# Properties

 Label 4205.2.a.f.1.2 Level $4205$ Weight $2$ Character 4205.1 Self dual yes Analytic conductor $33.577$ 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] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, 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("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.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: $$33.5770940499$$ 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: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 4205.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.903212 q^{6} -0.903212 q^{7} +1.21432 q^{8} +5.42864 q^{9} +O(q^{10})$$ $$q-0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.903212 q^{6} -0.903212 q^{7} +1.21432 q^{8} +5.42864 q^{9} -0.311108 q^{10} +1.52543 q^{11} +5.52543 q^{12} -0.622216 q^{13} +0.280996 q^{14} -2.90321 q^{15} +3.42864 q^{16} +7.95407 q^{17} -1.68889 q^{18} +1.09679 q^{19} -1.90321 q^{20} +2.62222 q^{21} -0.474572 q^{22} +7.52543 q^{23} -3.52543 q^{24} +1.00000 q^{25} +0.193576 q^{26} -7.05086 q^{27} +1.71900 q^{28} +0.903212 q^{30} +6.90321 q^{31} -3.49532 q^{32} -4.42864 q^{33} -2.47457 q^{34} -0.903212 q^{35} -10.3319 q^{36} -3.95407 q^{37} -0.341219 q^{38} +1.80642 q^{39} +1.21432 q^{40} -3.67307 q^{41} -0.815792 q^{42} +10.5161 q^{43} -2.90321 q^{44} +5.42864 q^{45} -2.34122 q^{46} -6.90321 q^{47} -9.95407 q^{48} -6.18421 q^{49} -0.311108 q^{50} -23.0923 q^{51} +1.18421 q^{52} +6.42864 q^{53} +2.19358 q^{54} +1.52543 q^{55} -1.09679 q^{56} -3.18421 q^{57} -1.67307 q^{59} +5.52543 q^{60} +1.86665 q^{61} -2.14764 q^{62} -4.90321 q^{63} -5.76986 q^{64} -0.622216 q^{65} +1.37778 q^{66} +11.5254 q^{67} -15.1383 q^{68} -21.8479 q^{69} +0.280996 q^{70} +13.6731 q^{71} +6.59210 q^{72} -10.1891 q^{73} +1.23014 q^{74} -2.90321 q^{75} -2.08742 q^{76} -1.37778 q^{77} -0.561993 q^{78} -9.13828 q^{79} +3.42864 q^{80} +4.18421 q^{81} +1.14272 q^{82} +10.7096 q^{83} -4.99063 q^{84} +7.95407 q^{85} -3.27163 q^{86} +1.85236 q^{88} +7.80642 q^{89} -1.68889 q^{90} +0.561993 q^{91} -14.3225 q^{92} -20.0415 q^{93} +2.14764 q^{94} +1.09679 q^{95} +10.1476 q^{96} +4.08742 q^{97} +1.92396 q^{98} +8.28100 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 2 * q^3 + q^4 + 3 * q^5 - 4 * q^6 + 4 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} - 2 q^{11} + 10 q^{12} - 2 q^{13} - 6 q^{14} - 2 q^{15} - 3 q^{16} + 4 q^{17} - 5 q^{18} + 10 q^{19} + q^{20} + 8 q^{21} - 8 q^{22} + 16 q^{23} - 4 q^{24} + 3 q^{25} + 14 q^{26} - 8 q^{27} + 12 q^{28} - 4 q^{30} + 14 q^{31} + 3 q^{32} - 14 q^{34} + 4 q^{35} - 11 q^{36} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 3 q^{40} + 2 q^{41} - 16 q^{42} - 2 q^{43} - 2 q^{44} + 3 q^{45} - 14 q^{46} - 14 q^{47} - 10 q^{48} - 5 q^{49} - q^{50} - 16 q^{51} - 10 q^{52} + 6 q^{53} + 20 q^{54} - 2 q^{55} - 10 q^{56} + 4 q^{57} + 8 q^{59} + 10 q^{60} + 6 q^{61} - 8 q^{63} - 11 q^{64} - 2 q^{65} + 4 q^{66} + 28 q^{67} - 12 q^{68} - 12 q^{69} - 6 q^{70} + 28 q^{71} + 13 q^{72} + 16 q^{73} + 10 q^{74} - 2 q^{75} + 14 q^{76} - 4 q^{77} + 12 q^{78} + 6 q^{79} - 3 q^{80} - q^{81} + 30 q^{82} + 12 q^{83} + 12 q^{84} + 4 q^{85} + 24 q^{86} + 12 q^{88} + 10 q^{89} - 5 q^{90} - 12 q^{91} + 4 q^{92} - 20 q^{93} + 10 q^{95} + 24 q^{96} - 8 q^{97} - 21 q^{98} + 18 q^{99}+O(q^{100})$$ 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 - 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 + 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 + 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 $$-0.535083\pi$$
−0.109993 + 0.993932i $$0.535083\pi$$
$$3$$ −2.90321 −1.67617 −0.838085 0.545540i $$-0.816325\pi$$
−0.838085 + 0.545540i $$0.816325\pi$$
$$4$$ −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 $$-0.426138\pi$$
0.229967 + 0.973198i $$0.426138\pi$$
$$12$$ 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 $$-0.0849820\pi$$
0.964572 + 0.263819i $$0.0849820\pi$$
$$18$$ −1.68889 −0.398076
$$19$$ 1.09679 0.251620 0.125810 0.992054i $$-0.459847\pi$$
0.125810 + 0.992054i $$0.459847\pi$$
$$20$$ −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
$$29$$ 0 0
$$30$$ 0.903212 0.164903
$$31$$ 6.90321 1.23985 0.619927 0.784660i $$-0.287162\pi$$
0.619927 + 0.784660i $$0.287162\pi$$
$$32$$ −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 $$-0.605372\pi$$
−0.325022 + 0.945706i $$0.605372\pi$$
$$38$$ −0.341219 −0.0553531
$$39$$ 1.80642 0.289259
$$40$$ 1.21432 0.192001
$$41$$ −3.67307 −0.573637 −0.286819 0.957985i $$-0.592598\pi$$
−0.286819 + 0.957985i $$0.592598\pi$$
$$42$$ −0.815792 −0.125879
$$43$$ 10.5161 1.60368 0.801842 0.597536i $$-0.203854\pi$$
0.801842 + 0.597536i $$0.203854\pi$$
$$44$$ −2.90321 −0.437676
$$45$$ 5.42864 0.809254
$$46$$ −2.34122 −0.345194
$$47$$ −6.90321 −1.00694 −0.503468 0.864014i $$-0.667943\pi$$
−0.503468 + 0.864014i $$0.667943\pi$$
$$48$$ −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 0
$$59$$ −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 $$-0.461871\pi$$
0.119500 + 0.992834i $$0.461871\pi$$
$$62$$ −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 $$-0.703353\pi$$
−0.596274 + 0.802781i $$0.703353\pi$$
$$74$$ 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 $$-0.671862\pi$$
−0.514068 + 0.857749i $$0.671862\pi$$
$$80$$ 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
$$87$$ 0 0
$$88$$ 1.85236 0.197462
$$89$$ 7.80642 0.827479 0.413740 0.910395i $$-0.364222\pi$$
0.413740 + 0.910395i $$0.364222\pi$$
$$90$$ −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 $$-0.433465\pi$$
0.207507 + 0.978233i $$0.433465\pi$$
$$98$$ 1.92396 0.194349
$$99$$ 8.28100 0.832271
$$100$$ −1.90321 −0.190321
$$101$$ −13.9081 −1.38391 −0.691956 0.721940i $$-0.743251\pi$$
−0.691956 + 0.721940i $$0.743251\pi$$
$$102$$ 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 $$-0.339337\pi$$
0.483577 + 0.875302i $$0.339337\pi$$
$$114$$ 0.990632 0.0927812
$$115$$ 7.52543 0.701750
$$116$$ 0 0
$$117$$ −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 $$-0.589010\pi$$
−0.276002 + 0.961157i $$0.589010\pi$$
$$128$$ 8.78568 0.776552
$$129$$ −30.5303 −2.68805
$$130$$ 0.193576 0.0169778
$$131$$ 11.7605 1.02752 0.513759 0.857934i $$-0.328252\pi$$
0.513759 + 0.857934i $$0.328252\pi$$
$$132$$ 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 $$-0.451309\pi$$
0.152371 + 0.988323i $$0.451309\pi$$
$$138$$ 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
$$145$$ 0 0
$$146$$ 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 $$-0.529013\pi$$
−0.0910217 + 0.995849i $$0.529013\pi$$
$$158$$ 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 $$-0.720750\pi$$
−0.639238 + 0.769009i $$0.720750\pi$$
$$164$$ 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 0
$$175$$ −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 $$-0.318167\pi$$
0.540680 + 0.841228i $$0.318167\pi$$
$$192$$ 16.7511 1.20891
$$193$$ 14.1476 1.01837 0.509185 0.860657i $$-0.329947\pi$$
0.509185 + 0.860657i $$0.329947\pi$$
$$194$$ −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 0
$$204$$ 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 $$-0.245311\pi$$
0.717445 + 0.696615i $$0.245311\pi$$
$$212$$ −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 $$-0.178842\pi$$
0.846272 + 0.532751i $$0.178842\pi$$
$$230$$ −2.34122 −0.154375
$$231$$ 4.00000 0.263181
$$232$$ 0 0
$$233$$ −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 $$-0.356651\pi$$
0.435276 + 0.900297i $$0.356651\pi$$
$$252$$ 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
$$261$$ 0 0
$$262$$ −3.65878 −0.226040
$$263$$ 0.442930 0.0273122 0.0136561 0.999907i $$-0.495653\pi$$
0.0136561 + 0.999907i $$0.495653\pi$$
$$264$$ −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 $$-0.538324\pi$$
−0.120106 + 0.992761i $$0.538324\pi$$
$$270$$ 2.19358 0.133497
$$271$$ −6.20787 −0.377101 −0.188551 0.982063i $$-0.560379\pi$$
−0.188551 + 0.982063i $$0.560379\pi$$
$$272$$ 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 0
$$291$$ −11.8666 −0.695635
$$292$$ 19.3921 1.13484
$$293$$ 18.8430 1.10082 0.550410 0.834895i $$-0.314472\pi$$
0.550410 + 0.834895i $$0.314472\pi$$
$$294$$ −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 $$-0.484927\pi$$
0.0473358 + 0.998879i $$0.484927\pi$$
$$308$$ 2.62222 0.149415
$$309$$ 37.5812 2.13792
$$310$$ −2.14764 −0.121978
$$311$$ −21.3002 −1.20782 −0.603912 0.797051i $$-0.706392\pi$$
−0.603912 + 0.797051i $$0.706392\pi$$
$$312$$ 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 $$-0.217771\pi$$
0.774959 + 0.632012i $$0.217771\pi$$
$$318$$ 5.80642 0.325608
$$319$$ 0 0
$$320$$ −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 $$-0.653878\pi$$
−0.464813 + 0.885409i $$0.653878\pi$$
$$332$$ −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 $$-0.605943\pi$$
−0.326720 + 0.945121i $$0.605943\pi$$
$$338$$ 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 $$0.447234\pi$$
$$348$$ 0 0
$$349$$ −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 $$-0.920188\pi$$
−0.968730 + 0.248116i $$0.920188\pi$$
$$360$$ 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 $$-0.570479\pi$$
−0.219613 + 0.975587i $$0.570479\pi$$
$$368$$ 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 0
$$378$$ −1.98126 −0.101905
$$379$$ 2.76986 0.142278 0.0711390 0.997466i $$-0.477337\pi$$
0.0711390 + 0.997466i $$0.477337\pi$$
$$380$$ −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 $$-0.453232\pi$$
0.146397 + 0.989226i $$0.453232\pi$$
$$390$$ −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 0
$$407$$ −6.03164 −0.298977
$$408$$ −28.0415 −1.38826
$$409$$ −5.09234 −0.251800 −0.125900 0.992043i $$-0.540182\pi$$
−0.125900 + 0.992043i $$0.540182\pi$$
$$410$$ 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 $$-0.703669\pi$$
−0.597072 + 0.802188i $$0.703669\pi$$
$$422$$ −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 $$-0.724765\pi$$
−0.648887 + 0.760885i $$0.724765\pi$$
$$434$$ 1.93978 0.0931123
$$435$$ 0 0
$$436$$ 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 $$-0.473768\pi$$
0.0823184 + 0.996606i $$0.473768\pi$$
$$444$$ −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 $$-0.843494\pi$$
−0.881541 + 0.472107i $$0.843494\pi$$
$$450$$ −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 $$-0.376255\pi$$
0.379037 + 0.925381i $$0.376255\pi$$
$$462$$ −1.24443 −0.0578962
$$463$$ −30.3926 −1.41246 −0.706231 0.707982i $$-0.749606\pi$$
−0.706231 + 0.707982i $$0.749606\pi$$
$$464$$ 0 0
$$465$$ −20.0415 −0.929402
$$466$$ 0.971896 0.0450222
$$467$$ 1.18865 0.0550043 0.0275022 0.999622i $$-0.491245\pi$$
0.0275022 + 0.999622i $$0.491245\pi$$
$$468$$ 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 $$-0.887076\pi$$
−0.937729 + 0.347367i $$0.887076\pi$$
$$480$$ 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 $$-0.270947\pi$$
0.659077 + 0.752075i $$0.270947\pi$$
$$492$$ −20.2953 −0.914982
$$493$$ 0 0
$$494$$ 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 $$-0.541126\pi$$
−0.128841 + 0.991665i $$0.541126\pi$$
$$504$$ −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 $$0.641820\pi$$
$$522$$ 0 0
$$523$$ −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 $$-0.518365\pi$$
−0.0576635 + 0.998336i $$0.518365\pi$$
$$542$$ 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
$$551$$ 0 0
$$552$$ −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 $$-0.303302\pi$$
0.579362 + 0.815070i $$0.303302\pi$$
$$564$$ −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 $$-0.618543\pi$$
−0.363865 + 0.931452i $$0.618543\pi$$
$$570$$ 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 $$-0.429064\pi$$
0.221012 + 0.975271i $$0.429064\pi$$
$$578$$ −14.3941 −0.598715
$$579$$ −41.0736 −1.70696
$$580$$ 0 0
$$581$$ −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 $$-0.347549\pi$$
0.460838 + 0.887484i $$0.347549\pi$$
$$600$$ −3.52543 −0.143925
$$601$$ 40.6133 1.65665 0.828326 0.560246i $$-0.189294\pi$$
0.828326 + 0.560246i $$0.189294\pi$$
$$602$$ 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 $$-0.588980\pi$$
−0.275912 + 0.961183i $$0.588980\pi$$
$$608$$ −3.83362 −0.155474
$$609$$ 0 0
$$610$$ −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 $$-0.735625\pi$$
−0.674464 + 0.738307i $$0.735625\pi$$
$$618$$ −11.6918 −0.470313
$$619$$ 14.6780 0.589958 0.294979 0.955504i $$-0.404687\pi$$
0.294979 + 0.955504i $$0.404687\pi$$
$$620$$ −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 0
$$639$$ 74.2262 2.93634
$$640$$ 8.78568 0.347285
$$641$$ −34.8988 −1.37842 −0.689209 0.724562i $$-0.742042\pi$$
−0.689209 + 0.724562i $$0.742042\pi$$
$$642$$ 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 $$-0.445108\pi$$
0.171596 + 0.985167i $$0.445108\pi$$
$$654$$ −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 $$-0.520414\pi$$
−0.0640872 + 0.997944i $$0.520414\pi$$
$$660$$ 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
$$667$$ 0 0
$$668$$ 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 $$-0.537382\pi$$
−0.117168 + 0.993112i $$0.537382\pi$$
$$678$$ 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
$$696$$ 0 0
$$697$$ −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
$$725$$ 0 0
$$726$$ −7.83362 −0.290733
$$727$$ −3.88094 −0.143936 −0.0719680 0.997407i $$-0.522928\pi$$
−0.0719680 + 0.997407i $$0.522928\pi$$
$$728$$ 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 $$-0.588640\pi$$
−0.274885 + 0.961477i $$0.588640\pi$$
$$734$$ 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 $$-0.513173\pi$$
−0.0413719 + 0.999144i $$0.513173\pi$$
$$740$$ 7.52543 0.276640
$$741$$ 1.98126 0.0727836
$$742$$ 1.80642 0.0663159
$$743$$ 3.46520 0.127126 0.0635630 0.997978i $$-0.479754\pi$$
0.0635630 + 0.997978i $$0.479754\pi$$
$$744$$ −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 $$-0.518420\pi$$
−0.0578360 + 0.998326i $$0.518420\pi$$
$$752$$ −23.6686 −0.863106
$$753$$ −40.0415 −1.45919
$$754$$ 0 0
$$755$$ 10.7971 0.392945
$$756$$ −12.1204 −0.440816
$$757$$ 52.0785 1.89283 0.946413 0.322958i $$-0.104677\pi$$
0.946413 + 0.322958i $$0.104677\pi$$
$$758$$ −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 $$-0.489047\pi$$
0.0344045 + 0.999408i $$0.489047\pi$$
$$770$$ 0.428639 0.0154471
$$771$$ 4.29529 0.154691
$$772$$ −26.9260 −0.969087
$$773$$ −21.7891 −0.783698 −0.391849 0.920029i $$-0.628165\pi$$
−0.391849 + 0.920029i $$0.628165\pi$$
$$774$$ −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
$$783$$ 0 0
$$784$$ −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 $$-0.516714\pi$$
−0.0524851 + 0.998622i $$0.516714\pi$$
$$798$$ −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 $$-0.347044\pi$$
0.462247 + 0.886751i $$0.347044\pi$$
$$810$$ −1.30174 −0.0457385
$$811$$ 24.3783 0.856037 0.428018 0.903770i $$-0.359212\pi$$
0.428018 + 0.903770i $$0.359212\pi$$
$$812$$ 0 0
$$813$$ 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 $$-0.802862\pi$$
−0.814269 + 0.580487i $$0.802862\pi$$
$$824$$ −15.7190 −0.547597
$$825$$ −4.42864 −0.154185
$$826$$ −0.470127 −0.0163578
$$827$$ 29.6499 1.03103 0.515514 0.856881i $$-0.327601\pi$$
0.515514 + 0.856881i $$0.327601\pi$$
$$828$$ −77.7516 −2.70205
$$829$$ 8.79706 0.305534 0.152767 0.988262i $$-0.451182\pi$$
0.152767 + 0.988262i $$0.451182\pi$$
$$830$$ −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 $$-0.562668\pi$$
−0.195609 + 0.980682i $$0.562668\pi$$
$$840$$ 3.18421 0.109866
$$841$$ 0 0
$$842$$ 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 $$-0.888807\pi$$
−0.939604 + 0.342263i $$0.888807\pi$$
$$854$$ 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 $$-0.509390\pi$$
−0.0294938 + 0.999565i $$0.509390\pi$$
$$860$$ −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 0
$$871$$ −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 $$-0.248874\pi$$
0.709605 + 0.704600i $$0.248874\pi$$
$$882$$ 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 $$-0.275337\pi$$
0.648643 + 0.761092i $$0.275337\pi$$
$$888$$ 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
$$899$$ 0 0
$$900$$ −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 $$-0.502826\pi$$
−0.00887880 + 0.999961i $$0.502826\pi$$
$$908$$ −36.9318 −1.22562
$$909$$ −75.5022 −2.50425
$$910$$ −0.174840 −0.00579590
$$911$$ 23.6686 0.784177 0.392088 0.919928i $$-0.371753\pi$$
0.392088 + 0.919928i $$0.371753\pi$$
$$912$$ −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
$$928$$ 0 0
$$929$$ −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 $$-0.549680\pi$$
−0.155441 + 0.987845i $$0.549680\pi$$
$$948$$ −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
$$957$$ 0 0
$$958$$ 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 $$-0.412927\pi$$
0.270150 + 0.962818i $$0.412927\pi$$
$$968$$ −10.5319 −0.338507
$$969$$ −25.3274 −0.813633
$$970$$ −1.27163 −0.0408295
$$971$$ 17.4465 0.559884 0.279942 0.960017i $$-0.409685\pi$$
0.279942 + 0.960017i $$0.409685\pi$$
$$972$$ −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 $$-0.415108\pi$$
0.263547 + 0.964646i $$0.415108\pi$$
$$984$$ 12.9491 0.412804
$$985$$ −5.70471 −0.181767
$$986$$ 0 0
$$987$$ −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 $$-0.418487\pi$$
0.253292 + 0.967390i $$0.418487\pi$$
$$998$$ −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 4205.2.a.f.1.2 3
29.28 even 2 145.2.a.c.1.2 3
87.86 odd 2 1305.2.a.p.1.2 3
116.115 odd 2 2320.2.a.n.1.1 3
145.28 odd 4 725.2.b.e.349.3 6
145.57 odd 4 725.2.b.e.349.4 6
145.144 even 2 725.2.a.e.1.2 3
203.202 odd 2 7105.2.a.o.1.2 3
232.115 odd 2 9280.2.a.br.1.3 3
232.173 even 2 9280.2.a.bj.1.1 3
435.434 odd 2 6525.2.a.be.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 29.28 even 2
725.2.a.e.1.2 3 145.144 even 2
725.2.b.e.349.3 6 145.28 odd 4
725.2.b.e.349.4 6 145.57 odd 4
1305.2.a.p.1.2 3 87.86 odd 2
2320.2.a.n.1.1 3 116.115 odd 2
4205.2.a.f.1.2 3 1.1 even 1 trivial
6525.2.a.be.1.2 3 435.434 odd 2
7105.2.a.o.1.2 3 203.202 odd 2
9280.2.a.bj.1.1 3 232.173 even 2
9280.2.a.br.1.3 3 232.115 odd 2