# Properties

 Label 1805.2.a.o.1.3 Level $1805$ Weight $2$ Character 1805.1 Self dual yes Analytic conductor $14.413$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, 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("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.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: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7537.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 4x + 3$$ x^4 - x^3 - 5*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.37933$$ of defining polynomial Character $$\chi$$ $$=$$ 1805.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.09744 q^{2} -0.379334 q^{3} -0.795629 q^{4} +1.00000 q^{5} -0.416295 q^{6} +1.89307 q^{7} -3.06803 q^{8} -2.85611 q^{9} +O(q^{10})$$ $$q+1.09744 q^{2} -0.379334 q^{3} -0.795629 q^{4} +1.00000 q^{5} -0.416295 q^{6} +1.89307 q^{7} -3.06803 q^{8} -2.85611 q^{9} +1.09744 q^{10} +0.134400 q^{11} +0.301809 q^{12} +3.51373 q^{13} +2.07752 q^{14} -0.379334 q^{15} -1.77572 q^{16} -1.66123 q^{17} -3.13440 q^{18} -0.795629 q^{20} -0.718104 q^{21} +0.147496 q^{22} +5.36984 q^{23} +1.16381 q^{24} +1.00000 q^{25} +3.85611 q^{26} +2.22142 q^{27} -1.50618 q^{28} +4.97059 q^{29} -0.416295 q^{30} +6.56472 q^{31} +4.18732 q^{32} -0.0509824 q^{33} -1.82310 q^{34} +1.89307 q^{35} +2.27240 q^{36} -1.69819 q^{37} -1.33288 q^{39} -3.06803 q^{40} +10.6327 q^{41} -0.788075 q^{42} +8.50784 q^{43} -0.106932 q^{44} -2.85611 q^{45} +5.89307 q^{46} -11.1154 q^{47} +0.673589 q^{48} -3.41630 q^{49} +1.09744 q^{50} +0.630160 q^{51} -2.79563 q^{52} -0.264847 q^{53} +2.43787 q^{54} +0.134400 q^{55} -5.80799 q^{56} +5.45492 q^{58} -6.89667 q^{59} +0.301809 q^{60} +9.17589 q^{61} +7.20437 q^{62} -5.40680 q^{63} +8.14676 q^{64} +3.51373 q^{65} -0.0559501 q^{66} -2.95354 q^{67} +1.32172 q^{68} -2.03696 q^{69} +2.07752 q^{70} +1.32835 q^{71} +8.76262 q^{72} -6.34237 q^{73} -1.86366 q^{74} -0.379334 q^{75} +0.254428 q^{77} -1.46275 q^{78} -1.46728 q^{79} -1.77572 q^{80} +7.72566 q^{81} +11.6688 q^{82} +7.44736 q^{83} +0.571345 q^{84} -1.66123 q^{85} +9.33683 q^{86} -1.88551 q^{87} -0.412343 q^{88} +9.73608 q^{89} -3.13440 q^{90} +6.65174 q^{91} -4.27240 q^{92} -2.49022 q^{93} -12.1985 q^{94} -1.58839 q^{96} +17.4689 q^{97} -3.74917 q^{98} -0.383860 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9}+O(q^{10})$$ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 4 * q^5 + 2 * q^6 - 4 * q^7 + 12 * q^8 + q^9 $$4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9} + q^{10} - 2 q^{11} + 6 q^{12} + 7 q^{13} - q^{14} + 3 q^{15} + 7 q^{16} - q^{17} - 10 q^{18} + 5 q^{20} - 4 q^{21} + 2 q^{22} + 2 q^{23} + 23 q^{24} + 4 q^{25} + 3 q^{26} + 12 q^{27} - 19 q^{28} - q^{29} + 2 q^{30} + 30 q^{32} + 19 q^{33} + 15 q^{34} - 4 q^{35} - 7 q^{36} - 2 q^{37} + 15 q^{39} + 12 q^{40} - 8 q^{41} - 15 q^{42} + q^{43} - 12 q^{44} + q^{45} + 12 q^{46} - 12 q^{47} + 23 q^{48} - 10 q^{49} + q^{50} + 22 q^{51} - 3 q^{52} - 5 q^{53} - 34 q^{54} - 2 q^{55} - 41 q^{56} - 27 q^{58} - 5 q^{59} + 6 q^{60} + 37 q^{62} - 3 q^{63} + 56 q^{64} + 7 q^{65} - 31 q^{66} + 4 q^{67} + 16 q^{68} - 9 q^{69} - q^{70} + 20 q^{71} + 17 q^{72} - 20 q^{73} + 25 q^{74} + 3 q^{75} + 14 q^{77} - 18 q^{78} + 17 q^{79} + 7 q^{80} + 12 q^{81} + 21 q^{82} + q^{83} - 20 q^{84} - q^{85} + 8 q^{86} - 16 q^{87} - 7 q^{88} + 11 q^{89} - 10 q^{90} + 6 q^{91} - q^{92} - 8 q^{93} - 31 q^{94} + 21 q^{96} + q^{97} + 9 q^{98} + 38 q^{99}+O(q^{100})$$ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 4 * q^5 + 2 * q^6 - 4 * q^7 + 12 * q^8 + q^9 + q^10 - 2 * q^11 + 6 * q^12 + 7 * q^13 - q^14 + 3 * q^15 + 7 * q^16 - q^17 - 10 * q^18 + 5 * q^20 - 4 * q^21 + 2 * q^22 + 2 * q^23 + 23 * q^24 + 4 * q^25 + 3 * q^26 + 12 * q^27 - 19 * q^28 - q^29 + 2 * q^30 + 30 * q^32 + 19 * q^33 + 15 * q^34 - 4 * q^35 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 12 * q^40 - 8 * q^41 - 15 * q^42 + q^43 - 12 * q^44 + q^45 + 12 * q^46 - 12 * q^47 + 23 * q^48 - 10 * q^49 + q^50 + 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 - 2 * q^55 - 41 * q^56 - 27 * q^58 - 5 * q^59 + 6 * q^60 + 37 * q^62 - 3 * q^63 + 56 * q^64 + 7 * q^65 - 31 * q^66 + 4 * q^67 + 16 * q^68 - 9 * q^69 - q^70 + 20 * q^71 + 17 * q^72 - 20 * q^73 + 25 * q^74 + 3 * q^75 + 14 * q^77 - 18 * q^78 + 17 * q^79 + 7 * q^80 + 12 * q^81 + 21 * q^82 + q^83 - 20 * q^84 - q^85 + 8 * q^86 - 16 * q^87 - 7 * q^88 + 11 * q^89 - 10 * q^90 + 6 * q^91 - q^92 - 8 * q^93 - 31 * q^94 + 21 * q^96 + q^97 + 9 * q^98 + 38 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.09744 0.776006 0.388003 0.921658i $$-0.373165\pi$$
0.388003 + 0.921658i $$0.373165\pi$$
$$3$$ −0.379334 −0.219008 −0.109504 0.993986i $$-0.534926\pi$$
−0.109504 + 0.993986i $$0.534926\pi$$
$$4$$ −0.795629 −0.397815
$$5$$ 1.00000 0.447214
$$6$$ −0.416295 −0.169952
$$7$$ 1.89307 0.715512 0.357756 0.933815i $$-0.383542\pi$$
0.357756 + 0.933815i $$0.383542\pi$$
$$8$$ −3.06803 −1.08471
$$9$$ −2.85611 −0.952035
$$10$$ 1.09744 0.347040
$$11$$ 0.134400 0.0405231 0.0202615 0.999795i $$-0.493550\pi$$
0.0202615 + 0.999795i $$0.493550\pi$$
$$12$$ 0.301809 0.0871248
$$13$$ 3.51373 0.974534 0.487267 0.873253i $$-0.337994\pi$$
0.487267 + 0.873253i $$0.337994\pi$$
$$14$$ 2.07752 0.555242
$$15$$ −0.379334 −0.0979436
$$16$$ −1.77572 −0.443929
$$17$$ −1.66123 −0.402907 −0.201454 0.979498i $$-0.564567\pi$$
−0.201454 + 0.979498i $$0.564567\pi$$
$$18$$ −3.13440 −0.738785
$$19$$ 0 0
$$20$$ −0.795629 −0.177908
$$21$$ −0.718104 −0.156703
$$22$$ 0.147496 0.0314462
$$23$$ 5.36984 1.11969 0.559844 0.828598i $$-0.310861\pi$$
0.559844 + 0.828598i $$0.310861\pi$$
$$24$$ 1.16381 0.237561
$$25$$ 1.00000 0.200000
$$26$$ 3.85611 0.756245
$$27$$ 2.22142 0.427512
$$28$$ −1.50618 −0.284641
$$29$$ 4.97059 0.923016 0.461508 0.887136i $$-0.347309\pi$$
0.461508 + 0.887136i $$0.347309\pi$$
$$30$$ −0.416295 −0.0760048
$$31$$ 6.56472 1.17906 0.589529 0.807747i $$-0.299313\pi$$
0.589529 + 0.807747i $$0.299313\pi$$
$$32$$ 4.18732 0.740221
$$33$$ −0.0509824 −0.00887490
$$34$$ −1.82310 −0.312658
$$35$$ 1.89307 0.319987
$$36$$ 2.27240 0.378734
$$37$$ −1.69819 −0.279181 −0.139590 0.990209i $$-0.544579\pi$$
−0.139590 + 0.990209i $$0.544579\pi$$
$$38$$ 0 0
$$39$$ −1.33288 −0.213431
$$40$$ −3.06803 −0.485098
$$41$$ 10.6327 1.66056 0.830278 0.557349i $$-0.188182\pi$$
0.830278 + 0.557349i $$0.188182\pi$$
$$42$$ −0.788075 −0.121603
$$43$$ 8.50784 1.29743 0.648717 0.761030i $$-0.275306\pi$$
0.648717 + 0.761030i $$0.275306\pi$$
$$44$$ −0.106932 −0.0161207
$$45$$ −2.85611 −0.425763
$$46$$ 5.89307 0.868885
$$47$$ −11.1154 −1.62135 −0.810675 0.585497i $$-0.800899\pi$$
−0.810675 + 0.585497i $$0.800899\pi$$
$$48$$ 0.673589 0.0972242
$$49$$ −3.41630 −0.488042
$$50$$ 1.09744 0.155201
$$51$$ 0.630160 0.0882401
$$52$$ −2.79563 −0.387684
$$53$$ −0.264847 −0.0363796 −0.0181898 0.999835i $$-0.505790\pi$$
−0.0181898 + 0.999835i $$0.505790\pi$$
$$54$$ 2.43787 0.331752
$$55$$ 0.134400 0.0181225
$$56$$ −5.80799 −0.776125
$$57$$ 0 0
$$58$$ 5.45492 0.716266
$$59$$ −6.89667 −0.897870 −0.448935 0.893564i $$-0.648196\pi$$
−0.448935 + 0.893564i $$0.648196\pi$$
$$60$$ 0.301809 0.0389634
$$61$$ 9.17589 1.17485 0.587426 0.809278i $$-0.300141\pi$$
0.587426 + 0.809278i $$0.300141\pi$$
$$62$$ 7.20437 0.914956
$$63$$ −5.40680 −0.681193
$$64$$ 8.14676 1.01834
$$65$$ 3.51373 0.435825
$$66$$ −0.0559501 −0.00688698
$$67$$ −2.95354 −0.360833 −0.180416 0.983590i $$-0.557745\pi$$
−0.180416 + 0.983590i $$0.557745\pi$$
$$68$$ 1.32172 0.160282
$$69$$ −2.03696 −0.245221
$$70$$ 2.07752 0.248312
$$71$$ 1.32835 0.157646 0.0788232 0.996889i $$-0.474884\pi$$
0.0788232 + 0.996889i $$0.474884\pi$$
$$72$$ 8.76262 1.03268
$$73$$ −6.34237 −0.742319 −0.371159 0.928569i $$-0.621040\pi$$
−0.371159 + 0.928569i $$0.621040\pi$$
$$74$$ −1.86366 −0.216646
$$75$$ −0.379334 −0.0438017
$$76$$ 0 0
$$77$$ 0.254428 0.0289948
$$78$$ −1.46275 −0.165624
$$79$$ −1.46728 −0.165082 −0.0825408 0.996588i $$-0.526303\pi$$
−0.0825408 + 0.996588i $$0.526303\pi$$
$$80$$ −1.77572 −0.198531
$$81$$ 7.72566 0.858406
$$82$$ 11.6688 1.28860
$$83$$ 7.44736 0.817454 0.408727 0.912657i $$-0.365973\pi$$
0.408727 + 0.912657i $$0.365973\pi$$
$$84$$ 0.571345 0.0623388
$$85$$ −1.66123 −0.180186
$$86$$ 9.33683 1.00682
$$87$$ −1.88551 −0.202148
$$88$$ −0.412343 −0.0439559
$$89$$ 9.73608 1.03202 0.516011 0.856582i $$-0.327416\pi$$
0.516011 + 0.856582i $$0.327416\pi$$
$$90$$ −3.13440 −0.330395
$$91$$ 6.65174 0.697291
$$92$$ −4.27240 −0.445429
$$93$$ −2.49022 −0.258224
$$94$$ −12.1985 −1.25818
$$95$$ 0 0
$$96$$ −1.58839 −0.162115
$$97$$ 17.4689 1.77370 0.886851 0.462055i $$-0.152888\pi$$
0.886851 + 0.462055i $$0.152888\pi$$
$$98$$ −3.74917 −0.378724
$$99$$ −0.383860 −0.0385794
$$100$$ −0.795629 −0.0795629
$$101$$ 5.39731 0.537052 0.268526 0.963272i $$-0.413463\pi$$
0.268526 + 0.963272i $$0.413463\pi$$
$$102$$ 0.691562 0.0684749
$$103$$ 2.14750 0.211599 0.105800 0.994387i $$-0.466260\pi$$
0.105800 + 0.994387i $$0.466260\pi$$
$$104$$ −10.7802 −1.05709
$$105$$ −0.718104 −0.0700798
$$106$$ −0.290654 −0.0282308
$$107$$ −1.00093 −0.0967631 −0.0483815 0.998829i $$-0.515406\pi$$
−0.0483815 + 0.998829i $$0.515406\pi$$
$$108$$ −1.76743 −0.170071
$$109$$ 16.2629 1.55770 0.778852 0.627208i $$-0.215802\pi$$
0.778852 + 0.627208i $$0.215802\pi$$
$$110$$ 0.147496 0.0140632
$$111$$ 0.644181 0.0611430
$$112$$ −3.36155 −0.317637
$$113$$ 0.843010 0.0793037 0.0396519 0.999214i $$-0.487375\pi$$
0.0396519 + 0.999214i $$0.487375\pi$$
$$114$$ 0 0
$$115$$ 5.36984 0.500740
$$116$$ −3.95475 −0.367189
$$117$$ −10.0356 −0.927791
$$118$$ −7.56867 −0.696752
$$119$$ −3.14482 −0.288285
$$120$$ 1.16381 0.106241
$$121$$ −10.9819 −0.998358
$$122$$ 10.0700 0.911692
$$123$$ −4.03336 −0.363676
$$124$$ −5.22308 −0.469046
$$125$$ 1.00000 0.0894427
$$126$$ −5.93363 −0.528610
$$127$$ 18.7397 1.66288 0.831439 0.555616i $$-0.187518\pi$$
0.831439 + 0.555616i $$0.187518\pi$$
$$128$$ 0.565920 0.0500208
$$129$$ −3.22731 −0.284149
$$130$$ 3.85611 0.338203
$$131$$ 2.88644 0.252189 0.126095 0.992018i $$-0.459756\pi$$
0.126095 + 0.992018i $$0.459756\pi$$
$$132$$ 0.0405631 0.00353057
$$133$$ 0 0
$$134$$ −3.24133 −0.280008
$$135$$ 2.22142 0.191189
$$136$$ 5.09670 0.437039
$$137$$ −18.8316 −1.60889 −0.804445 0.594027i $$-0.797537\pi$$
−0.804445 + 0.594027i $$0.797537\pi$$
$$138$$ −2.23544 −0.190293
$$139$$ −18.1795 −1.54196 −0.770982 0.636857i $$-0.780234\pi$$
−0.770982 + 0.636857i $$0.780234\pi$$
$$140$$ −1.50618 −0.127295
$$141$$ 4.21645 0.355089
$$142$$ 1.45778 0.122334
$$143$$ 0.472245 0.0394912
$$144$$ 5.07163 0.422636
$$145$$ 4.97059 0.412785
$$146$$ −6.96036 −0.576044
$$147$$ 1.29592 0.106885
$$148$$ 1.35113 0.111062
$$149$$ −22.2543 −1.82315 −0.911573 0.411138i $$-0.865131\pi$$
−0.911573 + 0.411138i $$0.865131\pi$$
$$150$$ −0.416295 −0.0339904
$$151$$ 3.33482 0.271384 0.135692 0.990751i $$-0.456674\pi$$
0.135692 + 0.990751i $$0.456674\pi$$
$$152$$ 0 0
$$153$$ 4.74465 0.383582
$$154$$ 0.279219 0.0225001
$$155$$ 6.56472 0.527291
$$156$$ 1.06048 0.0849061
$$157$$ 7.26291 0.579643 0.289822 0.957081i $$-0.406404\pi$$
0.289822 + 0.957081i $$0.406404\pi$$
$$158$$ −1.61025 −0.128104
$$159$$ 0.100466 0.00796744
$$160$$ 4.18732 0.331037
$$161$$ 10.1655 0.801151
$$162$$ 8.47843 0.666129
$$163$$ −19.7783 −1.54916 −0.774578 0.632478i $$-0.782038\pi$$
−0.774578 + 0.632478i $$0.782038\pi$$
$$164$$ −8.45972 −0.660593
$$165$$ −0.0509824 −0.00396898
$$166$$ 8.17302 0.634350
$$167$$ −3.40320 −0.263348 −0.131674 0.991293i $$-0.542035\pi$$
−0.131674 + 0.991293i $$0.542035\pi$$
$$168$$ 2.20317 0.169978
$$169$$ −0.653675 −0.0502827
$$170$$ −1.82310 −0.139825
$$171$$ 0 0
$$172$$ −6.76909 −0.516138
$$173$$ −10.5857 −0.804817 −0.402409 0.915460i $$-0.631827\pi$$
−0.402409 + 0.915460i $$0.631827\pi$$
$$174$$ −2.06923 −0.156868
$$175$$ 1.89307 0.143102
$$176$$ −0.238656 −0.0179894
$$177$$ 2.61614 0.196641
$$178$$ 10.6847 0.800855
$$179$$ −14.6024 −1.09144 −0.545718 0.837969i $$-0.683743\pi$$
−0.545718 + 0.837969i $$0.683743\pi$$
$$180$$ 2.27240 0.169375
$$181$$ −5.43261 −0.403803 −0.201901 0.979406i $$-0.564712\pi$$
−0.201901 + 0.979406i $$0.564712\pi$$
$$182$$ 7.29987 0.541102
$$183$$ −3.48072 −0.257303
$$184$$ −16.4748 −1.21454
$$185$$ −1.69819 −0.124853
$$186$$ −2.73286 −0.200383
$$187$$ −0.223269 −0.0163271
$$188$$ 8.84375 0.644996
$$189$$ 4.20530 0.305890
$$190$$ 0 0
$$191$$ 20.4758 1.48157 0.740787 0.671740i $$-0.234453\pi$$
0.740787 + 0.671740i $$0.234453\pi$$
$$192$$ −3.09034 −0.223026
$$193$$ 11.0235 0.793490 0.396745 0.917929i $$-0.370140\pi$$
0.396745 + 0.917929i $$0.370140\pi$$
$$194$$ 19.1711 1.37640
$$195$$ −1.33288 −0.0954494
$$196$$ 2.71810 0.194150
$$197$$ −19.8532 −1.41448 −0.707242 0.706971i $$-0.750061\pi$$
−0.707242 + 0.706971i $$0.750061\pi$$
$$198$$ −0.421263 −0.0299379
$$199$$ 21.0026 1.48883 0.744417 0.667715i $$-0.232728\pi$$
0.744417 + 0.667715i $$0.232728\pi$$
$$200$$ −3.06803 −0.216943
$$201$$ 1.12038 0.0790254
$$202$$ 5.92321 0.416756
$$203$$ 9.40967 0.660429
$$204$$ −0.501374 −0.0351032
$$205$$ 10.6327 0.742623
$$206$$ 2.35674 0.164202
$$207$$ −15.3368 −1.06598
$$208$$ −6.23939 −0.432624
$$209$$ 0 0
$$210$$ −0.788075 −0.0543824
$$211$$ −12.8257 −0.882956 −0.441478 0.897272i $$-0.645546\pi$$
−0.441478 + 0.897272i $$0.645546\pi$$
$$212$$ 0.210720 0.0144723
$$213$$ −0.503889 −0.0345259
$$214$$ −1.09845 −0.0750887
$$215$$ 8.50784 0.580230
$$216$$ −6.81538 −0.463728
$$217$$ 12.4275 0.843630
$$218$$ 17.8475 1.20879
$$219$$ 2.40588 0.162574
$$220$$ −0.106932 −0.00720939
$$221$$ −5.83712 −0.392647
$$222$$ 0.706949 0.0474473
$$223$$ 20.3944 1.36571 0.682856 0.730553i $$-0.260737\pi$$
0.682856 + 0.730553i $$0.260737\pi$$
$$224$$ 7.92688 0.529637
$$225$$ −2.85611 −0.190407
$$226$$ 0.925152 0.0615402
$$227$$ 25.4172 1.68700 0.843500 0.537129i $$-0.180491\pi$$
0.843500 + 0.537129i $$0.180491\pi$$
$$228$$ 0 0
$$229$$ 2.21553 0.146406 0.0732030 0.997317i $$-0.476678\pi$$
0.0732030 + 0.997317i $$0.476678\pi$$
$$230$$ 5.89307 0.388577
$$231$$ −0.0965132 −0.00635010
$$232$$ −15.2499 −1.00121
$$233$$ −14.1576 −0.927498 −0.463749 0.885967i $$-0.653496\pi$$
−0.463749 + 0.885967i $$0.653496\pi$$
$$234$$ −11.0134 −0.719972
$$235$$ −11.1154 −0.725089
$$236$$ 5.48719 0.357186
$$237$$ 0.556588 0.0361543
$$238$$ −3.45125 −0.223711
$$239$$ 3.01476 0.195008 0.0975042 0.995235i $$-0.468914\pi$$
0.0975042 + 0.995235i $$0.468914\pi$$
$$240$$ 0.673589 0.0434800
$$241$$ −23.7792 −1.53175 −0.765877 0.642987i $$-0.777695\pi$$
−0.765877 + 0.642987i $$0.777695\pi$$
$$242$$ −12.0520 −0.774732
$$243$$ −9.59486 −0.615511
$$244$$ −7.30060 −0.467373
$$245$$ −3.41630 −0.218259
$$246$$ −4.42636 −0.282215
$$247$$ 0 0
$$248$$ −20.1407 −1.27894
$$249$$ −2.82504 −0.179029
$$250$$ 1.09744 0.0694081
$$251$$ 17.1899 1.08502 0.542509 0.840050i $$-0.317475\pi$$
0.542509 + 0.840050i $$0.317475\pi$$
$$252$$ 4.30181 0.270988
$$253$$ 0.721706 0.0453733
$$254$$ 20.5656 1.29040
$$255$$ 0.630160 0.0394622
$$256$$ −15.6725 −0.979529
$$257$$ −19.5429 −1.21905 −0.609525 0.792767i $$-0.708640\pi$$
−0.609525 + 0.792767i $$0.708640\pi$$
$$258$$ −3.54178 −0.220501
$$259$$ −3.21479 −0.199757
$$260$$ −2.79563 −0.173378
$$261$$ −14.1965 −0.878744
$$262$$ 3.16769 0.195700
$$263$$ 8.81360 0.543470 0.271735 0.962372i $$-0.412403\pi$$
0.271735 + 0.962372i $$0.412403\pi$$
$$264$$ 0.156416 0.00962672
$$265$$ −0.264847 −0.0162694
$$266$$ 0 0
$$267$$ −3.69322 −0.226022
$$268$$ 2.34993 0.143545
$$269$$ 0.288362 0.0175818 0.00879088 0.999961i $$-0.497202\pi$$
0.00879088 + 0.999961i $$0.497202\pi$$
$$270$$ 2.43787 0.148364
$$271$$ −24.8712 −1.51082 −0.755409 0.655253i $$-0.772562\pi$$
−0.755409 + 0.655253i $$0.772562\pi$$
$$272$$ 2.94987 0.178862
$$273$$ −2.52323 −0.152713
$$274$$ −20.6665 −1.24851
$$275$$ 0.134400 0.00810462
$$276$$ 1.62067 0.0975526
$$277$$ −4.40486 −0.264662 −0.132331 0.991206i $$-0.542246\pi$$
−0.132331 + 0.991206i $$0.542246\pi$$
$$278$$ −19.9509 −1.19657
$$279$$ −18.7495 −1.12250
$$280$$ −5.80799 −0.347094
$$281$$ −32.7214 −1.95200 −0.975998 0.217778i $$-0.930119\pi$$
−0.975998 + 0.217778i $$0.930119\pi$$
$$282$$ 4.62729 0.275551
$$283$$ −1.32893 −0.0789964 −0.0394982 0.999220i $$-0.512576\pi$$
−0.0394982 + 0.999220i $$0.512576\pi$$
$$284$$ −1.05688 −0.0627140
$$285$$ 0 0
$$286$$ 0.518260 0.0306454
$$287$$ 20.1285 1.18815
$$288$$ −11.9594 −0.704717
$$289$$ −14.2403 −0.837666
$$290$$ 5.45492 0.320324
$$291$$ −6.62656 −0.388456
$$292$$ 5.04618 0.295305
$$293$$ −7.72365 −0.451220 −0.225610 0.974218i $$-0.572438\pi$$
−0.225610 + 0.974218i $$0.572438\pi$$
$$294$$ 1.42219 0.0829437
$$295$$ −6.89667 −0.401540
$$296$$ 5.21010 0.302831
$$297$$ 0.298558 0.0173241
$$298$$ −24.4228 −1.41477
$$299$$ 18.8682 1.09118
$$300$$ 0.301809 0.0174250
$$301$$ 16.1059 0.928330
$$302$$ 3.65976 0.210595
$$303$$ −2.04738 −0.117619
$$304$$ 0 0
$$305$$ 9.17589 0.525410
$$306$$ 5.20696 0.297662
$$307$$ −9.10002 −0.519366 −0.259683 0.965694i $$-0.583618\pi$$
−0.259683 + 0.965694i $$0.583618\pi$$
$$308$$ −0.202430 −0.0115345
$$309$$ −0.814618 −0.0463420
$$310$$ 7.20437 0.409181
$$311$$ −12.4569 −0.706364 −0.353182 0.935555i $$-0.614900\pi$$
−0.353182 + 0.935555i $$0.614900\pi$$
$$312$$ 4.08931 0.231512
$$313$$ 2.04553 0.115620 0.0578101 0.998328i $$-0.481588\pi$$
0.0578101 + 0.998328i $$0.481588\pi$$
$$314$$ 7.97059 0.449807
$$315$$ −5.40680 −0.304639
$$316$$ 1.16741 0.0656719
$$317$$ 23.5713 1.32389 0.661947 0.749551i $$-0.269730\pi$$
0.661947 + 0.749551i $$0.269730\pi$$
$$318$$ 0.110255 0.00618278
$$319$$ 0.668047 0.0374035
$$320$$ 8.14676 0.455418
$$321$$ 0.379685 0.0211919
$$322$$ 11.1560 0.621698
$$323$$ 0 0
$$324$$ −6.14676 −0.341487
$$325$$ 3.51373 0.194907
$$326$$ −21.7055 −1.20215
$$327$$ −6.16907 −0.341150
$$328$$ −32.6216 −1.80123
$$329$$ −21.0422 −1.16010
$$330$$ −0.0559501 −0.00307995
$$331$$ −18.7175 −1.02881 −0.514403 0.857549i $$-0.671986\pi$$
−0.514403 + 0.857549i $$0.671986\pi$$
$$332$$ −5.92534 −0.325195
$$333$$ 4.85021 0.265790
$$334$$ −3.73480 −0.204359
$$335$$ −2.95354 −0.161369
$$336$$ 1.27515 0.0695651
$$337$$ −32.6881 −1.78063 −0.890316 0.455343i $$-0.849517\pi$$
−0.890316 + 0.455343i $$0.849517\pi$$
$$338$$ −0.717368 −0.0390197
$$339$$ −0.319782 −0.0173682
$$340$$ 1.32172 0.0716805
$$341$$ 0.882297 0.0477791
$$342$$ 0 0
$$343$$ −19.7188 −1.06471
$$344$$ −26.1023 −1.40734
$$345$$ −2.03696 −0.109666
$$346$$ −11.6172 −0.624543
$$347$$ −2.56666 −0.137785 −0.0688927 0.997624i $$-0.521947\pi$$
−0.0688927 + 0.997624i $$0.521947\pi$$
$$348$$ 1.50017 0.0804175
$$349$$ 16.6195 0.889619 0.444810 0.895625i $$-0.353271\pi$$
0.444810 + 0.895625i $$0.353271\pi$$
$$350$$ 2.07752 0.111048
$$351$$ 7.80547 0.416625
$$352$$ 0.562776 0.0299961
$$353$$ 28.3629 1.50961 0.754803 0.655951i $$-0.227732\pi$$
0.754803 + 0.655951i $$0.227732\pi$$
$$354$$ 2.87105 0.152595
$$355$$ 1.32835 0.0705016
$$356$$ −7.74631 −0.410553
$$357$$ 1.19294 0.0631369
$$358$$ −16.0252 −0.846961
$$359$$ 17.3885 0.917733 0.458866 0.888505i $$-0.348256\pi$$
0.458866 + 0.888505i $$0.348256\pi$$
$$360$$ 8.76262 0.461831
$$361$$ 0 0
$$362$$ −5.96195 −0.313353
$$363$$ 4.16582 0.218649
$$364$$ −5.29231 −0.277393
$$365$$ −6.34237 −0.331975
$$366$$ −3.81988 −0.199668
$$367$$ 25.8048 1.34700 0.673501 0.739186i $$-0.264790\pi$$
0.673501 + 0.739186i $$0.264790\pi$$
$$368$$ −9.53531 −0.497062
$$369$$ −30.3683 −1.58091
$$370$$ −1.86366 −0.0968871
$$371$$ −0.501374 −0.0260300
$$372$$ 1.98129 0.102725
$$373$$ 27.0663 1.40144 0.700719 0.713437i $$-0.252862\pi$$
0.700719 + 0.713437i $$0.252862\pi$$
$$374$$ −0.245024 −0.0126699
$$375$$ −0.379334 −0.0195887
$$376$$ 34.1024 1.75870
$$377$$ 17.4653 0.899511
$$378$$ 4.61505 0.237373
$$379$$ 12.4028 0.637092 0.318546 0.947907i $$-0.396806\pi$$
0.318546 + 0.947907i $$0.396806\pi$$
$$380$$ 0 0
$$381$$ −7.10859 −0.364184
$$382$$ 22.4709 1.14971
$$383$$ −5.35942 −0.273854 −0.136927 0.990581i $$-0.543723\pi$$
−0.136927 + 0.990581i $$0.543723\pi$$
$$384$$ −0.214673 −0.0109550
$$385$$ 0.254428 0.0129669
$$386$$ 12.0976 0.615753
$$387$$ −24.2993 −1.23520
$$388$$ −13.8988 −0.705605
$$389$$ 8.56933 0.434482 0.217241 0.976118i $$-0.430294\pi$$
0.217241 + 0.976118i $$0.430294\pi$$
$$390$$ −1.46275 −0.0740693
$$391$$ −8.92053 −0.451131
$$392$$ 10.4813 0.529386
$$393$$ −1.09492 −0.0552316
$$394$$ −21.7877 −1.09765
$$395$$ −1.46728 −0.0738268
$$396$$ 0.305411 0.0153475
$$397$$ −10.6445 −0.534234 −0.267117 0.963664i $$-0.586071\pi$$
−0.267117 + 0.963664i $$0.586071\pi$$
$$398$$ 23.0490 1.15534
$$399$$ 0 0
$$400$$ −1.77572 −0.0887858
$$401$$ 7.65209 0.382127 0.191063 0.981578i $$-0.438806\pi$$
0.191063 + 0.981578i $$0.438806\pi$$
$$402$$ 1.22955 0.0613242
$$403$$ 23.0667 1.14903
$$404$$ −4.29426 −0.213647
$$405$$ 7.72566 0.383891
$$406$$ 10.3265 0.512497
$$407$$ −0.228237 −0.0113133
$$408$$ −1.93335 −0.0957152
$$409$$ −17.6887 −0.874650 −0.437325 0.899304i $$-0.644074\pi$$
−0.437325 + 0.899304i $$0.644074\pi$$
$$410$$ 11.6688 0.576280
$$411$$ 7.14345 0.352361
$$412$$ −1.70861 −0.0841772
$$413$$ −13.0559 −0.642437
$$414$$ −16.8312 −0.827210
$$415$$ 7.44736 0.365577
$$416$$ 14.7131 0.721371
$$417$$ 6.89609 0.337703
$$418$$ 0 0
$$419$$ 1.18732 0.0580045 0.0290023 0.999579i $$-0.490767\pi$$
0.0290023 + 0.999579i $$0.490767\pi$$
$$420$$ 0.571345 0.0278788
$$421$$ 33.3673 1.62622 0.813111 0.582109i $$-0.197772\pi$$
0.813111 + 0.582109i $$0.197772\pi$$
$$422$$ −14.0754 −0.685180
$$423$$ 31.7468 1.54358
$$424$$ 0.812560 0.0394614
$$425$$ −1.66123 −0.0805815
$$426$$ −0.552987 −0.0267923
$$427$$ 17.3706 0.840621
$$428$$ 0.796365 0.0384938
$$429$$ −0.179139 −0.00864890
$$430$$ 9.33683 0.450262
$$431$$ −6.17598 −0.297486 −0.148743 0.988876i $$-0.547523\pi$$
−0.148743 + 0.988876i $$0.547523\pi$$
$$432$$ −3.94461 −0.189785
$$433$$ −18.5552 −0.891707 −0.445854 0.895106i $$-0.647100\pi$$
−0.445854 + 0.895106i $$0.647100\pi$$
$$434$$ 13.6384 0.654662
$$435$$ −1.88551 −0.0904035
$$436$$ −12.9392 −0.619677
$$437$$ 0 0
$$438$$ 2.64030 0.126158
$$439$$ −0.227312 −0.0108490 −0.00542450 0.999985i $$-0.501727\pi$$
−0.00542450 + 0.999985i $$0.501727\pi$$
$$440$$ −0.412343 −0.0196577
$$441$$ 9.75730 0.464633
$$442$$ −6.40588 −0.304696
$$443$$ −34.9827 −1.66208 −0.831038 0.556215i $$-0.812253\pi$$
−0.831038 + 0.556215i $$0.812253\pi$$
$$444$$ −0.512529 −0.0243236
$$445$$ 9.73608 0.461534
$$446$$ 22.3816 1.05980
$$447$$ 8.44182 0.399285
$$448$$ 15.4224 0.728638
$$449$$ −16.9509 −0.799961 −0.399980 0.916524i $$-0.630983\pi$$
−0.399980 + 0.916524i $$0.630983\pi$$
$$450$$ −3.13440 −0.147757
$$451$$ 1.42904 0.0672909
$$452$$ −0.670724 −0.0315482
$$453$$ −1.26501 −0.0594353
$$454$$ 27.8938 1.30912
$$455$$ 6.65174 0.311838
$$456$$ 0 0
$$457$$ 1.60241 0.0749578 0.0374789 0.999297i $$-0.488067\pi$$
0.0374789 + 0.999297i $$0.488067\pi$$
$$458$$ 2.43140 0.113612
$$459$$ −3.69029 −0.172248
$$460$$ −4.27240 −0.199202
$$461$$ −8.74162 −0.407138 −0.203569 0.979061i $$-0.565254\pi$$
−0.203569 + 0.979061i $$0.565254\pi$$
$$462$$ −0.105917 −0.00492772
$$463$$ 21.1886 0.984718 0.492359 0.870392i $$-0.336135\pi$$
0.492359 + 0.870392i $$0.336135\pi$$
$$464$$ −8.82636 −0.409753
$$465$$ −2.49022 −0.115481
$$466$$ −15.5371 −0.719744
$$467$$ 20.4516 0.946388 0.473194 0.880958i $$-0.343101\pi$$
0.473194 + 0.880958i $$0.343101\pi$$
$$468$$ 7.98461 0.369089
$$469$$ −5.59126 −0.258180
$$470$$ −12.1985 −0.562674
$$471$$ −2.75507 −0.126947
$$472$$ 21.1592 0.973931
$$473$$ 1.14345 0.0525760
$$474$$ 0.610821 0.0280559
$$475$$ 0 0
$$476$$ 2.50211 0.114684
$$477$$ 0.756432 0.0346347
$$478$$ 3.30851 0.151328
$$479$$ 23.5491 1.07599 0.537994 0.842949i $$-0.319182\pi$$
0.537994 + 0.842949i $$0.319182\pi$$
$$480$$ −1.58839 −0.0724999
$$481$$ −5.96699 −0.272071
$$482$$ −26.0962 −1.18865
$$483$$ −3.85611 −0.175459
$$484$$ 8.73755 0.397161
$$485$$ 17.4689 0.793224
$$486$$ −10.5298 −0.477640
$$487$$ 36.0392 1.63309 0.816546 0.577280i $$-0.195886\pi$$
0.816546 + 0.577280i $$0.195886\pi$$
$$488$$ −28.1519 −1.27438
$$489$$ 7.50258 0.339278
$$490$$ −3.74917 −0.169370
$$491$$ 20.0595 0.905271 0.452635 0.891696i $$-0.350484\pi$$
0.452635 + 0.891696i $$0.350484\pi$$
$$492$$ 3.20906 0.144676
$$493$$ −8.25729 −0.371890
$$494$$ 0 0
$$495$$ −0.383860 −0.0172532
$$496$$ −11.6571 −0.523418
$$497$$ 2.51466 0.112798
$$498$$ −3.10030 −0.138928
$$499$$ −36.8729 −1.65066 −0.825328 0.564653i $$-0.809010\pi$$
−0.825328 + 0.564653i $$0.809010\pi$$
$$500$$ −0.795629 −0.0355816
$$501$$ 1.29095 0.0576753
$$502$$ 18.8649 0.841980
$$503$$ −21.6487 −0.965268 −0.482634 0.875822i $$-0.660320\pi$$
−0.482634 + 0.875822i $$0.660320\pi$$
$$504$$ 16.5882 0.738899
$$505$$ 5.39731 0.240177
$$506$$ 0.792028 0.0352099
$$507$$ 0.247961 0.0110123
$$508$$ −14.9098 −0.661517
$$509$$ −36.4558 −1.61588 −0.807938 0.589267i $$-0.799417\pi$$
−0.807938 + 0.589267i $$0.799417\pi$$
$$510$$ 0.691562 0.0306229
$$511$$ −12.0065 −0.531138
$$512$$ −18.3314 −0.810141
$$513$$ 0 0
$$514$$ −21.4471 −0.945990
$$515$$ 2.14750 0.0946300
$$516$$ 2.56774 0.113039
$$517$$ −1.49391 −0.0657021
$$518$$ −3.52803 −0.155013
$$519$$ 4.01552 0.176262
$$520$$ −10.7802 −0.472745
$$521$$ −22.6092 −0.990528 −0.495264 0.868742i $$-0.664929\pi$$
−0.495264 + 0.868742i $$0.664929\pi$$
$$522$$ −15.5798 −0.681910
$$523$$ 0.532911 0.0233026 0.0116513 0.999932i $$-0.496291\pi$$
0.0116513 + 0.999932i $$0.496291\pi$$
$$524$$ −2.29654 −0.100325
$$525$$ −0.718104 −0.0313406
$$526$$ 9.67238 0.421736
$$527$$ −10.9055 −0.475051
$$528$$ 0.0905303 0.00393983
$$529$$ 5.83518 0.253703
$$530$$ −0.290654 −0.0126252
$$531$$ 19.6976 0.854804
$$532$$ 0 0
$$533$$ 37.3606 1.61827
$$534$$ −4.05308 −0.175394
$$535$$ −1.00093 −0.0432738
$$536$$ 9.06156 0.391400
$$537$$ 5.53919 0.239034
$$538$$ 0.316460 0.0136436
$$539$$ −0.459150 −0.0197770
$$540$$ −1.76743 −0.0760579
$$541$$ 5.01640 0.215672 0.107836 0.994169i $$-0.465608\pi$$
0.107836 + 0.994169i $$0.465608\pi$$
$$542$$ −27.2946 −1.17240
$$543$$ 2.06077 0.0884362
$$544$$ −6.95610 −0.298240
$$545$$ 16.2629 0.696626
$$546$$ −2.76909 −0.118506
$$547$$ 22.6299 0.967584 0.483792 0.875183i $$-0.339259\pi$$
0.483792 + 0.875183i $$0.339259\pi$$
$$548$$ 14.9830 0.640040
$$549$$ −26.2073 −1.11850
$$550$$ 0.147496 0.00628923
$$551$$ 0 0
$$552$$ 6.24946 0.265995
$$553$$ −2.77766 −0.118118
$$554$$ −4.83406 −0.205380
$$555$$ 0.644181 0.0273440
$$556$$ 14.4641 0.613416
$$557$$ 35.2554 1.49382 0.746910 0.664925i $$-0.231537\pi$$
0.746910 + 0.664925i $$0.231537\pi$$
$$558$$ −20.5764 −0.871070
$$559$$ 29.8943 1.26439
$$560$$ −3.36155 −0.142051
$$561$$ 0.0846935 0.00357576
$$562$$ −35.9097 −1.51476
$$563$$ −24.6295 −1.03801 −0.519005 0.854771i $$-0.673698\pi$$
−0.519005 + 0.854771i $$0.673698\pi$$
$$564$$ −3.35473 −0.141260
$$565$$ 0.843010 0.0354657
$$566$$ −1.45841 −0.0613017
$$567$$ 14.6252 0.614200
$$568$$ −4.07542 −0.171001
$$569$$ −20.0193 −0.839252 −0.419626 0.907697i $$-0.637839\pi$$
−0.419626 + 0.907697i $$0.637839\pi$$
$$570$$ 0 0
$$571$$ −16.6121 −0.695195 −0.347597 0.937644i $$-0.613002\pi$$
−0.347597 + 0.937644i $$0.613002\pi$$
$$572$$ −0.375732 −0.0157102
$$573$$ −7.76715 −0.324477
$$574$$ 22.0898 0.922010
$$575$$ 5.36984 0.223938
$$576$$ −23.2680 −0.969500
$$577$$ 12.4486 0.518244 0.259122 0.965845i $$-0.416567\pi$$
0.259122 + 0.965845i $$0.416567\pi$$
$$578$$ −15.6279 −0.650034
$$579$$ −4.18159 −0.173781
$$580$$ −3.95475 −0.164212
$$581$$ 14.0984 0.584899
$$582$$ −7.27224 −0.301444
$$583$$ −0.0355955 −0.00147421
$$584$$ 19.4586 0.805202
$$585$$ −10.0356 −0.414921
$$586$$ −8.47623 −0.350150
$$587$$ 4.51144 0.186207 0.0931036 0.995656i $$-0.470321\pi$$
0.0931036 + 0.995656i $$0.470321\pi$$
$$588$$ −1.03107 −0.0425206
$$589$$ 0 0
$$590$$ −7.56867 −0.311597
$$591$$ 7.53101 0.309784
$$592$$ 3.01550 0.123936
$$593$$ 19.6082 0.805213 0.402606 0.915373i $$-0.368104\pi$$
0.402606 + 0.915373i $$0.368104\pi$$
$$594$$ 0.327650 0.0134436
$$595$$ −3.14482 −0.128925
$$596$$ 17.7062 0.725274
$$597$$ −7.96699 −0.326067
$$598$$ 20.7067 0.846759
$$599$$ 10.7759 0.440292 0.220146 0.975467i $$-0.429347\pi$$
0.220146 + 0.975467i $$0.429347\pi$$
$$600$$ 1.16381 0.0475122
$$601$$ 15.0244 0.612860 0.306430 0.951893i $$-0.400866\pi$$
0.306430 + 0.951893i $$0.400866\pi$$
$$602$$ 17.6753 0.720389
$$603$$ 8.43563 0.343526
$$604$$ −2.65328 −0.107960
$$605$$ −10.9819 −0.446479
$$606$$ −2.24687 −0.0912730
$$607$$ 25.1901 1.02243 0.511217 0.859452i $$-0.329195\pi$$
0.511217 + 0.859452i $$0.329195\pi$$
$$608$$ 0 0
$$609$$ −3.56940 −0.144640
$$610$$ 10.0700 0.407721
$$611$$ −39.0566 −1.58006
$$612$$ −3.77498 −0.152595
$$613$$ 16.2351 0.655728 0.327864 0.944725i $$-0.393671\pi$$
0.327864 + 0.944725i $$0.393671\pi$$
$$614$$ −9.98672 −0.403031
$$615$$ −4.03336 −0.162641
$$616$$ −0.780593 −0.0314510
$$617$$ 6.51826 0.262415 0.131208 0.991355i $$-0.458115\pi$$
0.131208 + 0.991355i $$0.458115\pi$$
$$618$$ −0.893993 −0.0359617
$$619$$ −4.39112 −0.176494 −0.0882470 0.996099i $$-0.528126\pi$$
−0.0882470 + 0.996099i $$0.528126\pi$$
$$620$$ −5.22308 −0.209764
$$621$$ 11.9287 0.478681
$$622$$ −13.6706 −0.548142
$$623$$ 18.4311 0.738425
$$624$$ 2.36681 0.0947483
$$625$$ 1.00000 0.0400000
$$626$$ 2.24484 0.0897220
$$627$$ 0 0
$$628$$ −5.77858 −0.230590
$$629$$ 2.82108 0.112484
$$630$$ −5.93363 −0.236402
$$631$$ −34.6209 −1.37824 −0.689118 0.724650i $$-0.742002\pi$$
−0.689118 + 0.724650i $$0.742002\pi$$
$$632$$ 4.50165 0.179066
$$633$$ 4.86521 0.193375
$$634$$ 25.8680 1.02735
$$635$$ 18.7397 0.743661
$$636$$ −0.0799333 −0.00316956
$$637$$ −12.0040 −0.475614
$$638$$ 0.733141 0.0290253
$$639$$ −3.79391 −0.150085
$$640$$ 0.565920 0.0223700
$$641$$ 7.41241 0.292773 0.146386 0.989227i $$-0.453236\pi$$
0.146386 + 0.989227i $$0.453236\pi$$
$$642$$ 0.416681 0.0164451
$$643$$ −16.5459 −0.652506 −0.326253 0.945282i $$-0.605786\pi$$
−0.326253 + 0.945282i $$0.605786\pi$$
$$644$$ −8.08794 −0.318710
$$645$$ −3.22731 −0.127075
$$646$$ 0 0
$$647$$ 29.4822 1.15907 0.579533 0.814949i $$-0.303235\pi$$
0.579533 + 0.814949i $$0.303235\pi$$
$$648$$ −23.7026 −0.931124
$$649$$ −0.926912 −0.0363845
$$650$$ 3.85611 0.151249
$$651$$ −4.71415 −0.184762
$$652$$ 15.7362 0.616277
$$653$$ 6.57421 0.257269 0.128634 0.991692i $$-0.458941\pi$$
0.128634 + 0.991692i $$0.458941\pi$$
$$654$$ −6.77017 −0.264735
$$655$$ 2.88644 0.112782
$$656$$ −18.8807 −0.737169
$$657$$ 18.1145 0.706713
$$658$$ −23.0925 −0.900241
$$659$$ −26.3370 −1.02594 −0.512972 0.858405i $$-0.671456\pi$$
−0.512972 + 0.858405i $$0.671456\pi$$
$$660$$ 0.0405631 0.00157892
$$661$$ 3.78419 0.147188 0.0735941 0.997288i $$-0.476553\pi$$
0.0735941 + 0.997288i $$0.476553\pi$$
$$662$$ −20.5413 −0.798359
$$663$$ 2.21422 0.0859930
$$664$$ −22.8487 −0.886703
$$665$$ 0 0
$$666$$ 5.32281 0.206255
$$667$$ 26.6913 1.03349
$$668$$ 2.70769 0.104763
$$669$$ −7.73630 −0.299103
$$670$$ −3.24133 −0.125224
$$671$$ 1.23324 0.0476086
$$672$$ −3.00694 −0.115995
$$673$$ −21.0431 −0.811150 −0.405575 0.914062i $$-0.632929\pi$$
−0.405575 + 0.914062i $$0.632929\pi$$
$$674$$ −35.8731 −1.38178
$$675$$ 2.22142 0.0855025
$$676$$ 0.520083 0.0200032
$$677$$ −15.2744 −0.587043 −0.293522 0.955952i $$-0.594827\pi$$
−0.293522 + 0.955952i $$0.594827\pi$$
$$678$$ −0.350941 −0.0134778
$$679$$ 33.0699 1.26911
$$680$$ 5.09670 0.195450
$$681$$ −9.64161 −0.369467
$$682$$ 0.968267 0.0370769
$$683$$ −8.60225 −0.329156 −0.164578 0.986364i $$-0.552626\pi$$
−0.164578 + 0.986364i $$0.552626\pi$$
$$684$$ 0 0
$$685$$ −18.8316 −0.719518
$$686$$ −21.6401 −0.826223
$$687$$ −0.840424 −0.0320642
$$688$$ −15.1075 −0.575968
$$689$$ −0.930603 −0.0354532
$$690$$ −2.23544 −0.0851017
$$691$$ 34.1079 1.29753 0.648763 0.760990i $$-0.275286\pi$$
0.648763 + 0.760990i $$0.275286\pi$$
$$692$$ 8.42231 0.320168
$$693$$ −0.726674 −0.0276040
$$694$$ −2.81675 −0.106922
$$695$$ −18.1795 −0.689587
$$696$$ 5.78481 0.219273
$$697$$ −17.6634 −0.669050
$$698$$ 18.2388 0.690350
$$699$$ 5.37047 0.203130
$$700$$ −1.50618 −0.0569282
$$701$$ 10.7293 0.405239 0.202619 0.979258i $$-0.435055\pi$$
0.202619 + 0.979258i $$0.435055\pi$$
$$702$$ 8.56603 0.323304
$$703$$ 0 0
$$704$$ 1.09492 0.0412665
$$705$$ 4.21645 0.158801
$$706$$ 31.1266 1.17146
$$707$$ 10.2175 0.384267
$$708$$ −2.08148 −0.0782267
$$709$$ −28.8476 −1.08339 −0.541697 0.840574i $$-0.682218\pi$$
−0.541697 + 0.840574i $$0.682218\pi$$
$$710$$ 1.45778 0.0547096
$$711$$ 4.19070 0.157164
$$712$$ −29.8706 −1.11945
$$713$$ 35.2515 1.32018
$$714$$ 1.30917 0.0489946
$$715$$ 0.472245 0.0176610
$$716$$ 11.6181 0.434189
$$717$$ −1.14360 −0.0427085
$$718$$ 19.0829 0.712166
$$719$$ −20.0555 −0.747944 −0.373972 0.927440i $$-0.622004\pi$$
−0.373972 + 0.927440i $$0.622004\pi$$
$$720$$ 5.07163 0.189009
$$721$$ 4.06535 0.151402
$$722$$ 0 0
$$723$$ 9.02026 0.335467
$$724$$ 4.32234 0.160639
$$725$$ 4.97059 0.184603
$$726$$ 4.57173 0.169673
$$727$$ −0.780521 −0.0289479 −0.0144740 0.999895i $$-0.504607\pi$$
−0.0144740 + 0.999895i $$0.504607\pi$$
$$728$$ −20.4077 −0.756361
$$729$$ −19.5373 −0.723604
$$730$$ −6.96036 −0.257615
$$731$$ −14.1335 −0.522745
$$732$$ 2.76937 0.102359
$$733$$ −26.8391 −0.991326 −0.495663 0.868515i $$-0.665075\pi$$
−0.495663 + 0.868515i $$0.665075\pi$$
$$734$$ 28.3192 1.04528
$$735$$ 1.29592 0.0478006
$$736$$ 22.4853 0.828817
$$737$$ −0.396956 −0.0146221
$$738$$ −33.3273 −1.22679
$$739$$ 20.6530 0.759735 0.379867 0.925041i $$-0.375970\pi$$
0.379867 + 0.925041i $$0.375970\pi$$
$$740$$ 1.35113 0.0496685
$$741$$ 0 0
$$742$$ −0.550227 −0.0201995
$$743$$ 1.47317 0.0540454 0.0270227 0.999635i $$-0.491397\pi$$
0.0270227 + 0.999635i $$0.491397\pi$$
$$744$$ 7.64007 0.280098
$$745$$ −22.2543 −0.815336
$$746$$ 29.7036 1.08752
$$747$$ −21.2705 −0.778245
$$748$$ 0.177639 0.00649514
$$749$$ −1.89482 −0.0692352
$$750$$ −0.416295 −0.0152010
$$751$$ 15.5624 0.567881 0.283940 0.958842i $$-0.408358\pi$$
0.283940 + 0.958842i $$0.408358\pi$$
$$752$$ 19.7378 0.719764
$$753$$ −6.52071 −0.237628
$$754$$ 19.1671 0.698026
$$755$$ 3.33482 0.121366
$$756$$ −3.34586 −0.121688
$$757$$ 20.1756 0.733295 0.366647 0.930360i $$-0.380506\pi$$
0.366647 + 0.930360i $$0.380506\pi$$
$$758$$ 13.6114 0.494387
$$759$$ −0.273767 −0.00993713
$$760$$ 0 0
$$761$$ −15.1076 −0.547649 −0.273825 0.961780i $$-0.588289\pi$$
−0.273825 + 0.961780i $$0.588289\pi$$
$$762$$ −7.80124 −0.282609
$$763$$ 30.7868 1.11456
$$764$$ −16.2911 −0.589392
$$765$$ 4.74465 0.171543
$$766$$ −5.88163 −0.212512
$$767$$ −24.2331 −0.875005
$$768$$ 5.94509 0.214525
$$769$$ 51.6580 1.86284 0.931418 0.363951i $$-0.118573\pi$$
0.931418 + 0.363951i $$0.118573\pi$$
$$770$$ 0.279219 0.0100624
$$771$$ 7.41327 0.266982
$$772$$ −8.77063 −0.315662
$$773$$ 30.1558 1.08463 0.542314 0.840176i $$-0.317548\pi$$
0.542314 + 0.840176i $$0.317548\pi$$
$$774$$ −26.6670 −0.958525
$$775$$ 6.56472 0.235812
$$776$$ −53.5952 −1.92396
$$777$$ 1.21948 0.0437485
$$778$$ 9.40431 0.337161
$$779$$ 0 0
$$780$$ 1.06048 0.0379712
$$781$$ 0.178530 0.00638832
$$782$$ −9.78974 −0.350080
$$783$$ 11.0418 0.394601
$$784$$ 6.06637 0.216656
$$785$$ 7.26291 0.259224
$$786$$ −1.20161 −0.0428601
$$787$$ 43.0969 1.53624 0.768119 0.640307i $$-0.221193\pi$$
0.768119 + 0.640307i $$0.221193\pi$$
$$788$$ 15.7958 0.562703
$$789$$ −3.34330 −0.119025
$$790$$ −1.61025 −0.0572900
$$791$$ 1.59588 0.0567428
$$792$$ 1.17770 0.0418476
$$793$$ 32.2416 1.14493
$$794$$ −11.6817 −0.414569
$$795$$ 0.100466 0.00356315
$$796$$ −16.7103 −0.592280
$$797$$ −50.5062 −1.78902 −0.894510 0.447048i $$-0.852475\pi$$
−0.894510 + 0.447048i $$0.852475\pi$$
$$798$$ 0 0
$$799$$ 18.4652 0.653253
$$800$$ 4.18732 0.148044
$$801$$ −27.8073 −0.982522
$$802$$ 8.39769 0.296533
$$803$$ −0.852414 −0.0300810
$$804$$ −0.891406 −0.0314375
$$805$$ 10.1655 0.358286
$$806$$ 25.3142 0.891656
$$807$$ −0.109386 −0.00385056
$$808$$ −16.5591 −0.582547
$$809$$ −30.0872 −1.05781 −0.528905 0.848681i $$-0.677397\pi$$
−0.528905 + 0.848681i $$0.677397\pi$$
$$810$$ 8.47843 0.297902
$$811$$ 22.9509 0.805916 0.402958 0.915218i $$-0.367982\pi$$
0.402958 + 0.915218i $$0.367982\pi$$
$$812$$ −7.48661 −0.262728
$$813$$ 9.43449 0.330882
$$814$$ −0.250476 −0.00877917
$$815$$ −19.7783 −0.692804
$$816$$ −1.11899 −0.0391723
$$817$$ 0 0
$$818$$ −19.4123 −0.678734
$$819$$ −18.9981 −0.663846
$$820$$ −8.45972 −0.295426
$$821$$ −38.0807 −1.32903 −0.664513 0.747277i $$-0.731361\pi$$
−0.664513 + 0.747277i $$0.731361\pi$$
$$822$$ 7.83950 0.273434
$$823$$ 53.7932 1.87511 0.937556 0.347835i $$-0.113083\pi$$
0.937556 + 0.347835i $$0.113083\pi$$
$$824$$ −6.58858 −0.229524
$$825$$ −0.0509824 −0.00177498
$$826$$ −14.3280 −0.498535
$$827$$ 32.4092 1.12698 0.563490 0.826123i $$-0.309458\pi$$
0.563490 + 0.826123i $$0.309458\pi$$
$$828$$ 12.2024 0.424064
$$829$$ −18.5305 −0.643592 −0.321796 0.946809i $$-0.604287\pi$$
−0.321796 + 0.946809i $$0.604287\pi$$
$$830$$ 8.17302 0.283690
$$831$$ 1.67091 0.0579633
$$832$$ 28.6255 0.992412
$$833$$ 5.67525 0.196636
$$834$$ 7.56804 0.262060
$$835$$ −3.40320 −0.117773
$$836$$ 0 0
$$837$$ 14.5830 0.504062
$$838$$ 1.30301 0.0450118
$$839$$ 0.826608 0.0285377 0.0142688 0.999898i $$-0.495458\pi$$
0.0142688 + 0.999898i $$0.495458\pi$$
$$840$$ 2.20317 0.0760165
$$841$$ −4.29321 −0.148042
$$842$$ 36.6185 1.26196
$$843$$ 12.4123 0.427504
$$844$$ 10.2045 0.351253
$$845$$ −0.653675 −0.0224871
$$846$$ 34.8401 1.19783
$$847$$ −20.7895 −0.714337
$$848$$ 0.470294 0.0161500
$$849$$ 0.504106 0.0173009
$$850$$ −1.82310 −0.0625317
$$851$$ −9.11901 −0.312596
$$852$$ 0.400908 0.0137349
$$853$$ −7.34938 −0.251638 −0.125819 0.992053i $$-0.540156\pi$$
−0.125819 + 0.992053i $$0.540156\pi$$
$$854$$ 19.0631 0.652327
$$855$$ 0 0
$$856$$ 3.07087 0.104960
$$857$$ −45.3496 −1.54911 −0.774556 0.632506i $$-0.782026\pi$$
−0.774556 + 0.632506i $$0.782026\pi$$
$$858$$ −0.196594 −0.00671160
$$859$$ 29.6285 1.01091 0.505456 0.862852i $$-0.331324\pi$$
0.505456 + 0.862852i $$0.331324\pi$$
$$860$$ −6.76909 −0.230824
$$861$$ −7.63542 −0.260215
$$862$$ −6.77775 −0.230851
$$863$$ 41.0807 1.39840 0.699201 0.714925i $$-0.253539\pi$$
0.699201 + 0.714925i $$0.253539\pi$$
$$864$$ 9.30180 0.316454
$$865$$ −10.5857 −0.359925
$$866$$ −20.3632 −0.691970
$$867$$ 5.40183 0.183456
$$868$$ −9.88764 −0.335608
$$869$$ −0.197202 −0.00668962
$$870$$ −2.06923 −0.0701536
$$871$$ −10.3780 −0.351644
$$872$$ −49.8951 −1.68966
$$873$$ −49.8931 −1.68863
$$874$$ 0 0
$$875$$ 1.89307 0.0639974
$$876$$ −1.91419 −0.0646743
$$877$$ 54.6307 1.84475 0.922374 0.386298i $$-0.126246\pi$$
0.922374 + 0.386298i $$0.126246\pi$$
$$878$$ −0.249460 −0.00841888
$$879$$ 2.92984 0.0988211
$$880$$ −0.238656 −0.00804509
$$881$$ −15.4805 −0.521552 −0.260776 0.965399i $$-0.583978\pi$$
−0.260776 + 0.965399i $$0.583978\pi$$
$$882$$ 10.7080 0.360558
$$883$$ −45.3495 −1.52613 −0.763066 0.646321i $$-0.776307\pi$$
−0.763066 + 0.646321i $$0.776307\pi$$
$$884$$ 4.64418 0.156201
$$885$$ 2.61614 0.0879406
$$886$$ −38.3913 −1.28978
$$887$$ −31.1399 −1.04558 −0.522788 0.852463i $$-0.675108\pi$$
−0.522788 + 0.852463i $$0.675108\pi$$
$$888$$ −1.97637 −0.0663226
$$889$$ 35.4755 1.18981
$$890$$ 10.6847 0.358153
$$891$$ 1.03833 0.0347853
$$892$$ −16.2264 −0.543301
$$893$$ 0 0
$$894$$ 9.26438 0.309847
$$895$$ −14.6024 −0.488105
$$896$$ 1.07133 0.0357905
$$897$$ −7.15734 −0.238977
$$898$$ −18.6025 −0.620775
$$899$$ 32.6305 1.08829
$$900$$ 2.27240 0.0757467
$$901$$ 0.439972 0.0146576
$$902$$ 1.56828 0.0522181
$$903$$ −6.10952 −0.203312
$$904$$ −2.58638 −0.0860218
$$905$$ −5.43261 −0.180586
$$906$$ −1.38827 −0.0461222
$$907$$ 43.5415 1.44577 0.722886 0.690967i $$-0.242815\pi$$
0.722886 + 0.690967i $$0.242815\pi$$
$$908$$ −20.2227 −0.671113
$$909$$ −15.4153 −0.511293
$$910$$ 7.29987 0.241988
$$911$$ 5.72789 0.189774 0.0948868 0.995488i $$-0.469751\pi$$
0.0948868 + 0.995488i $$0.469751\pi$$
$$912$$ 0 0
$$913$$ 1.00093 0.0331258
$$914$$ 1.75855 0.0581677
$$915$$ −3.48072 −0.115069
$$916$$ −1.76274 −0.0582425
$$917$$ 5.46422 0.180445
$$918$$ −4.04986 −0.133665
$$919$$ 7.93860 0.261870 0.130935 0.991391i $$-0.458202\pi$$
0.130935 + 0.991391i $$0.458202\pi$$
$$920$$ −16.4748 −0.543159
$$921$$ 3.45195 0.113746
$$922$$ −9.59339 −0.315941
$$923$$ 4.66747 0.153632
$$924$$ 0.0767887 0.00252616
$$925$$ −1.69819 −0.0558362
$$926$$ 23.2532 0.764147
$$927$$ −6.13347 −0.201450
$$928$$ 20.8135 0.683236
$$929$$ −28.9567 −0.950039 −0.475019 0.879975i $$-0.657559\pi$$
−0.475019 + 0.879975i $$0.657559\pi$$
$$930$$ −2.73286 −0.0896141
$$931$$ 0 0
$$932$$ 11.2642 0.368972
$$933$$ 4.72531 0.154700
$$934$$ 22.4444 0.734403
$$935$$ −0.223269 −0.00730168
$$936$$ 30.7895 1.00639
$$937$$ −14.6816 −0.479627 −0.239813 0.970819i $$-0.577086\pi$$
−0.239813 + 0.970819i $$0.577086\pi$$
$$938$$ −6.13606 −0.200349
$$939$$ −0.775939 −0.0253218
$$940$$ 8.84375 0.288451
$$941$$ −0.154755 −0.00504485 −0.00252243 0.999997i $$-0.500803\pi$$
−0.00252243 + 0.999997i $$0.500803\pi$$
$$942$$ −3.02352 −0.0985114
$$943$$ 57.0961 1.85931
$$944$$ 12.2465 0.398590
$$945$$ 4.20530 0.136798
$$946$$ 1.25487 0.0407993
$$947$$ −7.51807 −0.244304 −0.122152 0.992511i $$-0.538980\pi$$
−0.122152 + 0.992511i $$0.538980\pi$$
$$948$$ −0.442838 −0.0143827
$$949$$ −22.2854 −0.723415
$$950$$ 0 0
$$951$$ −8.94137 −0.289944
$$952$$ 9.64840 0.312706
$$953$$ 22.6743 0.734493 0.367247 0.930124i $$-0.380300\pi$$
0.367247 + 0.930124i $$0.380300\pi$$
$$954$$ 0.830138 0.0268767
$$955$$ 20.4758 0.662580
$$956$$ −2.39863 −0.0775772
$$957$$ −0.253413 −0.00819167
$$958$$ 25.8437 0.834973
$$959$$ −35.6494 −1.15118
$$960$$ −3.09034 −0.0997403
$$961$$ 12.0955 0.390177
$$962$$ −6.54840 −0.211129
$$963$$ 2.85875 0.0921219
$$964$$ 18.9194 0.609354
$$965$$ 11.0235 0.354860
$$966$$ −4.23184 −0.136157
$$967$$ 28.8344 0.927253 0.463627 0.886031i $$-0.346548\pi$$
0.463627 + 0.886031i $$0.346548\pi$$
$$968$$ 33.6929 1.08293
$$969$$ 0 0
$$970$$ 19.1711 0.615546
$$971$$ 26.3661 0.846130 0.423065 0.906099i $$-0.360954\pi$$
0.423065 + 0.906099i $$0.360954\pi$$
$$972$$ 7.63395 0.244859
$$973$$ −34.4150 −1.10329
$$974$$ 39.5508 1.26729
$$975$$ −1.33288 −0.0426863
$$976$$ −16.2938 −0.521551
$$977$$ 4.59218 0.146917 0.0734585 0.997298i $$-0.476596\pi$$
0.0734585 + 0.997298i $$0.476596\pi$$
$$978$$ 8.23362 0.263282
$$979$$ 1.30853 0.0418207
$$980$$ 2.71810 0.0868267
$$981$$ −46.4486 −1.48299
$$982$$ 22.0140 0.702496
$$983$$ 6.91473 0.220546 0.110273 0.993901i $$-0.464828\pi$$
0.110273 + 0.993901i $$0.464828\pi$$
$$984$$ 12.3745 0.394484
$$985$$ −19.8532 −0.632577
$$986$$ −9.06187 −0.288589
$$987$$ 7.98203 0.254071
$$988$$ 0 0
$$989$$ 45.6857 1.45272
$$990$$ −0.421263 −0.0133886
$$991$$ −38.0070 −1.20733 −0.603666 0.797237i $$-0.706294\pi$$
−0.603666 + 0.797237i $$0.706294\pi$$
$$992$$ 27.4886 0.872763
$$993$$ 7.10017 0.225317
$$994$$ 2.75968 0.0875318
$$995$$ 21.0026 0.665827
$$996$$ 2.24768 0.0712205
$$997$$ 16.2265 0.513899 0.256950 0.966425i $$-0.417283\pi$$
0.256950 + 0.966425i $$0.417283\pi$$
$$998$$ −40.4657 −1.28092
$$999$$ −3.77239 −0.119353
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 1805.2.a.o.1.3 4
5.4 even 2 9025.2.a.bg.1.2 4
19.7 even 3 95.2.e.c.11.2 8
19.11 even 3 95.2.e.c.26.2 yes 8
19.18 odd 2 1805.2.a.i.1.2 4
57.11 odd 6 855.2.k.h.406.3 8
57.26 odd 6 855.2.k.h.676.3 8
76.7 odd 6 1520.2.q.o.961.2 8
76.11 odd 6 1520.2.q.o.881.2 8
95.7 odd 12 475.2.j.c.49.4 16
95.49 even 6 475.2.e.e.26.3 8
95.64 even 6 475.2.e.e.201.3 8
95.68 odd 12 475.2.j.c.349.4 16
95.83 odd 12 475.2.j.c.49.5 16
95.87 odd 12 475.2.j.c.349.5 16
95.94 odd 2 9025.2.a.bp.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.2 8 19.7 even 3
95.2.e.c.26.2 yes 8 19.11 even 3
475.2.e.e.26.3 8 95.49 even 6
475.2.e.e.201.3 8 95.64 even 6
475.2.j.c.49.4 16 95.7 odd 12
475.2.j.c.49.5 16 95.83 odd 12
475.2.j.c.349.4 16 95.68 odd 12
475.2.j.c.349.5 16 95.87 odd 12
855.2.k.h.406.3 8 57.11 odd 6
855.2.k.h.676.3 8 57.26 odd 6
1520.2.q.o.881.2 8 76.11 odd 6
1520.2.q.o.961.2 8 76.7 odd 6
1805.2.a.i.1.2 4 19.18 odd 2
1805.2.a.o.1.3 4 1.1 even 1 trivial
9025.2.a.bg.1.2 4 5.4 even 2
9025.2.a.bp.1.3 4 95.94 odd 2