −0.173451
$$243$$ −9.00492 −0.577666
$$244$$ 3.55262 0.227433
$$245$$ −6.18421 −0.395095
$$246$$ 3.31756 0.211520
$$247$$ 0.682439 0.0434225
$$248$$ 8.38271 0.532302
$$249$$ 31.0923 1.97040
$$250$$ 0.311108 0.0196762
$$251$$ −13.7921 −0.870552 −0.435276 0.900297i $$-0.643349\pi$$
−0.435276 + 0.900297i $$0.643349\pi$$
$$252$$ 9.33185 0.587851
$$253$$ −11.4795 −0.721710
$$254$$ 1.93533 0.121433
$$255$$ −23.0923 −1.44610
$$256$$ 8.80642 0.550401
$$257$$ −1.47949 −0.0922883 −0.0461442 0.998935i $$-0.514693\pi$$
−0.0461442 + 0.998935i $$0.514693\pi$$
$$258$$ −9.49823 −0.591334
$$259$$ −3.57136 −0.221914
$$260$$ 1.18421 0.0734415
$$261$$ −5.42864 −0.336024
$$262$$ −3.65878 −0.226040
$$263$$ −0.442930 −0.0273122 −0.0136561 0.999907i $$-0.504347\pi$$
−0.0136561 + 0.999907i $$0.504347\pi$$
$$264$$ 5.37778 0.330980
$$265$$ 6.42864 0.394908
$$266$$ 0.308193 0.0188965
$$267$$ −22.6637 −1.38700
$$268$$ −21.9353 −1.33991
$$269$$ 3.93978 0.240212 0.120106 0.992761i $$-0.461676\pi$$
0.120106 + 0.992761i $$0.461676\pi$$
$$270$$ 2.19358 0.133497
$$271$$ 6.20787 0.377101 0.188551 0.982063i $$-0.439621\pi$$
0.188551 + 0.982063i $$0.439621\pi$$
$$272$$ −27.2716 −1.65359
$$273$$ 1.63158 0.0987479
$$274$$ −1.10970 −0.0670391
$$275$$ −1.52543 −0.0919867
$$276$$ −41.5812 −2.50289
$$277$$ 5.57136 0.334751 0.167375 0.985893i $$-0.446471\pi$$
0.167375 + 0.985893i $$0.446471\pi$$
$$278$$ −2.66370 −0.159758
$$279$$ −37.4750 −2.24357
$$280$$ 1.09679 0.0655456
$$281$$ 6.69535 0.399411 0.199705 0.979856i $$-0.436001\pi$$
0.199705 + 0.979856i $$0.436001\pi$$
$$282$$ 6.23506 0.371293
$$283$$ 25.8020 1.53377 0.766884 0.641785i $$-0.221806\pi$$
0.766884 + 0.641785i $$0.221806\pi$$
$$284$$ −26.0228 −1.54417
$$285$$ −3.18421 −0.188616
$$286$$ 0.295286 0.0174607
$$287$$ −3.31756 −0.195829
$$288$$ 18.9748 1.11810
$$289$$ 46.2672 2.72160
$$290$$ −0.311108 −0.0182689
$$291$$ −11.8666 −0.695635
$$292$$ −19.3921 −1.13484
$$293$$ −18.8430 −1.10082 −0.550410 0.834895i $$-0.685528\pi$$
−0.550410 + 0.834895i $$0.685528\pi$$
$$294$$ −5.58565 −0.325762
$$295$$ −1.67307 −0.0974099
$$296$$ −4.80150 −0.279082
$$297$$ −10.7556 −0.624101
$$298$$ −1.74620 −0.101155
$$299$$ −4.68244 −0.270792
$$300$$ −5.52543 −0.319011
$$301$$ 9.49823 0.547469
$$302$$ 3.35905 0.193292
$$303$$ 40.3783 2.31967
$$304$$ −3.76049 −0.215679
$$305$$ −1.86665 −0.106884
$$306$$ −13.4336 −0.767946
$$307$$ −1.65878 −0.0946716 −0.0473358 0.998879i $$-0.515073\pi$$
−0.0473358 + 0.998879i $$0.515073\pi$$
$$308$$ −2.62222 −0.149415
$$309$$ −37.5812 −2.13792
$$310$$ −2.14764 −0.121978
$$311$$ 21.3002 1.20782 0.603912 0.797051i $$-0.293608\pi$$
0.603912 + 0.797051i $$0.293608\pi$$
$$312$$ 2.19358 0.124187
$$313$$ 8.62222 0.487356 0.243678 0.969856i $$-0.421646\pi$$
0.243678 + 0.969856i $$0.421646\pi$$
$$314$$ 0.709636 0.0400471
$$315$$ −4.90321 −0.276265
$$316$$ −17.3921 −0.978381
$$317$$ −27.5955 −1.54992 −0.774959 0.632012i $$-0.782229\pi$$
−0.774959 + 0.632012i $$0.782229\pi$$
$$318$$ 5.80642 0.325608
$$319$$ 1.52543 0.0854075
$$320$$ −5.76986 −0.322545
$$321$$ 31.9496 1.78325
$$322$$ −2.11462 −0.117843
$$323$$ 8.72393 0.485412
$$324$$ −7.96343 −0.442413
$$325$$ −0.622216 −0.0345143
$$326$$ 5.07805 0.281247
$$327$$ −52.3783 −2.89652
$$328$$ −4.46028 −0.246278
$$329$$ −6.23506 −0.343750
$$330$$ −1.37778 −0.0758445
$$331$$ 16.9131 0.929626 0.464813 0.885409i $$-0.346122\pi$$
0.464813 + 0.885409i $$0.346122\pi$$
$$332$$ −20.3827 −1.11865
$$333$$ 21.4652 1.17629
$$334$$ −1.48394 −0.0811976
$$335$$ 11.5254 0.629701
$$336$$ −8.99063 −0.490479
$$337$$ 11.9956 0.653439 0.326720 0.945121i $$-0.394057\pi$$
0.326720 + 0.945121i $$0.394057\pi$$
$$338$$ −3.92396 −0.213435
$$339$$ −29.8479 −1.62112
$$340$$ 15.1383 0.820988
$$341$$ 10.5303 0.570250
$$342$$ −1.85236 −0.100164
$$343$$ 11.9081 0.642979
$$344$$ 12.7699 0.688505
$$345$$ 21.8479 1.17625
$$346$$ 1.31756 0.0708325
$$347$$ 6.14764 0.330023 0.165011 0.986292i $$-0.447234\pi$$
0.165011 + 0.986292i $$0.447234\pi$$
$$348$$ 5.52543 0.296194
$$349$$ −7.12399 −0.381338 −0.190669 0.981654i $$-0.561066\pi$$
−0.190669 + 0.981654i $$0.561066\pi$$
$$350$$ −0.280996 −0.0150199
$$351$$ −4.38715 −0.234169
$$352$$ −5.33185 −0.284189
$$353$$ −16.9175 −0.900428 −0.450214 0.892921i $$-0.648652\pi$$
−0.450214 + 0.892921i $$0.648652\pi$$
$$354$$ −1.51114 −0.0803160
$$355$$ 13.6731 0.725691
$$356$$ 14.8573 0.787434
$$357$$ 20.8573 1.10388
$$358$$ 3.02227 0.159732
$$359$$ 36.7096 1.93746 0.968730 0.248116i $$-0.0798115\pi$$
0.968730 + 0.248116i $$0.0798115\pi$$
$$360$$ −6.59210 −0.347434
$$361$$ −17.7971 −0.936687
$$362$$ 0.101710 0.00534577
$$363$$ −25.1798 −1.32159
$$364$$ −1.06959 −0.0560618
$$365$$ 10.1891 0.533323
$$366$$ −1.68598 −0.0881275
$$367$$ 8.41435 0.439225 0.219613 0.975587i $$-0.429521\pi$$
0.219613 + 0.975587i $$0.429521\pi$$
$$368$$ 25.8020 1.34502
$$369$$ 19.9398 1.03802
$$370$$ 1.23014 0.0639520
$$371$$ −5.80642 −0.301455
$$372$$ 38.1432 1.97763
$$373$$ −8.66370 −0.448590 −0.224295 0.974521i $$-0.572008\pi$$
−0.224295 + 0.974521i $$0.572008\pi$$
$$374$$ 3.77478 0.195189
$$375$$ 2.90321 0.149921
$$376$$ −8.38271 −0.432305
$$377$$ 0.622216 0.0320457
$$378$$ −1.98126 −0.101905
$$379$$ −2.76986 −0.142278 −0.0711390 0.997466i $$-0.522663\pi$$
−0.0711390 + 0.997466i $$0.522663\pi$$
$$380$$ 2.08742 0.107082
$$381$$ 18.0602 0.925253
$$382$$ −4.64941 −0.237885
$$383$$ 1.67752 0.0857171 0.0428585 0.999081i $$-0.486354\pi$$
0.0428585 + 0.999081i $$0.486354\pi$$
$$384$$ −25.5067 −1.30163
$$385$$ 1.37778 0.0702184
$$386$$ −4.40144 −0.224028
$$387$$ −57.0879 −2.90194
$$388$$ 7.77923 0.394930
$$389$$ −5.77478 −0.292793 −0.146397 0.989226i $$-0.546768\pi$$
−0.146397 + 0.989226i $$0.546768\pi$$
$$390$$ −0.561993 −0.0284576
$$391$$ −59.8578 −3.02714
$$392$$ 7.50961 0.379292
$$393$$ −34.1432 −1.72230
$$394$$ −1.77478 −0.0894122
$$395$$ 9.13828 0.459797
$$396$$ 15.7605 0.791994
$$397$$ 29.9081 1.50105 0.750523 0.660844i $$-0.229802\pi$$
0.750523 + 0.660844i $$0.229802\pi$$
$$398$$ −6.88892 −0.345310
$$399$$ 2.87601 0.143981
$$400$$ 3.42864 0.171432
$$401$$ 8.53035 0.425985 0.212993 0.977054i $$-0.431679\pi$$
0.212993 + 0.977054i $$0.431679\pi$$
$$402$$ 10.4099 0.519199
$$403$$ 4.29529 0.213963
$$404$$ −26.4701 −1.31694
$$405$$ 4.18421 0.207915
$$406$$ 0.280996 0.0139456
$$407$$ −6.03164 −0.298977
$$408$$ 28.0415 1.38826
$$409$$ 5.09234 0.251800 0.125900 0.992043i $$-0.459818\pi$$
0.125900 + 0.992043i $$0.459818\pi$$
$$410$$ 1.14272 0.0564350
$$411$$ −10.3555 −0.510800
$$412$$ 24.6365 1.21375
$$413$$ 1.51114 0.0743582
$$414$$ 12.7096 0.624645
$$415$$ 10.7096 0.525715
$$416$$ −2.17484 −0.106630
$$417$$ −24.8573 −1.21727
$$418$$ 0.520505 0.0254588
$$419$$ −24.3368 −1.18893 −0.594465 0.804122i $$-0.702636\pi$$
−0.594465 + 0.804122i $$0.702636\pi$$
$$420$$ 4.99063 0.243518
$$421$$ 24.5018 1.19414 0.597072 0.802188i $$-0.296331\pi$$
0.597072 + 0.802188i $$0.296331\pi$$
$$422$$ −6.48442 −0.315656
$$423$$ 37.4750 1.82210
$$424$$ −7.80642 −0.379113
$$425$$ −7.95407 −0.385829
$$426$$ 12.3497 0.598344
$$427$$ 1.68598 0.0815902
$$428$$ −20.9447 −1.01240
$$429$$ 2.75557 0.133040
$$430$$ −3.27163 −0.157772
$$431$$ 4.26671 0.205520 0.102760 0.994706i $$-0.467233\pi$$
0.102760 + 0.994706i $$0.467233\pi$$
$$432$$ 24.1748 1.16311
$$433$$ 27.0049 1.29777 0.648887 0.760885i $$-0.275235\pi$$
0.648887 + 0.760885i $$0.275235\pi$$
$$434$$ 1.93978 0.0931123
$$435$$ −2.90321 −0.139198
$$436$$ 34.3368 1.64443
$$437$$ −8.25380 −0.394833
$$438$$ 9.20294 0.439734
$$439$$ −2.03164 −0.0969650 −0.0484825 0.998824i $$-0.515439\pi$$
−0.0484825 + 0.998824i $$0.515439\pi$$
$$440$$ 1.85236 0.0883076
$$441$$ −33.5718 −1.59866
$$442$$ 1.53972 0.0732369
$$443$$ −3.46520 −0.164637 −0.0823184 0.996606i $$-0.526232\pi$$
−0.0823184 + 0.996606i $$0.526232\pi$$
$$444$$ −21.8479 −1.03686
$$445$$ −7.80642 −0.370060
$$446$$ 2.81135 0.133121
$$447$$ −16.2953 −0.770741
$$448$$ 5.21141 0.246216
$$449$$ 37.3590 1.76308 0.881541 0.472107i $$-0.156506\pi$$
0.881541 + 0.472107i $$0.156506\pi$$
$$450$$ 1.68889 0.0796151
$$451$$ −5.60300 −0.263835
$$452$$ 19.5669 0.920350
$$453$$ 31.3461 1.47277
$$454$$ 6.03704 0.283332
$$455$$ 0.561993 0.0263466
$$456$$ 3.86665 0.181072
$$457$$ 13.4509 0.629207 0.314604 0.949223i $$-0.398128\pi$$
0.314604 + 0.949223i $$0.398128\pi$$
$$458$$ −7.96836 −0.372337
$$459$$ −56.0830 −2.61773
$$460$$ −14.3225 −0.667789
$$461$$ −16.2766 −0.758075 −0.379037 0.925381i $$-0.623745\pi$$
−0.379037 + 0.925381i $$0.623745\pi$$
$$462$$ 1.24443 0.0578962
$$463$$ −30.3926 −1.41246 −0.706231 0.707982i $$-0.749606\pi$$
−0.706231 + 0.707982i $$0.749606\pi$$
$$464$$ −3.42864 −0.159171
$$465$$ −20.0415 −0.929402
$$466$$ −0.971896 −0.0450222
$$467$$ −1.18865 −0.0550043 −0.0275022 0.999622i $$-0.508755\pi$$
−0.0275022 + 0.999622i $$0.508755\pi$$
$$468$$ 6.42864 0.297164
$$469$$ −10.4099 −0.480685
$$470$$ 2.14764 0.0990634
$$471$$ 6.62222 0.305136
$$472$$ 2.03164 0.0935139
$$473$$ 16.0415 0.737588
$$474$$ 8.25380 0.379110
$$475$$ −1.09679 −0.0503241
$$476$$ −13.6731 −0.626704
$$477$$ 34.8988 1.59790
$$478$$ 4.33677 0.198359
$$479$$ 41.0464 1.87546 0.937729 0.347367i $$-0.112924\pi$$
0.937729 + 0.347367i $$0.112924\pi$$
$$480$$ 10.1476 0.463174
$$481$$ −2.46028 −0.112179
$$482$$ −5.74620 −0.261732
$$483$$ −19.7333 −0.897896
$$484$$ 16.5067 0.750304
$$485$$ −4.08742 −0.185600
$$486$$ −2.80150 −0.127079
$$487$$ −10.1476 −0.459834 −0.229917 0.973210i $$-0.573845\pi$$
−0.229917 + 0.973210i $$0.573845\pi$$
$$488$$ 2.26671 0.102609
$$489$$ 47.3876 2.14294
$$490$$ −1.92396 −0.0869155
$$491$$ −29.2083 −1.31815 −0.659077 0.752075i $$-0.729053\pi$$
−0.659077 + 0.752075i $$0.729053\pi$$
$$492$$ −20.2953 −0.914982
$$493$$ 7.95407 0.358233
$$494$$ 0.212312 0.00955237
$$495$$ −8.28100 −0.372203
$$496$$ −23.6686 −1.06275
$$497$$ −12.3497 −0.553959
$$498$$ 9.67307 0.433461
$$499$$ 21.9813 0.984017 0.492008 0.870591i $$-0.336263\pi$$
0.492008 + 0.870591i $$0.336263\pi$$
$$500$$ −1.90321 −0.0851142
$$501$$ −13.8479 −0.618679
$$502$$ −4.29084 −0.191510
$$503$$ 5.77923 0.257683 0.128841 0.991665i $$-0.458874\pi$$
0.128841 + 0.991665i $$0.458874\pi$$
$$504$$ 5.95407 0.265215
$$505$$ 13.9081 0.618904
$$506$$ −3.57136 −0.158766
$$507$$ −36.6178 −1.62625
$$508$$ −11.8394 −0.525291
$$509$$ 13.6543 0.605218 0.302609 0.953115i $$-0.402142\pi$$
0.302609 + 0.953115i $$0.402142\pi$$
$$510$$ −7.18421 −0.318122
$$511$$ −9.20294 −0.407114
$$512$$ 20.3111 0.897633
$$513$$ −7.73329 −0.341433
$$514$$ −0.460282 −0.0203022
$$515$$ −12.9447 −0.570412
$$516$$ 58.1057 2.55796
$$517$$ −10.5303 −0.463124
$$518$$ −1.11108 −0.0488180
$$519$$ 12.2953 0.539703
$$520$$ 0.755569 0.0331339
$$521$$ −19.6731 −0.861893 −0.430946 0.902378i $$-0.641820\pi$$
−0.430946 + 0.902378i $$0.641820\pi$$
$$522$$ −1.68889 −0.0739208
$$523$$ −15.1383 −0.661951 −0.330975 0.943639i $$-0.607378\pi$$
−0.330975 + 0.943639i $$0.607378\pi$$
$$524$$ 22.3827 0.977793
$$525$$ −2.62222 −0.114443
$$526$$ −0.137799 −0.00600832
$$527$$ 54.9086 2.39186
$$528$$ −15.1842 −0.660808
$$529$$ 33.6321 1.46226
$$530$$ 2.00000 0.0868744
$$531$$ −9.08250 −0.394147
$$532$$ −1.88538 −0.0817417
$$533$$ −2.28544 −0.0989935
$$534$$ −7.05086 −0.305120
$$535$$ 11.0049 0.475784
$$536$$ −13.9956 −0.604516
$$537$$ 28.2034 1.21707
$$538$$ 1.22570 0.0528435
$$539$$ 9.43356 0.406332
$$540$$ −13.4193 −0.577474
$$541$$ 2.68244 0.115327 0.0576635 0.998336i $$-0.481635\pi$$
0.0576635 + 0.998336i $$0.481635\pi$$
$$542$$ 1.93132 0.0829571
$$543$$ 0.949145 0.0407317
$$544$$ −27.8020 −1.19200
$$545$$ −18.0415 −0.772812
$$546$$ 0.507598 0.0217232
$$547$$ −15.3635 −0.656896 −0.328448 0.944522i $$-0.606525\pi$$
−0.328448 + 0.944522i $$0.606525\pi$$
$$548$$ 6.78859 0.289994
$$549$$ −10.1334 −0.432481
$$550$$ −0.474572 −0.0202358
$$551$$ 1.09679 0.0467247
$$552$$ −26.5303 −1.12921
$$553$$ −8.25380 −0.350987
$$554$$ 1.73329 0.0736406
$$555$$ 11.4795 0.487277
$$556$$ 16.2953 0.691074
$$557$$ −9.87955 −0.418610 −0.209305 0.977850i $$-0.567120\pi$$
−0.209305 + 0.977850i $$0.567120\pi$$
$$558$$ −11.6588 −0.493556
$$559$$ 6.54326 0.276750
$$560$$ −3.09679 −0.130863
$$561$$ 35.2257 1.48723
$$562$$ 2.08297 0.0878650
$$563$$ −27.4938 −1.15872 −0.579362 0.815070i $$-0.696698\pi$$
−0.579362 + 0.815070i $$0.696698\pi$$
$$564$$ −38.1432 −1.60612
$$565$$ −10.2810 −0.432525
$$566$$ 8.02720 0.337408
$$567$$ −3.77923 −0.158713
$$568$$ −16.6035 −0.696667
$$569$$ 17.3590 0.727729 0.363865 0.931452i $$-0.381457\pi$$
0.363865 + 0.931452i $$0.381457\pi$$
$$570$$ −0.990632 −0.0414930
$$571$$ −25.4479 −1.06496 −0.532480 0.846443i $$-0.678740\pi$$
−0.532480 + 0.846443i $$0.678740\pi$$
$$572$$ −1.80642 −0.0755304
$$573$$ −43.3876 −1.81254
$$574$$ −1.03212 −0.0430798
$$575$$ 7.52543 0.313832
$$576$$ −31.3225 −1.30510
$$577$$ −10.6178 −0.442024 −0.221012 0.975271i $$-0.570936\pi$$
−0.221012 + 0.975271i $$0.570936\pi$$
$$578$$ 14.3941 0.598715
$$579$$ −41.0736 −1.70696
$$580$$ 1.90321 0.0790266
$$581$$ −9.67307 −0.401307
$$582$$ −3.69181 −0.153030
$$583$$ −9.80642 −0.406141
$$584$$ −12.3729 −0.511993
$$585$$ −3.37778 −0.139654
$$586$$ −5.86220 −0.242165
$$587$$ −8.94470 −0.369187 −0.184594 0.982815i $$-0.559097\pi$$
−0.184594 + 0.982815i $$0.559097\pi$$
$$588$$ 34.1704 1.40916
$$589$$ 7.57136 0.311972
$$590$$ −0.520505 −0.0214289
$$591$$ −16.5620 −0.681269
$$592$$ 13.5571 0.557192
$$593$$ 14.1619 0.581561 0.290780 0.956790i $$-0.406085\pi$$
0.290780 + 0.956790i $$0.406085\pi$$
$$594$$ −3.34614 −0.137294
$$595$$ 7.18421 0.294524
$$596$$ 10.6824 0.437570
$$597$$ −64.2864 −2.63107
$$598$$ −1.45674 −0.0595707
$$599$$ −22.5575 −0.921676 −0.460838 0.887484i $$-0.652451\pi$$
−0.460838 + 0.887484i $$0.652451\pi$$
$$600$$ −3.52543 −0.143925
$$601$$ −40.6133 −1.65665 −0.828326 0.560246i $$-0.810706\pi$$
−0.828326 + 0.560246i $$0.810706\pi$$
$$602$$ 2.95497 0.120436
$$603$$ 62.5674 2.54794
$$604$$ −20.5491 −0.836130
$$605$$ −8.67307 −0.352610
$$606$$ 12.5620 0.510296
$$607$$ 13.5955 0.551824 0.275912 0.961183i $$-0.411020\pi$$
0.275912 + 0.961183i $$0.411020\pi$$
$$608$$ −3.83362 −0.155474
$$609$$ 2.62222 0.106258
$$610$$ −0.580728 −0.0235130
$$611$$ −4.29529 −0.173769
$$612$$ 82.1802 3.32194
$$613$$ −42.0830 −1.69972 −0.849858 0.527012i $$-0.823312\pi$$
−0.849858 + 0.527012i $$0.823312\pi$$
$$614$$ −0.516060 −0.0208265
$$615$$ 10.6637 0.430002
$$616$$ −1.67307 −0.0674099
$$617$$ 33.5067 1.34893 0.674464 0.738307i $$-0.264375\pi$$
0.674464 + 0.738307i $$0.264375\pi$$
$$618$$ −11.6918 −0.470313
$$619$$ −14.6780 −0.589958 −0.294979 0.955504i $$-0.595313\pi$$
−0.294979 + 0.955504i $$0.595313\pi$$
$$620$$ 13.1383 0.527646
$$621$$ 53.0607 2.12925
$$622$$ 6.62666 0.265705
$$623$$ 7.05086 0.282487
$$624$$ −6.19358 −0.247941
$$625$$ 1.00000 0.0400000
$$626$$ 2.68244 0.107212
$$627$$ 4.85728 0.193981
$$628$$ −4.34122 −0.173234
$$629$$ −31.4509 −1.25403
$$630$$ −1.52543 −0.0607745
$$631$$ 11.3176 0.450545 0.225273 0.974296i $$-0.427673\pi$$
0.225273 + 0.974296i $$0.427673\pi$$
$$632$$ −11.0968 −0.441407
$$633$$ −60.5116 −2.40512
$$634$$ −8.58517 −0.340961
$$635$$ 6.22077 0.246864
$$636$$ −35.5210 −1.40850
$$637$$ 3.84791 0.152460
$$638$$ 0.474572 0.0187885
$$639$$ 74.2262 2.93634
$$640$$ −8.78568 −0.347285
$$641$$ 34.8988 1.37842 0.689209 0.724562i $$-0.257958\pi$$
0.689209 + 0.724562i $$0.257958\pi$$
$$642$$ 9.93978 0.392292
$$643$$ −41.9768 −1.65540 −0.827702 0.561168i $$-0.810352\pi$$
−0.827702 + 0.561168i $$0.810352\pi$$
$$644$$ 12.9362 0.509759
$$645$$ −30.5303 −1.20213
$$646$$ 2.71408 0.106784
$$647$$ 5.46520 0.214859 0.107430 0.994213i $$-0.465738\pi$$
0.107430 + 0.994213i $$0.465738\pi$$
$$648$$ −5.08097 −0.199599
$$649$$ 2.55215 0.100181
$$650$$ −0.193576 −0.00759268
$$651$$ 18.1017 0.709462
$$652$$ −31.0651 −1.21660
$$653$$ −8.76986 −0.343191 −0.171596 0.985167i $$-0.554892\pi$$
−0.171596 + 0.985167i $$0.554892\pi$$
$$654$$ −16.2953 −0.637196
$$655$$ −11.7605 −0.459520
$$656$$ 12.5936 0.491699
$$657$$ 55.3131 2.15797
$$658$$ −1.93978 −0.0756204
$$659$$ 3.29036 0.128174 0.0640872 0.997944i $$-0.479586\pi$$
0.0640872 + 0.997944i $$0.479586\pi$$
$$660$$ 8.42864 0.328084
$$661$$ 19.7560 0.768421 0.384211 0.923246i $$-0.374474\pi$$
0.384211 + 0.923246i $$0.374474\pi$$
$$662$$ 5.26178 0.204505
$$663$$ 14.3684 0.558023
$$664$$ −13.0049 −0.504689
$$665$$ 0.990632 0.0384151
$$666$$ 6.67799 0.258767
$$667$$ −7.52543 −0.291386
$$668$$ 9.07805 0.351240
$$669$$ 26.2351 1.01431
$$670$$ 3.58565 0.138526
$$671$$ 2.84743 0.109924
$$672$$ −9.16547 −0.353566
$$673$$ 44.3970 1.71138 0.855689 0.517490i $$-0.173134\pi$$
0.855689 + 0.517490i $$0.173134\pi$$
$$674$$ 3.73191 0.143748
$$675$$ 7.05086 0.271388
$$676$$ 24.0049 0.923266
$$677$$ 6.09726 0.234337 0.117168 0.993112i $$-0.462618\pi$$
0.117168 + 0.993112i $$0.462618\pi$$
$$678$$ −9.28592 −0.356624
$$679$$ 3.69181 0.141679
$$680$$ 9.65878 0.370397
$$681$$ 56.3368 2.15883
$$682$$ 3.27607 0.125447
$$683$$ 37.9224 1.45106 0.725531 0.688190i $$-0.241594\pi$$
0.725531 + 0.688190i $$0.241594\pi$$
$$684$$ 11.3319 0.433284
$$685$$ −3.56691 −0.136285
$$686$$ 3.70471 0.141447
$$687$$ −74.3595 −2.83699
$$688$$ −36.0558 −1.37461
$$689$$ −4.00000 −0.152388
$$690$$ 6.79706 0.258759
$$691$$ 13.3145 0.506507 0.253254 0.967400i $$-0.418499\pi$$
0.253254 + 0.967400i $$0.418499\pi$$
$$692$$ −8.06022 −0.306404
$$693$$ 7.47949 0.284123
$$694$$ 1.91258 0.0726005
$$695$$ −8.56199 −0.324775
$$696$$ 3.52543 0.133631
$$697$$ −29.2159 −1.10663
$$698$$ −2.21633 −0.0838892
$$699$$ −9.06959 −0.343043
$$700$$ 1.71900 0.0649722
$$701$$ −23.4893 −0.887180 −0.443590 0.896230i $$-0.646295\pi$$
−0.443590 + 0.896230i $$0.646295\pi$$
$$702$$ −1.36488 −0.0515140
$$703$$ −4.33677 −0.163565
$$704$$ 8.80150 0.331719
$$705$$ 20.0415 0.754806
$$706$$ −5.26317 −0.198082
$$707$$ −12.5620 −0.472442
$$708$$ 9.24443 0.347427
$$709$$ 11.6731 0.438391 0.219196 0.975681i $$-0.429657\pi$$
0.219196 + 0.975681i $$0.429657\pi$$
$$710$$ 4.25380 0.159642
$$711$$ 49.6084 1.86046
$$712$$ 9.47949 0.355259
$$713$$ −51.9496 −1.94553
$$714$$ 6.48886 0.242840
$$715$$ 0.949145 0.0354960
$$716$$ −18.4889 −0.690961
$$717$$ 40.4701 1.51138
$$718$$ 11.4207 0.426215
$$719$$ 29.5526 1.10213 0.551063 0.834463i $$-0.314222\pi$$
0.551063 + 0.834463i $$0.314222\pi$$
$$720$$ 18.6128 0.693660
$$721$$ 11.6918 0.435426
$$722$$ −5.53680 −0.206058
$$723$$ −53.6227 −1.99425
$$724$$ −0.622216 −0.0231245
$$725$$ −1.00000 −0.0371391
$$726$$ −7.83362 −0.290733
$$727$$ 3.88094 0.143936 0.0719680 0.997407i $$-0.477072\pi$$
0.0719680 + 0.997407i $$0.477072\pi$$
$$728$$ −0.682439 −0.0252929
$$729$$ −38.6958 −1.43318
$$730$$ 3.16992 0.117324
$$731$$ 83.6454 3.09374
$$732$$ 10.3140 0.381217
$$733$$ 14.8845 0.549771 0.274885 0.961477i $$-0.411360\pi$$
0.274885 + 0.961477i $$0.411360\pi$$
$$734$$ 2.61777 0.0966236
$$735$$ −17.9541 −0.662246
$$736$$ 26.3037 0.969569
$$737$$ −17.5812 −0.647612
$$738$$ 6.20342 0.228351
$$739$$ 2.24935 0.0827438 0.0413719 0.999144i $$-0.486827\pi$$
0.0413719 + 0.999144i $$0.486827\pi$$
$$740$$ −7.52543 −0.276640
$$741$$ 1.98126 0.0727836
$$742$$ −1.80642 −0.0663159
$$743$$ −3.46520 −0.127126 −0.0635630 0.997978i $$-0.520246\pi$$
−0.0635630 + 0.997978i $$0.520246\pi$$
$$744$$ 24.3368 0.892229
$$745$$ −5.61285 −0.205639
$$746$$ −2.69535 −0.0986836
$$747$$ 58.1388 2.12719
$$748$$ −23.0923 −0.844340
$$749$$ −9.93978 −0.363192
$$750$$ 0.903212 0.0329806
$$751$$ 3.16992 0.115672 0.0578360 0.998326i $$-0.481580\pi$$
0.0578360 + 0.998326i $$0.481580\pi$$
$$752$$ 23.6686 0.863106
$$753$$ −40.0415 −1.45919
$$754$$ 0.193576 0.00704963
$$755$$ 10.7971 0.392945
$$756$$ 12.1204 0.440816
$$757$$ −52.0785 −1.89283 −0.946413 0.322958i $$-0.895323\pi$$
−0.946413 + 0.322958i $$0.895323\pi$$
$$758$$ −0.861725 −0.0312993
$$759$$ −33.3274 −1.20971
$$760$$ 1.33185 0.0483113
$$761$$ −14.9777 −0.542942 −0.271471 0.962447i $$-0.587510\pi$$
−0.271471 + 0.962447i $$0.587510\pi$$
$$762$$ 5.61868 0.203543
$$763$$ 16.2953 0.589929
$$764$$ 28.4429 1.02903
$$765$$ −43.1798 −1.56117
$$766$$ 0.521889 0.0188566
$$767$$ 1.04101 0.0375887
$$768$$ 25.5669 0.922567
$$769$$ −1.90813 −0.0688091 −0.0344045 0.999408i $$-0.510953\pi$$
−0.0344045 + 0.999408i $$0.510953\pi$$
$$770$$ 0.428639 0.0154471
$$771$$ −4.29529 −0.154691
$$772$$ 26.9260 0.969087
$$773$$ 21.7891 0.783698 0.391849 0.920029i $$-0.371835\pi$$
0.391849 + 0.920029i $$0.371835\pi$$
$$774$$ −17.7605 −0.638388
$$775$$ −6.90321 −0.247971
$$776$$ 4.96343 0.178177
$$777$$ −10.3684 −0.371965
$$778$$ −1.79658 −0.0644105
$$779$$ −4.02858 −0.144339
$$780$$ 3.43801 0.123100
$$781$$ −20.8573 −0.746332
$$782$$ −18.6222 −0.665929
$$783$$ −7.05086 −0.251977
$$784$$ −21.2034 −0.757265
$$785$$ 2.28100 0.0814122
$$786$$ −10.6222 −0.378882
$$787$$ −18.1388 −0.646577 −0.323288 0.946301i $$-0.604788\pi$$
−0.323288 + 0.946301i $$0.604788\pi$$
$$788$$ 10.8573 0.386775
$$789$$ −1.28592 −0.0457799
$$790$$ 2.84299 0.101149
$$791$$ 9.28592 0.330169
$$792$$ 10.0558 0.357316
$$793$$ 1.16146 0.0412445
$$794$$ 9.30465 0.330210
$$795$$ 18.6637 0.661933
$$796$$ 42.1432 1.49373
$$797$$ 2.96343 0.104970 0.0524851 0.998622i $$-0.483286\pi$$
0.0524851 + 0.998622i $$0.483286\pi$$
$$798$$ 0.894751 0.0316738
$$799$$ −54.9086 −1.94253
$$800$$ 3.49532 0.123578
$$801$$ −42.3783 −1.49736
$$802$$ 2.65386 0.0937110
$$803$$ −15.5428 −0.548493
$$804$$ −63.6829 −2.24592
$$805$$ −6.79706 −0.239565
$$806$$ 1.33630 0.0470691
$$807$$ 11.4380 0.402637
$$808$$ −16.8889 −0.594150
$$809$$ −26.2953 −0.924493 −0.462247 0.886751i $$-0.652956\pi$$
−0.462247 + 0.886751i $$0.652956\pi$$
$$810$$ 1.30174 0.0457385
$$811$$ 24.3783 0.856037 0.428018 0.903770i $$-0.359212\pi$$
0.428018 + 0.903770i $$0.359212\pi$$
$$812$$ −1.71900 −0.0603252
$$813$$ 18.0228 0.632085
$$814$$ −1.87649 −0.0657710
$$815$$ 16.3225 0.571752
$$816$$ −79.1753 −2.77169
$$817$$ 11.5339 0.403520
$$818$$ 1.58427 0.0553926
$$819$$ 3.05086 0.106606
$$820$$ −6.99063 −0.244123
$$821$$ 1.52987 0.0533929 0.0266965 0.999644i $$-0.491501\pi$$
0.0266965 + 0.999644i $$0.491501\pi$$
$$822$$ −3.22168 −0.112369
$$823$$ 46.7195 1.62854 0.814269 0.580487i $$-0.197138\pi$$
0.814269 + 0.580487i $$0.197138\pi$$
$$824$$ 15.7190 0.547597
$$825$$ −4.42864 −0.154185
$$826$$ 0.470127 0.0163578
$$827$$ −29.6499 −1.03103 −0.515514 0.856881i $$-0.672399\pi$$
−0.515514 + 0.856881i $$0.672399\pi$$
$$828$$ −77.7516 −2.70205
$$829$$ −8.79706 −0.305534 −0.152767 0.988262i $$-0.548818\pi$$
−0.152767 + 0.988262i $$0.548818\pi$$
$$830$$ 3.33185 0.115650
$$831$$ 16.1748 0.561099
$$832$$ 3.59010 0.124464
$$833$$ 49.1896 1.70432
$$834$$ −7.73329 −0.267782
$$835$$ −4.76986 −0.165068
$$836$$ −3.18421 −0.110128
$$837$$ −48.6735 −1.68240
$$838$$ −7.57136 −0.261548
$$839$$ 11.3319 0.391219 0.195609 0.980682i $$-0.437332\pi$$
0.195609 + 0.980682i $$0.437332\pi$$
$$840$$ 3.18421 0.109866
$$841$$ 1.00000 0.0344828
$$842$$ 7.62269 0.262695
$$843$$ 19.4380 0.669481
$$844$$ 39.6686 1.36545
$$845$$ −12.6128 −0.433895
$$846$$ 11.6588 0.400837
$$847$$ 7.83362 0.269166
$$848$$ 22.0415 0.756908
$$849$$ 74.9086 2.57086
$$850$$ −2.47457 −0.0848771
$$851$$ 29.7560 1.02002
$$852$$ −75.5496 −2.58829
$$853$$ 54.8845 1.87921 0.939604 0.342263i $$-0.111193\pi$$
0.939604 + 0.342263i $$0.111193\pi$$
$$854$$ 0.524521 0.0179487
$$855$$ −5.95407 −0.203625
$$856$$ −13.3635 −0.456755
$$857$$ 36.4385 1.24471 0.622357 0.782733i $$-0.286175\pi$$
0.622357 + 0.782733i $$0.286175\pi$$
$$858$$ 0.857279 0.0292670
$$859$$ 1.72885 0.0589875 0.0294938 0.999565i $$-0.490610\pi$$
0.0294938 + 0.999565i $$0.490610\pi$$
$$860$$ 20.0143 0.682482
$$861$$ −9.63158 −0.328243
$$862$$ 1.32741 0.0452116
$$863$$ −9.40192 −0.320045 −0.160023 0.987113i $$-0.551157\pi$$
−0.160023 + 0.987113i $$0.551157\pi$$
$$864$$ 24.6450 0.838439
$$865$$ 4.23506 0.143996
$$866$$ 8.40144 0.285493
$$867$$ 134.323 4.56186
$$868$$ −11.8666 −0.402780
$$869$$ −13.9398 −0.472875
$$870$$ −0.903212 −0.0306218
$$871$$ −7.17130 −0.242990
$$872$$ 21.9081 0.741903
$$873$$ −22.1891 −0.750988
$$874$$ −2.56782 −0.0868579
$$875$$ −0.903212 −0.0305341
$$876$$ −56.2993 −1.90218
$$877$$ 8.91750 0.301123 0.150561 0.988601i $$-0.451892\pi$$
0.150561 + 0.988601i $$0.451892\pi$$
$$878$$ −0.632060 −0.0213310
$$879$$ −54.7052 −1.84516
$$880$$ −5.23014 −0.176308
$$881$$ −42.1245 −1.41921 −0.709605 0.704600i $$-0.751126\pi$$
−0.709605 + 0.704600i $$0.751126\pi$$
$$882$$ −10.4445 −0.351683
$$883$$ −38.4340 −1.29341 −0.646704 0.762741i $$-0.723853\pi$$
−0.646704 + 0.762741i $$0.723853\pi$$
$$884$$ −9.41927 −0.316804
$$885$$ −4.85728 −0.163276
$$886$$ −1.07805 −0.0362179
$$887$$ −38.6365 −1.29729 −0.648643 0.761092i $$-0.724663\pi$$
−0.648643 + 0.761092i $$0.724663\pi$$
$$888$$ −13.9398 −0.467788
$$889$$ −5.61868 −0.188444
$$890$$ −2.42864 −0.0814082
$$891$$ −6.38271 −0.213829
$$892$$ −17.1985 −0.575848
$$893$$ −7.57136 −0.253366
$$894$$ −5.06959 −0.169552
$$895$$ 9.71456 0.324722
$$896$$ 7.93533 0.265101
$$897$$ −13.5941 −0.453894
$$898$$ 11.6227 0.387854
$$899$$ 6.90321 0.230235
$$900$$ −10.3319 −0.344395
$$901$$ −51.1338 −1.70351
$$902$$ −1.74314 −0.0580402
$$903$$ 27.5754 0.917651
$$904$$ 12.4844 0.415226
$$905$$ 0.326929 0.0108675
$$906$$ 9.75203 0.323989
$$907$$ 0.534795 0.0177576 0.00887880 0.999961i $$-0.497174\pi$$
0.00887880 + 0.999961i $$0.497174\pi$$
$$908$$ −36.9318 −1.22562
$$909$$ 75.5022 2.50425
$$910$$ 0.174840 0.00579590
$$911$$ −23.6686 −0.784177 −0.392088 0.919928i $$-0.628247\pi$$
−0.392088 + 0.919928i $$0.628247\pi$$
$$912$$ −10.9175 −0.361515
$$913$$ −16.3368 −0.540668
$$914$$ 4.18468 0.138417
$$915$$ −5.41927 −0.179156
$$916$$ 48.7467 1.61064
$$917$$ 10.6222 0.350776
$$918$$ −17.4479 −0.575865
$$919$$ 35.7748 1.18010 0.590051 0.807366i $$-0.299108\pi$$
0.590051 + 0.807366i $$0.299108\pi$$
$$920$$ −9.13828 −0.301280
$$921$$ −4.81579 −0.158686
$$922$$ −5.06376 −0.166766
$$923$$ −8.50760 −0.280031
$$924$$ −7.61285 −0.250444
$$925$$ 3.95407 0.130009
$$926$$ −9.45536 −0.310722
$$927$$ −70.2721 −2.30804
$$928$$ −3.49532 −0.114739
$$929$$ −52.7753 −1.73150 −0.865750 0.500477i $$-0.833158\pi$$
−0.865750 + 0.500477i $$0.833158\pi$$
$$930$$ −6.23506 −0.204456
$$931$$ 6.78277 0.222296
$$932$$ 5.94561 0.194755
$$933$$ 61.8390 2.02452
$$934$$ −0.369800 −0.0121002
$$935$$ 12.1334 0.396803
$$936$$ 4.10171 0.134069
$$937$$ 42.1245 1.37615 0.688073 0.725641i $$-0.258457\pi$$
0.688073 + 0.725641i $$0.258457\pi$$
$$938$$ −3.23860 −0.105744
$$939$$ 25.0321 0.816892
$$940$$ −13.1383 −0.428523
$$941$$ −3.89829 −0.127081 −0.0635403 0.997979i $$-0.520239\pi$$
−0.0635403 + 0.997979i $$0.520239\pi$$
$$942$$ 2.06022 0.0671257
$$943$$ 27.6414 0.900129
$$944$$ −5.73636 −0.186703
$$945$$ −6.36842 −0.207165
$$946$$ 4.99063 0.162259
$$947$$ 9.56691 0.310883 0.155441 0.987845i $$-0.450320\pi$$
0.155441 + 0.987845i $$0.450320\pi$$
$$948$$ −50.4929 −1.63993
$$949$$ −6.33984 −0.205800
$$950$$ −0.341219 −0.0110706
$$951$$ −80.1156 −2.59793
$$952$$ −8.72393 −0.282744
$$953$$ 27.2070 0.881320 0.440660 0.897674i $$-0.354744\pi$$
0.440660 + 0.897674i $$0.354744\pi$$
$$954$$ 10.8573 0.351517
$$955$$ −14.9447 −0.483599
$$956$$ −26.5303 −0.858053
$$957$$ 4.42864 0.143158
$$958$$ 12.7699 0.412575
$$959$$ 3.22168 0.104033
$$960$$ −16.7511 −0.540640
$$961$$ 16.6543 0.537237
$$962$$ −0.765413 −0.0246779
$$963$$ 59.7418 1.92515
$$964$$ 35.1526 1.13219
$$965$$ −14.1476 −0.455429
$$966$$ −6.13918 −0.197525
$$967$$ −16.8015 −0.540300 −0.270150 0.962818i $$-0.587073\pi$$
−0.270150 + 0.962818i $$0.587073\pi$$
$$968$$ 10.5319 0.338507
$$969$$ 25.3274 0.813633
$$970$$ −1.27163 −0.0408295
$$971$$ −17.4465 −0.559884 −0.279942 0.960017i $$-0.590315\pi$$
−0.279942 + 0.960017i $$0.590315\pi$$
$$972$$ 17.1383 0.549710
$$973$$ 7.73329 0.247918
$$974$$ −3.15701 −0.101157
$$975$$ −1.80642 −0.0578519
$$976$$ −6.40006 −0.204861
$$977$$ 32.0513 1.02541 0.512706 0.858564i $$-0.328643\pi$$
0.512706 + 0.858564i $$0.328643\pi$$
$$978$$ 14.7427 0.471418
$$979$$ 11.9081 0.380586
$$980$$ 11.7699 0.375974
$$981$$ −97.9407 −3.12701
$$982$$ −9.08694 −0.289976
$$983$$ −16.5259 −0.527094 −0.263547 0.964646i $$-0.584892\pi$$
−0.263547 + 0.964646i $$0.584892\pi$$
$$984$$ −12.9491 −0.412804
$$985$$ −5.70471 −0.181767
$$986$$ 2.47457 0.0788064
$$987$$ −18.1017 −0.576184
$$988$$ −1.29883 −0.0413211
$$989$$ −79.1378 −2.51644
$$990$$ −2.57628 −0.0818796
$$991$$ −9.34920 −0.296987 −0.148494 0.988913i $$-0.547442\pi$$
−0.148494 + 0.988913i $$0.547442\pi$$
$$992$$ −24.1289 −0.766094
$$993$$ 49.1022 1.55821
$$994$$ −3.84208 −0.121863
$$995$$ −22.1432 −0.701987
$$996$$ −59.1753 −1.87504
$$997$$ −15.9956 −0.506584 −0.253292 0.967390i $$-0.581513\pi$$
−0.253292 + 0.967390i $$0.581513\pi$$
$$998$$ 6.83854 0.216470
$$999$$ 27.8796 0.882070
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 145.2.a.c.1.2 3
3.2 odd 2 1305.2.a.p.1.2 3
4.3 odd 2 2320.2.a.n.1.1 3
5.2 odd 4 725.2.b.e.349.4 6
5.3 odd 4 725.2.b.e.349.3 6
5.4 even 2 725.2.a.e.1.2 3
7.6 odd 2 7105.2.a.o.1.2 3
8.3 odd 2 9280.2.a.br.1.3 3
8.5 even 2 9280.2.a.bj.1.1 3
15.14 odd 2 6525.2.a.be.1.2 3
29.28 even 2 4205.2.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 1.1 even 1 trivial
725.2.a.e.1.2 3 5.4 even 2
725.2.b.e.349.3 6 5.3 odd 4
725.2.b.e.349.4 6 5.2 odd 4
1305.2.a.p.1.2 3 3.2 odd 2
2320.2.a.n.1.1 3 4.3 odd 2
4205.2.a.f.1.2 3 29.28 even 2
6525.2.a.be.1.2 3 15.14 odd 2
7105.2.a.o.1.2 3 7.6 odd 2
9280.2.a.bj.1.1 3 8.5 even 2
9280.2.a.br.1.3 3 8.3 odd 2