# Properties

 Label 4205.2.a.r.1.11 Level $4205$ Weight $2$ Character 4205.1 Self dual yes Analytic conductor $33.577$ Analytic rank $1$ Dimension $12$ 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: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 62 x^{5} + 156 x^{4} + \cdots - 1$$ x^12 - x^11 - 14*x^10 + 11*x^9 + 72*x^8 - 41*x^7 - 164*x^6 + 62*x^5 + 156*x^4 - 43*x^3 - 46*x^2 + 15*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 1.11 Root $$2.19433$$ 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)

 $$f(q)$$ $$=$$ $$q+2.19433 q^{2} -2.38477 q^{3} +2.81508 q^{4} -1.00000 q^{5} -5.23297 q^{6} +1.90240 q^{7} +1.78854 q^{8} +2.68713 q^{9} +O(q^{10})$$ $$q+2.19433 q^{2} -2.38477 q^{3} +2.81508 q^{4} -1.00000 q^{5} -5.23297 q^{6} +1.90240 q^{7} +1.78854 q^{8} +2.68713 q^{9} -2.19433 q^{10} -0.278924 q^{11} -6.71331 q^{12} -5.08002 q^{13} +4.17449 q^{14} +2.38477 q^{15} -1.70550 q^{16} +5.24937 q^{17} +5.89644 q^{18} +0.614162 q^{19} -2.81508 q^{20} -4.53679 q^{21} -0.612051 q^{22} +3.94079 q^{23} -4.26526 q^{24} +1.00000 q^{25} -11.1472 q^{26} +0.746132 q^{27} +5.35540 q^{28} +5.23297 q^{30} +5.88337 q^{31} -7.31952 q^{32} +0.665170 q^{33} +11.5189 q^{34} -1.90240 q^{35} +7.56446 q^{36} -10.7441 q^{37} +1.34767 q^{38} +12.1147 q^{39} -1.78854 q^{40} -12.3385 q^{41} -9.95519 q^{42} -4.89423 q^{43} -0.785193 q^{44} -2.68713 q^{45} +8.64739 q^{46} +4.17785 q^{47} +4.06723 q^{48} -3.38087 q^{49} +2.19433 q^{50} -12.5185 q^{51} -14.3007 q^{52} -2.85801 q^{53} +1.63726 q^{54} +0.278924 q^{55} +3.40252 q^{56} -1.46463 q^{57} -9.74727 q^{59} +6.71331 q^{60} +3.95531 q^{61} +12.9100 q^{62} +5.11199 q^{63} -12.6504 q^{64} +5.08002 q^{65} +1.45960 q^{66} -15.0867 q^{67} +14.7774 q^{68} -9.39788 q^{69} -4.17449 q^{70} -2.34551 q^{71} +4.80604 q^{72} +5.96936 q^{73} -23.5761 q^{74} -2.38477 q^{75} +1.72891 q^{76} -0.530625 q^{77} +26.5836 q^{78} +14.6401 q^{79} +1.70550 q^{80} -9.84073 q^{81} -27.0747 q^{82} +6.75685 q^{83} -12.7714 q^{84} -5.24937 q^{85} -10.7395 q^{86} -0.498868 q^{88} -11.3728 q^{89} -5.89644 q^{90} -9.66424 q^{91} +11.0936 q^{92} -14.0305 q^{93} +9.16757 q^{94} -0.614162 q^{95} +17.4554 q^{96} +4.74472 q^{97} -7.41875 q^{98} -0.749505 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{2} - q^{3} + 5 q^{4} - 12 q^{5} - 9 q^{6} - 5 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10})$$ 12 * q + q^2 - q^3 + 5 * q^4 - 12 * q^5 - 9 * q^6 - 5 * q^7 + 6 * q^8 + 7 * q^9 $$12 q + q^{2} - q^{3} + 5 q^{4} - 12 q^{5} - 9 q^{6} - 5 q^{7} + 6 q^{8} + 7 q^{9} - q^{10} + q^{11} - q^{12} - 5 q^{13} - 5 q^{14} + q^{15} - 9 q^{16} + 20 q^{17} - 5 q^{18} - 8 q^{19} - 5 q^{20} + 3 q^{22} - 8 q^{23} + 10 q^{24} + 12 q^{25} - 17 q^{26} - 25 q^{27} + 10 q^{28} + 9 q^{30} - 7 q^{31} + 3 q^{32} + q^{33} - 8 q^{34} + 5 q^{35} - 8 q^{36} + 14 q^{38} + 29 q^{39} - 6 q^{40} - 4 q^{41} + 13 q^{42} - 15 q^{43} + 2 q^{44} - 7 q^{45} + 24 q^{46} - 39 q^{47} + 2 q^{48} - 19 q^{49} + q^{50} - 32 q^{51} - 32 q^{52} + 12 q^{53} - 34 q^{54} - q^{55} - 19 q^{56} + 10 q^{57} - 19 q^{59} + q^{60} + 28 q^{61} - 13 q^{62} - 40 q^{63} - 34 q^{64} + 5 q^{65} + 48 q^{66} - 38 q^{67} + 18 q^{68} + 18 q^{69} + 5 q^{70} - 20 q^{71} - 6 q^{72} - 5 q^{73} - 12 q^{74} - q^{75} - 19 q^{76} + 32 q^{77} - 14 q^{78} + 13 q^{79} + 9 q^{80} - 32 q^{81} - 34 q^{82} + 15 q^{83} - 32 q^{84} - 20 q^{85} - 9 q^{86} - 10 q^{88} - 22 q^{89} + 5 q^{90} - 46 q^{91} - 31 q^{92} - 6 q^{93} - 13 q^{94} + 8 q^{95} - 3 q^{96} + 53 q^{97} - 8 q^{98} - 44 q^{99}+O(q^{100})$$ 12 * q + q^2 - q^3 + 5 * q^4 - 12 * q^5 - 9 * q^6 - 5 * q^7 + 6 * q^8 + 7 * q^9 - q^10 + q^11 - q^12 - 5 * q^13 - 5 * q^14 + q^15 - 9 * q^16 + 20 * q^17 - 5 * q^18 - 8 * q^19 - 5 * q^20 + 3 * q^22 - 8 * q^23 + 10 * q^24 + 12 * q^25 - 17 * q^26 - 25 * q^27 + 10 * q^28 + 9 * q^30 - 7 * q^31 + 3 * q^32 + q^33 - 8 * q^34 + 5 * q^35 - 8 * q^36 + 14 * q^38 + 29 * q^39 - 6 * q^40 - 4 * q^41 + 13 * q^42 - 15 * q^43 + 2 * q^44 - 7 * q^45 + 24 * q^46 - 39 * q^47 + 2 * q^48 - 19 * q^49 + q^50 - 32 * q^51 - 32 * q^52 + 12 * q^53 - 34 * q^54 - q^55 - 19 * q^56 + 10 * q^57 - 19 * q^59 + q^60 + 28 * q^61 - 13 * q^62 - 40 * q^63 - 34 * q^64 + 5 * q^65 + 48 * q^66 - 38 * q^67 + 18 * q^68 + 18 * q^69 + 5 * q^70 - 20 * q^71 - 6 * q^72 - 5 * q^73 - 12 * q^74 - q^75 - 19 * q^76 + 32 * q^77 - 14 * q^78 + 13 * q^79 + 9 * q^80 - 32 * q^81 - 34 * q^82 + 15 * q^83 - 32 * q^84 - 20 * q^85 - 9 * q^86 - 10 * q^88 - 22 * q^89 + 5 * q^90 - 46 * q^91 - 31 * q^92 - 6 * q^93 - 13 * q^94 + 8 * q^95 - 3 * q^96 + 53 * q^97 - 8 * q^98 - 44 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.19433 1.55162 0.775812 0.630964i $$-0.217340\pi$$
0.775812 + 0.630964i $$0.217340\pi$$
$$3$$ −2.38477 −1.37685 −0.688424 0.725309i $$-0.741697\pi$$
−0.688424 + 0.725309i $$0.741697\pi$$
$$4$$ 2.81508 1.40754
$$5$$ −1.00000 −0.447214
$$6$$ −5.23297 −2.13635
$$7$$ 1.90240 0.719040 0.359520 0.933137i $$-0.382941\pi$$
0.359520 + 0.933137i $$0.382941\pi$$
$$8$$ 1.78854 0.632345
$$9$$ 2.68713 0.895709
$$10$$ −2.19433 −0.693907
$$11$$ −0.278924 −0.0840988 −0.0420494 0.999116i $$-0.513389\pi$$
−0.0420494 + 0.999116i $$0.513389\pi$$
$$12$$ −6.71331 −1.93796
$$13$$ −5.08002 −1.40895 −0.704473 0.709731i $$-0.748816\pi$$
−0.704473 + 0.709731i $$0.748816\pi$$
$$14$$ 4.17449 1.11568
$$15$$ 2.38477 0.615745
$$16$$ −1.70550 −0.426375
$$17$$ 5.24937 1.27316 0.636580 0.771210i $$-0.280348\pi$$
0.636580 + 0.771210i $$0.280348\pi$$
$$18$$ 5.89644 1.38980
$$19$$ 0.614162 0.140898 0.0704492 0.997515i $$-0.477557\pi$$
0.0704492 + 0.997515i $$0.477557\pi$$
$$20$$ −2.81508 −0.629470
$$21$$ −4.53679 −0.990008
$$22$$ −0.612051 −0.130490
$$23$$ 3.94079 0.821712 0.410856 0.911700i $$-0.365230\pi$$
0.410856 + 0.911700i $$0.365230\pi$$
$$24$$ −4.26526 −0.870643
$$25$$ 1.00000 0.200000
$$26$$ −11.1472 −2.18615
$$27$$ 0.746132 0.143593
$$28$$ 5.35540 1.01208
$$29$$ 0 0
$$30$$ 5.23297 0.955405
$$31$$ 5.88337 1.05668 0.528342 0.849032i $$-0.322814\pi$$
0.528342 + 0.849032i $$0.322814\pi$$
$$32$$ −7.31952 −1.29392
$$33$$ 0.665170 0.115791
$$34$$ 11.5189 1.97547
$$35$$ −1.90240 −0.321564
$$36$$ 7.56446 1.26074
$$37$$ −10.7441 −1.76632 −0.883161 0.469070i $$-0.844589\pi$$
−0.883161 + 0.469070i $$0.844589\pi$$
$$38$$ 1.34767 0.218621
$$39$$ 12.1147 1.93990
$$40$$ −1.78854 −0.282793
$$41$$ −12.3385 −1.92695 −0.963474 0.267801i $$-0.913703\pi$$
−0.963474 + 0.267801i $$0.913703\pi$$
$$42$$ −9.95519 −1.53612
$$43$$ −4.89423 −0.746362 −0.373181 0.927758i $$-0.621733\pi$$
−0.373181 + 0.927758i $$0.621733\pi$$
$$44$$ −0.785193 −0.118372
$$45$$ −2.68713 −0.400573
$$46$$ 8.64739 1.27499
$$47$$ 4.17785 0.609402 0.304701 0.952448i $$-0.401444\pi$$
0.304701 + 0.952448i $$0.401444\pi$$
$$48$$ 4.06723 0.587054
$$49$$ −3.38087 −0.482982
$$50$$ 2.19433 0.310325
$$51$$ −12.5185 −1.75295
$$52$$ −14.3007 −1.98314
$$53$$ −2.85801 −0.392577 −0.196289 0.980546i $$-0.562889\pi$$
−0.196289 + 0.980546i $$0.562889\pi$$
$$54$$ 1.63726 0.222803
$$55$$ 0.278924 0.0376101
$$56$$ 3.40252 0.454681
$$57$$ −1.46463 −0.193996
$$58$$ 0 0
$$59$$ −9.74727 −1.26899 −0.634493 0.772928i $$-0.718791\pi$$
−0.634493 + 0.772928i $$0.718791\pi$$
$$60$$ 6.71331 0.866684
$$61$$ 3.95531 0.506426 0.253213 0.967411i $$-0.418513\pi$$
0.253213 + 0.967411i $$0.418513\pi$$
$$62$$ 12.9100 1.63958
$$63$$ 5.11199 0.644050
$$64$$ −12.6504 −1.58130
$$65$$ 5.08002 0.630099
$$66$$ 1.45960 0.179665
$$67$$ −15.0867 −1.84314 −0.921568 0.388218i $$-0.873091\pi$$
−0.921568 + 0.388218i $$0.873091\pi$$
$$68$$ 14.7774 1.79202
$$69$$ −9.39788 −1.13137
$$70$$ −4.17449 −0.498947
$$71$$ −2.34551 −0.278361 −0.139181 0.990267i $$-0.544447\pi$$
−0.139181 + 0.990267i $$0.544447\pi$$
$$72$$ 4.80604 0.566397
$$73$$ 5.96936 0.698661 0.349330 0.937000i $$-0.386409\pi$$
0.349330 + 0.937000i $$0.386409\pi$$
$$74$$ −23.5761 −2.74067
$$75$$ −2.38477 −0.275369
$$76$$ 1.72891 0.198320
$$77$$ −0.530625 −0.0604704
$$78$$ 26.5836 3.01000
$$79$$ 14.6401 1.64714 0.823572 0.567212i $$-0.191978\pi$$
0.823572 + 0.567212i $$0.191978\pi$$
$$80$$ 1.70550 0.190681
$$81$$ −9.84073 −1.09341
$$82$$ −27.0747 −2.98990
$$83$$ 6.75685 0.741661 0.370830 0.928701i $$-0.379073\pi$$
0.370830 + 0.928701i $$0.379073\pi$$
$$84$$ −12.7714 −1.39347
$$85$$ −5.24937 −0.569375
$$86$$ −10.7395 −1.15807
$$87$$ 0 0
$$88$$ −0.498868 −0.0531795
$$89$$ −11.3728 −1.20552 −0.602760 0.797923i $$-0.705932\pi$$
−0.602760 + 0.797923i $$0.705932\pi$$
$$90$$ −5.89644 −0.621539
$$91$$ −9.66424 −1.01309
$$92$$ 11.0936 1.15659
$$93$$ −14.0305 −1.45489
$$94$$ 9.16757 0.945562
$$95$$ −0.614162 −0.0630117
$$96$$ 17.4554 1.78153
$$97$$ 4.74472 0.481753 0.240877 0.970556i $$-0.422565\pi$$
0.240877 + 0.970556i $$0.422565\pi$$
$$98$$ −7.41875 −0.749407
$$99$$ −0.749505 −0.0753281
$$100$$ 2.81508 0.281508
$$101$$ −9.35820 −0.931176 −0.465588 0.885002i $$-0.654157\pi$$
−0.465588 + 0.885002i $$0.654157\pi$$
$$102$$ −27.4698 −2.71992
$$103$$ −5.60012 −0.551796 −0.275898 0.961187i $$-0.588975\pi$$
−0.275898 + 0.961187i $$0.588975\pi$$
$$104$$ −9.08584 −0.890940
$$105$$ 4.53679 0.442745
$$106$$ −6.27140 −0.609133
$$107$$ −14.7188 −1.42292 −0.711461 0.702726i $$-0.751966\pi$$
−0.711461 + 0.702726i $$0.751966\pi$$
$$108$$ 2.10042 0.202113
$$109$$ 4.98580 0.477553 0.238777 0.971075i $$-0.423254\pi$$
0.238777 + 0.971075i $$0.423254\pi$$
$$110$$ 0.612051 0.0583568
$$111$$ 25.6222 2.43196
$$112$$ −3.24455 −0.306581
$$113$$ 3.69371 0.347475 0.173738 0.984792i $$-0.444416\pi$$
0.173738 + 0.984792i $$0.444416\pi$$
$$114$$ −3.21389 −0.301008
$$115$$ −3.94079 −0.367481
$$116$$ 0 0
$$117$$ −13.6507 −1.26200
$$118$$ −21.3887 −1.96899
$$119$$ 9.98641 0.915453
$$120$$ 4.26526 0.389363
$$121$$ −10.9222 −0.992927
$$122$$ 8.67925 0.785782
$$123$$ 29.4245 2.65311
$$124$$ 16.5621 1.48732
$$125$$ −1.00000 −0.0894427
$$126$$ 11.2174 0.999324
$$127$$ 1.44963 0.128634 0.0643169 0.997930i $$-0.479513\pi$$
0.0643169 + 0.997930i $$0.479513\pi$$
$$128$$ −13.1201 −1.15967
$$129$$ 11.6716 1.02763
$$130$$ 11.1472 0.977678
$$131$$ 1.40660 0.122895 0.0614475 0.998110i $$-0.480428\pi$$
0.0614475 + 0.998110i $$0.480428\pi$$
$$132$$ 1.87250 0.162981
$$133$$ 1.16838 0.101312
$$134$$ −33.1052 −2.85985
$$135$$ −0.746132 −0.0642168
$$136$$ 9.38873 0.805077
$$137$$ 1.07830 0.0921253 0.0460626 0.998939i $$-0.485333\pi$$
0.0460626 + 0.998939i $$0.485333\pi$$
$$138$$ −20.6220 −1.75546
$$139$$ 5.31906 0.451157 0.225578 0.974225i $$-0.427573\pi$$
0.225578 + 0.974225i $$0.427573\pi$$
$$140$$ −5.35540 −0.452614
$$141$$ −9.96320 −0.839053
$$142$$ −5.14683 −0.431912
$$143$$ 1.41694 0.118491
$$144$$ −4.58290 −0.381908
$$145$$ 0 0
$$146$$ 13.0987 1.08406
$$147$$ 8.06261 0.664993
$$148$$ −30.2455 −2.48616
$$149$$ 0.677308 0.0554873 0.0277436 0.999615i $$-0.491168\pi$$
0.0277436 + 0.999615i $$0.491168\pi$$
$$150$$ −5.23297 −0.427270
$$151$$ −19.7723 −1.60904 −0.804522 0.593923i $$-0.797579\pi$$
−0.804522 + 0.593923i $$0.797579\pi$$
$$152$$ 1.09845 0.0890964
$$153$$ 14.1057 1.14038
$$154$$ −1.16437 −0.0938273
$$155$$ −5.88337 −0.472564
$$156$$ 34.1038 2.73049
$$157$$ −2.99894 −0.239342 −0.119671 0.992814i $$-0.538184\pi$$
−0.119671 + 0.992814i $$0.538184\pi$$
$$158$$ 32.1253 2.55575
$$159$$ 6.81569 0.540519
$$160$$ 7.31952 0.578658
$$161$$ 7.49696 0.590843
$$162$$ −21.5938 −1.69657
$$163$$ −15.9286 −1.24763 −0.623814 0.781573i $$-0.714418\pi$$
−0.623814 + 0.781573i $$0.714418\pi$$
$$164$$ −34.7338 −2.71225
$$165$$ −0.665170 −0.0517834
$$166$$ 14.8268 1.15078
$$167$$ −21.9924 −1.70182 −0.850910 0.525312i $$-0.823949\pi$$
−0.850910 + 0.525312i $$0.823949\pi$$
$$168$$ −8.11423 −0.626027
$$169$$ 12.8066 0.985127
$$170$$ −11.5189 −0.883455
$$171$$ 1.65033 0.126204
$$172$$ −13.7776 −1.05053
$$173$$ −5.36001 −0.407514 −0.203757 0.979021i $$-0.565315\pi$$
−0.203757 + 0.979021i $$0.565315\pi$$
$$174$$ 0 0
$$175$$ 1.90240 0.143808
$$176$$ 0.475706 0.0358577
$$177$$ 23.2450 1.74720
$$178$$ −24.9558 −1.87051
$$179$$ −10.4707 −0.782614 −0.391307 0.920260i $$-0.627977\pi$$
−0.391307 + 0.920260i $$0.627977\pi$$
$$180$$ −7.56446 −0.563822
$$181$$ −12.7994 −0.951370 −0.475685 0.879616i $$-0.657800\pi$$
−0.475685 + 0.879616i $$0.657800\pi$$
$$182$$ −21.2065 −1.57193
$$183$$ −9.43251 −0.697271
$$184$$ 7.04827 0.519605
$$185$$ 10.7441 0.789923
$$186$$ −30.7875 −2.25745
$$187$$ −1.46418 −0.107071
$$188$$ 11.7610 0.857756
$$189$$ 1.41944 0.103249
$$190$$ −1.34767 −0.0977704
$$191$$ −4.59594 −0.332550 −0.166275 0.986079i $$-0.553174\pi$$
−0.166275 + 0.986079i $$0.553174\pi$$
$$192$$ 30.1683 2.17721
$$193$$ 16.7814 1.20795 0.603975 0.797003i $$-0.293583\pi$$
0.603975 + 0.797003i $$0.293583\pi$$
$$194$$ 10.4115 0.747500
$$195$$ −12.1147 −0.867551
$$196$$ −9.51742 −0.679816
$$197$$ −11.6929 −0.833088 −0.416544 0.909116i $$-0.636759\pi$$
−0.416544 + 0.909116i $$0.636759\pi$$
$$198$$ −1.64466 −0.116881
$$199$$ −1.69617 −0.120238 −0.0601191 0.998191i $$-0.519148\pi$$
−0.0601191 + 0.998191i $$0.519148\pi$$
$$200$$ 1.78854 0.126469
$$201$$ 35.9783 2.53772
$$202$$ −20.5350 −1.44483
$$203$$ 0 0
$$204$$ −35.2407 −2.46734
$$205$$ 12.3385 0.861757
$$206$$ −12.2885 −0.856180
$$207$$ 10.5894 0.736014
$$208$$ 8.66399 0.600740
$$209$$ −0.171305 −0.0118494
$$210$$ 9.95519 0.686974
$$211$$ 10.3599 0.713204 0.356602 0.934256i $$-0.383935\pi$$
0.356602 + 0.934256i $$0.383935\pi$$
$$212$$ −8.04550 −0.552567
$$213$$ 5.59351 0.383261
$$214$$ −32.2979 −2.20784
$$215$$ 4.89423 0.333783
$$216$$ 1.33449 0.0908005
$$217$$ 11.1925 0.759798
$$218$$ 10.9405 0.740983
$$219$$ −14.2356 −0.961949
$$220$$ 0.785193 0.0529377
$$221$$ −26.6670 −1.79381
$$222$$ 56.2236 3.77348
$$223$$ 20.0329 1.34150 0.670750 0.741684i $$-0.265972\pi$$
0.670750 + 0.741684i $$0.265972\pi$$
$$224$$ −13.9246 −0.930379
$$225$$ 2.68713 0.179142
$$226$$ 8.10522 0.539151
$$227$$ 0.649740 0.0431248 0.0215624 0.999768i $$-0.493136\pi$$
0.0215624 + 0.999768i $$0.493136\pi$$
$$228$$ −4.12306 −0.273056
$$229$$ −3.10682 −0.205304 −0.102652 0.994717i $$-0.532733\pi$$
−0.102652 + 0.994717i $$0.532733\pi$$
$$230$$ −8.64739 −0.570192
$$231$$ 1.26542 0.0832585
$$232$$ 0 0
$$233$$ 22.2022 1.45451 0.727257 0.686365i $$-0.240795\pi$$
0.727257 + 0.686365i $$0.240795\pi$$
$$234$$ −29.9540 −1.95816
$$235$$ −4.17785 −0.272533
$$236$$ −27.4393 −1.78615
$$237$$ −34.9134 −2.26787
$$238$$ 21.9135 1.42044
$$239$$ 4.21375 0.272565 0.136282 0.990670i $$-0.456485\pi$$
0.136282 + 0.990670i $$0.456485\pi$$
$$240$$ −4.06723 −0.262538
$$241$$ −16.4052 −1.05675 −0.528377 0.849010i $$-0.677199\pi$$
−0.528377 + 0.849010i $$0.677199\pi$$
$$242$$ −23.9669 −1.54065
$$243$$ 21.2295 1.36187
$$244$$ 11.1345 0.712813
$$245$$ 3.38087 0.215996
$$246$$ 64.5669 4.11664
$$247$$ −3.11996 −0.198518
$$248$$ 10.5227 0.668189
$$249$$ −16.1135 −1.02115
$$250$$ −2.19433 −0.138781
$$251$$ 8.05930 0.508699 0.254349 0.967112i $$-0.418139\pi$$
0.254349 + 0.967112i $$0.418139\pi$$
$$252$$ 14.3906 0.906525
$$253$$ −1.09918 −0.0691050
$$254$$ 3.18096 0.199591
$$255$$ 12.5185 0.783942
$$256$$ −3.48903 −0.218064
$$257$$ 23.1144 1.44184 0.720918 0.693020i $$-0.243720\pi$$
0.720918 + 0.693020i $$0.243720\pi$$
$$258$$ 25.6113 1.59449
$$259$$ −20.4396 −1.27006
$$260$$ 14.3007 0.886889
$$261$$ 0 0
$$262$$ 3.08654 0.190687
$$263$$ −28.5353 −1.75956 −0.879781 0.475380i $$-0.842311\pi$$
−0.879781 + 0.475380i $$0.842311\pi$$
$$264$$ 1.18968 0.0732200
$$265$$ 2.85801 0.175566
$$266$$ 2.56381 0.157197
$$267$$ 27.1216 1.65982
$$268$$ −42.4702 −2.59428
$$269$$ −2.50846 −0.152944 −0.0764718 0.997072i $$-0.524366\pi$$
−0.0764718 + 0.997072i $$0.524366\pi$$
$$270$$ −1.63726 −0.0996404
$$271$$ 6.57485 0.399394 0.199697 0.979858i $$-0.436004\pi$$
0.199697 + 0.979858i $$0.436004\pi$$
$$272$$ −8.95282 −0.542844
$$273$$ 23.0470 1.39487
$$274$$ 2.36614 0.142944
$$275$$ −0.278924 −0.0168198
$$276$$ −26.4557 −1.59245
$$277$$ −7.85666 −0.472061 −0.236030 0.971746i $$-0.575847\pi$$
−0.236030 + 0.971746i $$0.575847\pi$$
$$278$$ 11.6718 0.700026
$$279$$ 15.8094 0.946481
$$280$$ −3.40252 −0.203340
$$281$$ −28.3025 −1.68838 −0.844192 0.536041i $$-0.819919\pi$$
−0.844192 + 0.536041i $$0.819919\pi$$
$$282$$ −21.8625 −1.30189
$$283$$ 28.0445 1.66707 0.833535 0.552467i $$-0.186313\pi$$
0.833535 + 0.552467i $$0.186313\pi$$
$$284$$ −6.60280 −0.391804
$$285$$ 1.46463 0.0867575
$$286$$ 3.10924 0.183853
$$287$$ −23.4727 −1.38555
$$288$$ −19.6685 −1.15898
$$289$$ 10.5559 0.620937
$$290$$ 0 0
$$291$$ −11.3151 −0.663301
$$292$$ 16.8042 0.983392
$$293$$ 24.5161 1.43225 0.716123 0.697974i $$-0.245915\pi$$
0.716123 + 0.697974i $$0.245915\pi$$
$$294$$ 17.6920 1.03182
$$295$$ 9.74727 0.567508
$$296$$ −19.2163 −1.11693
$$297$$ −0.208114 −0.0120760
$$298$$ 1.48624 0.0860954
$$299$$ −20.0193 −1.15775
$$300$$ −6.71331 −0.387593
$$301$$ −9.31077 −0.536664
$$302$$ −43.3868 −2.49663
$$303$$ 22.3171 1.28209
$$304$$ −1.04745 −0.0600756
$$305$$ −3.95531 −0.226480
$$306$$ 30.9526 1.76944
$$307$$ 14.1917 0.809964 0.404982 0.914325i $$-0.367278\pi$$
0.404982 + 0.914325i $$0.367278\pi$$
$$308$$ −1.49375 −0.0851143
$$309$$ 13.3550 0.759739
$$310$$ −12.9100 −0.733241
$$311$$ −14.4874 −0.821507 −0.410754 0.911746i $$-0.634734\pi$$
−0.410754 + 0.911746i $$0.634734\pi$$
$$312$$ 21.6676 1.22669
$$313$$ 0.295540 0.0167049 0.00835245 0.999965i $$-0.497341\pi$$
0.00835245 + 0.999965i $$0.497341\pi$$
$$314$$ −6.58067 −0.371368
$$315$$ −5.11199 −0.288028
$$316$$ 41.2131 2.31842
$$317$$ −25.0561 −1.40729 −0.703645 0.710551i $$-0.748446\pi$$
−0.703645 + 0.710551i $$0.748446\pi$$
$$318$$ 14.9559 0.838683
$$319$$ 0 0
$$320$$ 12.6504 0.707180
$$321$$ 35.1010 1.95915
$$322$$ 16.4508 0.916766
$$323$$ 3.22397 0.179386
$$324$$ −27.7024 −1.53902
$$325$$ −5.08002 −0.281789
$$326$$ −34.9527 −1.93585
$$327$$ −11.8900 −0.657518
$$328$$ −22.0679 −1.21850
$$329$$ 7.94794 0.438184
$$330$$ −1.45960 −0.0803484
$$331$$ 6.99239 0.384336 0.192168 0.981362i $$-0.438448\pi$$
0.192168 + 0.981362i $$0.438448\pi$$
$$332$$ 19.0210 1.04392
$$333$$ −28.8708 −1.58211
$$334$$ −48.2584 −2.64058
$$335$$ 15.0867 0.824275
$$336$$ 7.73750 0.422115
$$337$$ 26.5197 1.44462 0.722310 0.691570i $$-0.243081\pi$$
0.722310 + 0.691570i $$0.243081\pi$$
$$338$$ 28.1020 1.52855
$$339$$ −8.80866 −0.478421
$$340$$ −14.7774 −0.801416
$$341$$ −1.64101 −0.0888659
$$342$$ 3.62137 0.195821
$$343$$ −19.7486 −1.06632
$$344$$ −8.75353 −0.471959
$$345$$ 9.39788 0.505965
$$346$$ −11.7616 −0.632309
$$347$$ 9.55347 0.512857 0.256428 0.966563i $$-0.417454\pi$$
0.256428 + 0.966563i $$0.417454\pi$$
$$348$$ 0 0
$$349$$ −7.52546 −0.402829 −0.201414 0.979506i $$-0.564554\pi$$
−0.201414 + 0.979506i $$0.564554\pi$$
$$350$$ 4.17449 0.223136
$$351$$ −3.79037 −0.202315
$$352$$ 2.04159 0.108817
$$353$$ 5.81234 0.309360 0.154680 0.987965i $$-0.450565\pi$$
0.154680 + 0.987965i $$0.450565\pi$$
$$354$$ 51.0072 2.71100
$$355$$ 2.34551 0.124487
$$356$$ −32.0154 −1.69681
$$357$$ −23.8153 −1.26044
$$358$$ −22.9761 −1.21432
$$359$$ −9.24500 −0.487933 −0.243966 0.969784i $$-0.578449\pi$$
−0.243966 + 0.969784i $$0.578449\pi$$
$$360$$ −4.80604 −0.253300
$$361$$ −18.6228 −0.980148
$$362$$ −28.0860 −1.47617
$$363$$ 26.0469 1.36711
$$364$$ −27.2056 −1.42596
$$365$$ −5.96936 −0.312451
$$366$$ −20.6980 −1.08190
$$367$$ −21.5144 −1.12304 −0.561521 0.827463i $$-0.689784\pi$$
−0.561521 + 0.827463i $$0.689784\pi$$
$$368$$ −6.72102 −0.350358
$$369$$ −33.1551 −1.72598
$$370$$ 23.5761 1.22566
$$371$$ −5.43707 −0.282279
$$372$$ −39.4969 −2.04782
$$373$$ 6.92805 0.358721 0.179360 0.983783i $$-0.442597\pi$$
0.179360 + 0.983783i $$0.442597\pi$$
$$374$$ −3.21289 −0.166134
$$375$$ 2.38477 0.123149
$$376$$ 7.47226 0.385352
$$377$$ 0 0
$$378$$ 3.11472 0.160204
$$379$$ 12.4943 0.641789 0.320895 0.947115i $$-0.396016\pi$$
0.320895 + 0.947115i $$0.396016\pi$$
$$380$$ −1.72891 −0.0886913
$$381$$ −3.45703 −0.177109
$$382$$ −10.0850 −0.515993
$$383$$ −0.143189 −0.00731659 −0.00365830 0.999993i $$-0.501164\pi$$
−0.00365830 + 0.999993i $$0.501164\pi$$
$$384$$ 31.2885 1.59668
$$385$$ 0.530625 0.0270432
$$386$$ 36.8238 1.87428
$$387$$ −13.1514 −0.668523
$$388$$ 13.3567 0.678086
$$389$$ 21.5588 1.09307 0.546536 0.837435i $$-0.315946\pi$$
0.546536 + 0.837435i $$0.315946\pi$$
$$390$$ −26.5836 −1.34611
$$391$$ 20.6867 1.04617
$$392$$ −6.04684 −0.305411
$$393$$ −3.35441 −0.169208
$$394$$ −25.6582 −1.29264
$$395$$ −14.6401 −0.736625
$$396$$ −2.10991 −0.106027
$$397$$ 21.0868 1.05832 0.529158 0.848523i $$-0.322508\pi$$
0.529158 + 0.848523i $$0.322508\pi$$
$$398$$ −3.72195 −0.186564
$$399$$ −2.78632 −0.139490
$$400$$ −1.70550 −0.0852751
$$401$$ −1.09145 −0.0545045 −0.0272522 0.999629i $$-0.508676\pi$$
−0.0272522 + 0.999629i $$0.508676\pi$$
$$402$$ 78.9483 3.93758
$$403$$ −29.8877 −1.48881
$$404$$ −26.3440 −1.31066
$$405$$ 9.84073 0.488990
$$406$$ 0 0
$$407$$ 2.99679 0.148546
$$408$$ −22.3900 −1.10847
$$409$$ 32.4609 1.60509 0.802543 0.596594i $$-0.203480\pi$$
0.802543 + 0.596594i $$0.203480\pi$$
$$410$$ 27.0747 1.33712
$$411$$ −2.57149 −0.126842
$$412$$ −15.7647 −0.776673
$$413$$ −18.5432 −0.912452
$$414$$ 23.2366 1.14202
$$415$$ −6.75685 −0.331681
$$416$$ 37.1833 1.82306
$$417$$ −12.6847 −0.621174
$$418$$ −0.375899 −0.0183858
$$419$$ 32.5363 1.58950 0.794751 0.606936i $$-0.207601\pi$$
0.794751 + 0.606936i $$0.207601\pi$$
$$420$$ 12.7714 0.623180
$$421$$ 13.9080 0.677832 0.338916 0.940817i $$-0.389940\pi$$
0.338916 + 0.940817i $$0.389940\pi$$
$$422$$ 22.7330 1.10662
$$423$$ 11.2264 0.545846
$$424$$ −5.11167 −0.248244
$$425$$ 5.24937 0.254632
$$426$$ 12.2740 0.594677
$$427$$ 7.52458 0.364140
$$428$$ −41.4346 −2.00282
$$429$$ −3.37908 −0.163144
$$430$$ 10.7395 0.517906
$$431$$ 2.84230 0.136909 0.0684544 0.997654i $$-0.478193\pi$$
0.0684544 + 0.997654i $$0.478193\pi$$
$$432$$ −1.27253 −0.0612246
$$433$$ −22.9138 −1.10117 −0.550584 0.834780i $$-0.685595\pi$$
−0.550584 + 0.834780i $$0.685595\pi$$
$$434$$ 24.5601 1.17892
$$435$$ 0 0
$$436$$ 14.0354 0.672174
$$437$$ 2.42028 0.115778
$$438$$ −31.2375 −1.49258
$$439$$ −16.3250 −0.779148 −0.389574 0.920995i $$-0.627378\pi$$
−0.389574 + 0.920995i $$0.627378\pi$$
$$440$$ 0.498868 0.0237826
$$441$$ −9.08484 −0.432611
$$442$$ −58.5160 −2.78332
$$443$$ −16.7440 −0.795529 −0.397765 0.917488i $$-0.630214\pi$$
−0.397765 + 0.917488i $$0.630214\pi$$
$$444$$ 72.1285 3.42307
$$445$$ 11.3728 0.539125
$$446$$ 43.9587 2.08150
$$447$$ −1.61522 −0.0763975
$$448$$ −24.0661 −1.13702
$$449$$ −20.4370 −0.964480 −0.482240 0.876039i $$-0.660177\pi$$
−0.482240 + 0.876039i $$0.660177\pi$$
$$450$$ 5.89644 0.277961
$$451$$ 3.44150 0.162054
$$452$$ 10.3981 0.489085
$$453$$ 47.1523 2.21541
$$454$$ 1.42574 0.0669134
$$455$$ 9.66424 0.453066
$$456$$ −2.61956 −0.122672
$$457$$ 34.4661 1.61226 0.806128 0.591741i $$-0.201559\pi$$
0.806128 + 0.591741i $$0.201559\pi$$
$$458$$ −6.81738 −0.318555
$$459$$ 3.91673 0.182817
$$460$$ −11.0936 −0.517243
$$461$$ 24.8506 1.15741 0.578704 0.815538i $$-0.303559\pi$$
0.578704 + 0.815538i $$0.303559\pi$$
$$462$$ 2.77675 0.129186
$$463$$ −21.6764 −1.00739 −0.503693 0.863883i $$-0.668026\pi$$
−0.503693 + 0.863883i $$0.668026\pi$$
$$464$$ 0 0
$$465$$ 14.0305 0.650648
$$466$$ 48.7189 2.25686
$$467$$ 4.01458 0.185773 0.0928863 0.995677i $$-0.470391\pi$$
0.0928863 + 0.995677i $$0.470391\pi$$
$$468$$ −38.4277 −1.77632
$$469$$ −28.7010 −1.32529
$$470$$ −9.16757 −0.422868
$$471$$ 7.15179 0.329537
$$472$$ −17.4334 −0.802438
$$473$$ 1.36512 0.0627682
$$474$$ −76.6113 −3.51888
$$475$$ 0.614162 0.0281797
$$476$$ 28.1125 1.28853
$$477$$ −7.67982 −0.351635
$$478$$ 9.24635 0.422918
$$479$$ 18.3640 0.839074 0.419537 0.907738i $$-0.362192\pi$$
0.419537 + 0.907738i $$0.362192\pi$$
$$480$$ −17.4554 −0.796724
$$481$$ 54.5804 2.48865
$$482$$ −35.9985 −1.63969
$$483$$ −17.8785 −0.813501
$$484$$ −30.7468 −1.39758
$$485$$ −4.74472 −0.215447
$$486$$ 46.5844 2.11311
$$487$$ 9.09280 0.412034 0.206017 0.978548i $$-0.433950\pi$$
0.206017 + 0.978548i $$0.433950\pi$$
$$488$$ 7.07424 0.320236
$$489$$ 37.9861 1.71779
$$490$$ 7.41875 0.335145
$$491$$ 15.2185 0.686799 0.343400 0.939189i $$-0.388422\pi$$
0.343400 + 0.939189i $$0.388422\pi$$
$$492$$ 82.8321 3.73436
$$493$$ 0 0
$$494$$ −6.84621 −0.308026
$$495$$ 0.749505 0.0336877
$$496$$ −10.0341 −0.450544
$$497$$ −4.46211 −0.200153
$$498$$ −35.3584 −1.58445
$$499$$ −38.9176 −1.74219 −0.871096 0.491113i $$-0.836590\pi$$
−0.871096 + 0.491113i $$0.836590\pi$$
$$500$$ −2.81508 −0.125894
$$501$$ 52.4467 2.34315
$$502$$ 17.6848 0.789309
$$503$$ 30.4089 1.35586 0.677932 0.735124i $$-0.262876\pi$$
0.677932 + 0.735124i $$0.262876\pi$$
$$504$$ 9.14301 0.407262
$$505$$ 9.35820 0.416434
$$506$$ −2.41197 −0.107225
$$507$$ −30.5409 −1.35637
$$508$$ 4.08082 0.181057
$$509$$ 35.9878 1.59513 0.797565 0.603234i $$-0.206121\pi$$
0.797565 + 0.603234i $$0.206121\pi$$
$$510$$ 27.4698 1.21638
$$511$$ 11.3561 0.502365
$$512$$ 18.5842 0.821312
$$513$$ 0.458246 0.0202321
$$514$$ 50.7206 2.23719
$$515$$ 5.60012 0.246771
$$516$$ 32.8564 1.44642
$$517$$ −1.16530 −0.0512500
$$518$$ −44.8512 −1.97065
$$519$$ 12.7824 0.561085
$$520$$ 9.08584 0.398440
$$521$$ 14.2540 0.624477 0.312239 0.950004i $$-0.398921\pi$$
0.312239 + 0.950004i $$0.398921\pi$$
$$522$$ 0 0
$$523$$ 11.6757 0.510542 0.255271 0.966870i $$-0.417835\pi$$
0.255271 + 0.966870i $$0.417835\pi$$
$$524$$ 3.95968 0.172979
$$525$$ −4.53679 −0.198002
$$526$$ −62.6158 −2.73018
$$527$$ 30.8840 1.34533
$$528$$ −1.13445 −0.0493705
$$529$$ −7.47017 −0.324790
$$530$$ 6.27140 0.272412
$$531$$ −26.1921 −1.13664
$$532$$ 3.28908 0.142600
$$533$$ 62.6798 2.71496
$$534$$ 59.5137 2.57541
$$535$$ 14.7188 0.636350
$$536$$ −26.9832 −1.16550
$$537$$ 24.9701 1.07754
$$538$$ −5.50439 −0.237311
$$539$$ 0.943008 0.0406182
$$540$$ −2.10042 −0.0903876
$$541$$ −17.9562 −0.771996 −0.385998 0.922500i $$-0.626143\pi$$
−0.385998 + 0.922500i $$0.626143\pi$$
$$542$$ 14.4274 0.619709
$$543$$ 30.5236 1.30989
$$544$$ −38.4229 −1.64737
$$545$$ −4.98580 −0.213568
$$546$$ 50.5726 2.16431
$$547$$ −0.159361 −0.00681379 −0.00340689 0.999994i $$-0.501084\pi$$
−0.00340689 + 0.999994i $$0.501084\pi$$
$$548$$ 3.03549 0.129670
$$549$$ 10.6284 0.453610
$$550$$ −0.612051 −0.0260980
$$551$$ 0 0
$$552$$ −16.8085 −0.715417
$$553$$ 27.8514 1.18436
$$554$$ −17.2401 −0.732461
$$555$$ −25.6222 −1.08760
$$556$$ 14.9736 0.635020
$$557$$ 36.5737 1.54968 0.774838 0.632160i $$-0.217831\pi$$
0.774838 + 0.632160i $$0.217831\pi$$
$$558$$ 34.6909 1.46858
$$559$$ 24.8628 1.05158
$$560$$ 3.24455 0.137107
$$561$$ 3.49173 0.147421
$$562$$ −62.1049 −2.61974
$$563$$ 17.6202 0.742604 0.371302 0.928512i $$-0.378911\pi$$
0.371302 + 0.928512i $$0.378911\pi$$
$$564$$ −28.0472 −1.18100
$$565$$ −3.69371 −0.155396
$$566$$ 61.5387 2.58667
$$567$$ −18.7210 −0.786208
$$568$$ −4.19505 −0.176020
$$569$$ 28.6186 1.19975 0.599876 0.800093i $$-0.295216\pi$$
0.599876 + 0.800093i $$0.295216\pi$$
$$570$$ 3.21389 0.134615
$$571$$ −41.6130 −1.74145 −0.870725 0.491771i $$-0.836350\pi$$
−0.870725 + 0.491771i $$0.836350\pi$$
$$572$$ 3.98880 0.166780
$$573$$ 10.9603 0.457871
$$574$$ −51.5069 −2.14986
$$575$$ 3.94079 0.164342
$$576$$ −33.9933 −1.41639
$$577$$ −6.79516 −0.282886 −0.141443 0.989946i $$-0.545174\pi$$
−0.141443 + 0.989946i $$0.545174\pi$$
$$578$$ 23.1632 0.963462
$$579$$ −40.0197 −1.66316
$$580$$ 0 0
$$581$$ 12.8542 0.533284
$$582$$ −24.8290 −1.02919
$$583$$ 0.797167 0.0330153
$$584$$ 10.6765 0.441795
$$585$$ 13.6507 0.564386
$$586$$ 53.7964 2.22231
$$587$$ 4.82377 0.199098 0.0995491 0.995033i $$-0.468260\pi$$
0.0995491 + 0.995033i $$0.468260\pi$$
$$588$$ 22.6968 0.936002
$$589$$ 3.61334 0.148885
$$590$$ 21.3887 0.880559
$$591$$ 27.8850 1.14704
$$592$$ 18.3241 0.753116
$$593$$ 15.6524 0.642767 0.321384 0.946949i $$-0.395852\pi$$
0.321384 + 0.946949i $$0.395852\pi$$
$$594$$ −0.456671 −0.0187374
$$595$$ −9.98641 −0.409403
$$596$$ 1.90667 0.0781004
$$597$$ 4.04497 0.165550
$$598$$ −43.9289 −1.79639
$$599$$ −34.9062 −1.42623 −0.713114 0.701048i $$-0.752716\pi$$
−0.713114 + 0.701048i $$0.752716\pi$$
$$600$$ −4.26526 −0.174129
$$601$$ −38.0509 −1.55213 −0.776065 0.630653i $$-0.782787\pi$$
−0.776065 + 0.630653i $$0.782787\pi$$
$$602$$ −20.4309 −0.832701
$$603$$ −40.5399 −1.65091
$$604$$ −55.6604 −2.26479
$$605$$ 10.9222 0.444051
$$606$$ 48.9711 1.98932
$$607$$ 2.57433 0.104489 0.0522444 0.998634i $$-0.483363\pi$$
0.0522444 + 0.998634i $$0.483363\pi$$
$$608$$ −4.49537 −0.182311
$$609$$ 0 0
$$610$$ −8.67925 −0.351413
$$611$$ −21.2236 −0.858613
$$612$$ 39.7087 1.60513
$$613$$ 6.76425 0.273205 0.136603 0.990626i $$-0.456382\pi$$
0.136603 + 0.990626i $$0.456382\pi$$
$$614$$ 31.1413 1.25676
$$615$$ −29.4245 −1.18651
$$616$$ −0.949046 −0.0382382
$$617$$ 45.7702 1.84264 0.921320 0.388804i $$-0.127112\pi$$
0.921320 + 0.388804i $$0.127112\pi$$
$$618$$ 29.3052 1.17883
$$619$$ −26.0033 −1.04516 −0.522581 0.852590i $$-0.675031\pi$$
−0.522581 + 0.852590i $$0.675031\pi$$
$$620$$ −16.5621 −0.665151
$$621$$ 2.94035 0.117992
$$622$$ −31.7902 −1.27467
$$623$$ −21.6357 −0.866816
$$624$$ −20.6616 −0.827127
$$625$$ 1.00000 0.0400000
$$626$$ 0.648512 0.0259197
$$627$$ 0.408522 0.0163148
$$628$$ −8.44225 −0.336883
$$629$$ −56.3999 −2.24881
$$630$$ −11.2174 −0.446911
$$631$$ −47.6495 −1.89690 −0.948448 0.316934i $$-0.897347\pi$$
−0.948448 + 0.316934i $$0.897347\pi$$
$$632$$ 26.1845 1.04156
$$633$$ −24.7059 −0.981972
$$634$$ −54.9813 −2.18359
$$635$$ −1.44963 −0.0575268
$$636$$ 19.1867 0.760801
$$637$$ 17.1749 0.680495
$$638$$ 0 0
$$639$$ −6.30269 −0.249331
$$640$$ 13.1201 0.518619
$$641$$ 5.76089 0.227541 0.113771 0.993507i $$-0.463707\pi$$
0.113771 + 0.993507i $$0.463707\pi$$
$$642$$ 77.0231 3.03986
$$643$$ −5.48613 −0.216352 −0.108176 0.994132i $$-0.534501\pi$$
−0.108176 + 0.994132i $$0.534501\pi$$
$$644$$ 21.1045 0.831634
$$645$$ −11.6716 −0.459569
$$646$$ 7.07444 0.278340
$$647$$ 24.6781 0.970196 0.485098 0.874460i $$-0.338784\pi$$
0.485098 + 0.874460i $$0.338784\pi$$
$$648$$ −17.6006 −0.691416
$$649$$ 2.71875 0.106720
$$650$$ −11.1472 −0.437231
$$651$$ −26.6916 −1.04613
$$652$$ −44.8403 −1.75608
$$653$$ −28.0235 −1.09665 −0.548323 0.836267i $$-0.684733\pi$$
−0.548323 + 0.836267i $$0.684733\pi$$
$$654$$ −26.0905 −1.02022
$$655$$ −1.40660 −0.0549603
$$656$$ 21.0433 0.821603
$$657$$ 16.0404 0.625797
$$658$$ 17.4404 0.679897
$$659$$ 11.7014 0.455821 0.227910 0.973682i $$-0.426811\pi$$
0.227910 + 0.973682i $$0.426811\pi$$
$$660$$ −1.87250 −0.0728871
$$661$$ 14.4561 0.562277 0.281139 0.959667i $$-0.409288\pi$$
0.281139 + 0.959667i $$0.409288\pi$$
$$662$$ 15.3436 0.596346
$$663$$ 63.5945 2.46981
$$664$$ 12.0849 0.468986
$$665$$ −1.16838 −0.0453079
$$666$$ −63.3520 −2.45484
$$667$$ 0 0
$$668$$ −61.9101 −2.39537
$$669$$ −47.7738 −1.84704
$$670$$ 33.1052 1.27897
$$671$$ −1.10323 −0.0425898
$$672$$ 33.2071 1.28099
$$673$$ 38.0616 1.46717 0.733584 0.679599i $$-0.237846\pi$$
0.733584 + 0.679599i $$0.237846\pi$$
$$674$$ 58.1929 2.24151
$$675$$ 0.746132 0.0287186
$$676$$ 36.0517 1.38660
$$677$$ 37.2140 1.43025 0.715125 0.698996i $$-0.246370\pi$$
0.715125 + 0.698996i $$0.246370\pi$$
$$678$$ −19.3291 −0.742329
$$679$$ 9.02636 0.346400
$$680$$ −9.38873 −0.360041
$$681$$ −1.54948 −0.0593762
$$682$$ −3.60092 −0.137886
$$683$$ −23.7197 −0.907611 −0.453805 0.891101i $$-0.649934\pi$$
−0.453805 + 0.891101i $$0.649934\pi$$
$$684$$ 4.64580 0.177637
$$685$$ −1.07830 −0.0411997
$$686$$ −43.3348 −1.65453
$$687$$ 7.40905 0.282673
$$688$$ 8.34711 0.318231
$$689$$ 14.5187 0.553120
$$690$$ 20.6220 0.785067
$$691$$ −11.7426 −0.446710 −0.223355 0.974737i $$-0.571701\pi$$
−0.223355 + 0.974737i $$0.571701\pi$$
$$692$$ −15.0888 −0.573592
$$693$$ −1.42586 −0.0541638
$$694$$ 20.9634 0.795761
$$695$$ −5.31906 −0.201763
$$696$$ 0 0
$$697$$ −64.7694 −2.45331
$$698$$ −16.5133 −0.625039
$$699$$ −52.9471 −2.00264
$$700$$ 5.35540 0.202415
$$701$$ −6.99171 −0.264073 −0.132037 0.991245i $$-0.542152\pi$$
−0.132037 + 0.991245i $$0.542152\pi$$
$$702$$ −8.31732 −0.313917
$$703$$ −6.59863 −0.248872
$$704$$ 3.52851 0.132986
$$705$$ 9.96320 0.375236
$$706$$ 12.7542 0.480010
$$707$$ −17.8030 −0.669552
$$708$$ 65.4364 2.45925
$$709$$ −4.43231 −0.166459 −0.0832295 0.996530i $$-0.526523\pi$$
−0.0832295 + 0.996530i $$0.526523\pi$$
$$710$$ 5.14683 0.193157
$$711$$ 39.3399 1.47536
$$712$$ −20.3408 −0.762305
$$713$$ 23.1851 0.868290
$$714$$ −52.2586 −1.95573
$$715$$ −1.41694 −0.0529906
$$716$$ −29.4757 −1.10156
$$717$$ −10.0488 −0.375280
$$718$$ −20.2866 −0.757088
$$719$$ −2.21360 −0.0825532 −0.0412766 0.999148i $$-0.513142\pi$$
−0.0412766 + 0.999148i $$0.513142\pi$$
$$720$$ 4.58290 0.170795
$$721$$ −10.6537 −0.396763
$$722$$ −40.8645 −1.52082
$$723$$ 39.1227 1.45499
$$724$$ −36.0312 −1.33909
$$725$$ 0 0
$$726$$ 57.1555 2.12124
$$727$$ −38.5631 −1.43023 −0.715113 0.699009i $$-0.753625\pi$$
−0.715113 + 0.699009i $$0.753625\pi$$
$$728$$ −17.2849 −0.640621
$$729$$ −21.1052 −0.781675
$$730$$ −13.0987 −0.484806
$$731$$ −25.6916 −0.950239
$$732$$ −26.5532 −0.981435
$$733$$ −3.34900 −0.123698 −0.0618491 0.998086i $$-0.519700\pi$$
−0.0618491 + 0.998086i $$0.519700\pi$$
$$734$$ −47.2096 −1.74254
$$735$$ −8.06261 −0.297394
$$736$$ −28.8447 −1.06323
$$737$$ 4.20805 0.155006
$$738$$ −72.7531 −2.67808
$$739$$ 13.4141 0.493447 0.246723 0.969086i $$-0.420646\pi$$
0.246723 + 0.969086i $$0.420646\pi$$
$$740$$ 30.2455 1.11185
$$741$$ 7.44038 0.273329
$$742$$ −11.9307 −0.437990
$$743$$ 9.01319 0.330662 0.165331 0.986238i $$-0.447131\pi$$
0.165331 + 0.986238i $$0.447131\pi$$
$$744$$ −25.0941 −0.919995
$$745$$ −0.677308 −0.0248147
$$746$$ 15.2024 0.556600
$$747$$ 18.1565 0.664312
$$748$$ −4.12177 −0.150707
$$749$$ −28.0011 −1.02314
$$750$$ 5.23297 0.191081
$$751$$ −35.0914 −1.28050 −0.640252 0.768165i $$-0.721170\pi$$
−0.640252 + 0.768165i $$0.721170\pi$$
$$752$$ −7.12533 −0.259834
$$753$$ −19.2196 −0.700401
$$754$$ 0 0
$$755$$ 19.7723 0.719587
$$756$$ 3.99584 0.145327
$$757$$ −27.0450 −0.982966 −0.491483 0.870887i $$-0.663545\pi$$
−0.491483 + 0.870887i $$0.663545\pi$$
$$758$$ 27.4166 0.995816
$$759$$ 2.62130 0.0951470
$$760$$ −1.09845 −0.0398451
$$761$$ 2.00129 0.0725468 0.0362734 0.999342i $$-0.488451\pi$$
0.0362734 + 0.999342i $$0.488451\pi$$
$$762$$ −7.58586 −0.274807
$$763$$ 9.48499 0.343380
$$764$$ −12.9379 −0.468077
$$765$$ −14.1057 −0.509994
$$766$$ −0.314203 −0.0113526
$$767$$ 49.5164 1.78793
$$768$$ 8.32054 0.300242
$$769$$ −21.4788 −0.774545 −0.387273 0.921965i $$-0.626583\pi$$
−0.387273 + 0.921965i $$0.626583\pi$$
$$770$$ 1.16437 0.0419608
$$771$$ −55.1225 −1.98519
$$772$$ 47.2408 1.70023
$$773$$ −48.9146 −1.75934 −0.879668 0.475587i $$-0.842236\pi$$
−0.879668 + 0.475587i $$0.842236\pi$$
$$774$$ −28.8585 −1.03730
$$775$$ 5.88337 0.211337
$$776$$ 8.48613 0.304634
$$777$$ 48.7437 1.74867
$$778$$ 47.3070 1.69604
$$779$$ −7.57783 −0.271504
$$780$$ −34.1038 −1.22111
$$781$$ 0.654221 0.0234099
$$782$$ 45.3934 1.62326
$$783$$ 0 0
$$784$$ 5.76609 0.205932
$$785$$ 2.99894 0.107037
$$786$$ −7.36068 −0.262547
$$787$$ 40.3535 1.43845 0.719223 0.694779i $$-0.244498\pi$$
0.719223 + 0.694779i $$0.244498\pi$$
$$788$$ −32.9165 −1.17260
$$789$$ 68.0501 2.42265
$$790$$ −32.1253 −1.14297
$$791$$ 7.02692 0.249849
$$792$$ −1.34052 −0.0476333
$$793$$ −20.0931 −0.713526
$$794$$ 46.2714 1.64211
$$795$$ −6.81569 −0.241727
$$796$$ −4.77484 −0.169240
$$797$$ −3.92627 −0.139076 −0.0695378 0.997579i $$-0.522152\pi$$
−0.0695378 + 0.997579i $$0.522152\pi$$
$$798$$ −6.11410 −0.216437
$$799$$ 21.9311 0.775866
$$800$$ −7.31952 −0.258784
$$801$$ −30.5603 −1.07979
$$802$$ −2.39500 −0.0845705
$$803$$ −1.66500 −0.0587566
$$804$$ 101.282 3.57193
$$805$$ −7.49696 −0.264233
$$806$$ −65.5833 −2.31007
$$807$$ 5.98210 0.210580
$$808$$ −16.7375 −0.588824
$$809$$ 7.93781 0.279079 0.139539 0.990217i $$-0.455438\pi$$
0.139539 + 0.990217i $$0.455438\pi$$
$$810$$ 21.5938 0.758729
$$811$$ −21.1570 −0.742924 −0.371462 0.928448i $$-0.621143\pi$$
−0.371462 + 0.928448i $$0.621143\pi$$
$$812$$ 0 0
$$813$$ −15.6795 −0.549904
$$814$$ 6.57595 0.230487
$$815$$ 15.9286 0.557956
$$816$$ 21.3504 0.747414
$$817$$ −3.00585 −0.105161
$$818$$ 71.2298 2.49049
$$819$$ −25.9690 −0.907431
$$820$$ 34.7338 1.21296
$$821$$ 31.9265 1.11424 0.557121 0.830431i $$-0.311906\pi$$
0.557121 + 0.830431i $$0.311906\pi$$
$$822$$ −5.64270 −0.196812
$$823$$ −37.2104 −1.29707 −0.648536 0.761184i $$-0.724619\pi$$
−0.648536 + 0.761184i $$0.724619\pi$$
$$824$$ −10.0160 −0.348925
$$825$$ 0.665170 0.0231583
$$826$$ −40.6899 −1.41578
$$827$$ 7.49098 0.260487 0.130243 0.991482i $$-0.458424\pi$$
0.130243 + 0.991482i $$0.458424\pi$$
$$828$$ 29.8100 1.03597
$$829$$ −29.1578 −1.01269 −0.506346 0.862331i $$-0.669004\pi$$
−0.506346 + 0.862331i $$0.669004\pi$$
$$830$$ −14.8268 −0.514644
$$831$$ 18.7363 0.649956
$$832$$ 64.2644 2.22797
$$833$$ −17.7475 −0.614914
$$834$$ −27.8345 −0.963828
$$835$$ 21.9924 0.761077
$$836$$ −0.482235 −0.0166785
$$837$$ 4.38977 0.151733
$$838$$ 71.3953 2.46631
$$839$$ −2.95936 −0.102169 −0.0510843 0.998694i $$-0.516268\pi$$
−0.0510843 + 0.998694i $$0.516268\pi$$
$$840$$ 8.11423 0.279968
$$841$$ 0 0
$$842$$ 30.5186 1.05174
$$843$$ 67.4949 2.32465
$$844$$ 29.1638 1.00386
$$845$$ −12.8066 −0.440562
$$846$$ 24.6344 0.846948
$$847$$ −20.7784 −0.713954
$$848$$ 4.87433 0.167385
$$849$$ −66.8796 −2.29530
$$850$$ 11.5189 0.395093
$$851$$ −42.3403 −1.45141
$$852$$ 15.7462 0.539454
$$853$$ −27.2968 −0.934626 −0.467313 0.884092i $$-0.654778\pi$$
−0.467313 + 0.884092i $$0.654778\pi$$
$$854$$ 16.5114 0.565009
$$855$$ −1.65033 −0.0564401
$$856$$ −26.3252 −0.899777
$$857$$ −5.32546 −0.181914 −0.0909572 0.995855i $$-0.528993\pi$$
−0.0909572 + 0.995855i $$0.528993\pi$$
$$858$$ −7.41481 −0.253137
$$859$$ 17.6002 0.600511 0.300255 0.953859i $$-0.402928\pi$$
0.300255 + 0.953859i $$0.402928\pi$$
$$860$$ 13.7776 0.469813
$$861$$ 55.9771 1.90769
$$862$$ 6.23694 0.212431
$$863$$ 23.2283 0.790699 0.395350 0.918531i $$-0.370623\pi$$
0.395350 + 0.918531i $$0.370623\pi$$
$$864$$ −5.46133 −0.185798
$$865$$ 5.36001 0.182246
$$866$$ −50.2804 −1.70860
$$867$$ −25.1735 −0.854936
$$868$$ 31.5078 1.06944
$$869$$ −4.08349 −0.138523
$$870$$ 0 0
$$871$$ 76.6409 2.59688
$$872$$ 8.91732 0.301979
$$873$$ 12.7497 0.431511
$$874$$ 5.31089 0.179644
$$875$$ −1.90240 −0.0643129
$$876$$ −40.0742 −1.35398
$$877$$ −15.0318 −0.507589 −0.253794 0.967258i $$-0.581679\pi$$
−0.253794 + 0.967258i $$0.581679\pi$$
$$878$$ −35.8223 −1.20895
$$879$$ −58.4653 −1.97198
$$880$$ −0.475706 −0.0160360
$$881$$ −17.3625 −0.584957 −0.292479 0.956272i $$-0.594480\pi$$
−0.292479 + 0.956272i $$0.594480\pi$$
$$882$$ −19.9351 −0.671250
$$883$$ −12.6701 −0.426381 −0.213191 0.977011i $$-0.568386\pi$$
−0.213191 + 0.977011i $$0.568386\pi$$
$$884$$ −75.0695 −2.52486
$$885$$ −23.2450 −0.781372
$$886$$ −36.7417 −1.23436
$$887$$ 19.9338 0.669311 0.334655 0.942341i $$-0.391380\pi$$
0.334655 + 0.942341i $$0.391380\pi$$
$$888$$ 45.8265 1.53784
$$889$$ 2.75777 0.0924928
$$890$$ 24.9558 0.836519
$$891$$ 2.74482 0.0919549
$$892$$ 56.3940 1.88821
$$893$$ 2.56587 0.0858637
$$894$$ −3.54433 −0.118540
$$895$$ 10.4707 0.349996
$$896$$ −24.9597 −0.833846
$$897$$ 47.7414 1.59404
$$898$$ −44.8454 −1.49651
$$899$$ 0 0
$$900$$ 7.56446 0.252149
$$901$$ −15.0027 −0.499814
$$902$$ 7.55179 0.251447
$$903$$ 22.2041 0.738905
$$904$$ 6.60636 0.219724
$$905$$ 12.7994 0.425465
$$906$$ 103.468 3.43748
$$907$$ −7.09724 −0.235660 −0.117830 0.993034i $$-0.537594\pi$$
−0.117830 + 0.993034i $$0.537594\pi$$
$$908$$ 1.82907 0.0606997
$$909$$ −25.1467 −0.834062
$$910$$ 21.2065 0.702989
$$911$$ −25.7940 −0.854595 −0.427297 0.904111i $$-0.640534\pi$$
−0.427297 + 0.904111i $$0.640534\pi$$
$$912$$ 2.49794 0.0827150
$$913$$ −1.88465 −0.0623728
$$914$$ 75.6299 2.50162
$$915$$ 9.43251 0.311829
$$916$$ −8.74593 −0.288974
$$917$$ 2.67591 0.0883664
$$918$$ 8.59459 0.283664
$$919$$ 14.8805 0.490864 0.245432 0.969414i $$-0.421070\pi$$
0.245432 + 0.969414i $$0.421070\pi$$
$$920$$ −7.04827 −0.232375
$$921$$ −33.8440 −1.11520
$$922$$ 54.5303 1.79586
$$923$$ 11.9153 0.392196
$$924$$ 3.56225 0.117189
$$925$$ −10.7441 −0.353264
$$926$$ −47.5651 −1.56309
$$927$$ −15.0482 −0.494248
$$928$$ 0 0
$$929$$ −20.6522 −0.677576 −0.338788 0.940863i $$-0.610017\pi$$
−0.338788 + 0.940863i $$0.610017\pi$$
$$930$$ 30.7875 1.00956
$$931$$ −2.07640 −0.0680514
$$932$$ 62.5009 2.04728
$$933$$ 34.5492 1.13109
$$934$$ 8.80931 0.288249
$$935$$ 1.46418 0.0478837
$$936$$ −24.4148 −0.798023
$$937$$ −19.0828 −0.623408 −0.311704 0.950179i $$-0.600900\pi$$
−0.311704 + 0.950179i $$0.600900\pi$$
$$938$$ −62.9793 −2.05635
$$939$$ −0.704795 −0.0230001
$$940$$ −11.7610 −0.383600
$$941$$ 12.1701 0.396734 0.198367 0.980128i $$-0.436436\pi$$
0.198367 + 0.980128i $$0.436436\pi$$
$$942$$ 15.6934 0.511318
$$943$$ −48.6234 −1.58340
$$944$$ 16.6240 0.541065
$$945$$ −1.41944 −0.0461745
$$946$$ 2.99552 0.0973927
$$947$$ 38.9052 1.26425 0.632125 0.774867i $$-0.282183\pi$$
0.632125 + 0.774867i $$0.282183\pi$$
$$948$$ −98.2837 −3.19211
$$949$$ −30.3245 −0.984375
$$950$$ 1.34767 0.0437243
$$951$$ 59.7530 1.93762
$$952$$ 17.8611 0.578882
$$953$$ −56.5011 −1.83025 −0.915125 0.403170i $$-0.867908\pi$$
−0.915125 + 0.403170i $$0.867908\pi$$
$$954$$ −16.8521 −0.545605
$$955$$ 4.59594 0.148721
$$956$$ 11.8620 0.383645
$$957$$ 0 0
$$958$$ 40.2967 1.30193
$$959$$ 2.05136 0.0662417
$$960$$ −30.1683 −0.973678
$$961$$ 3.61402 0.116581
$$962$$ 119.767 3.86145
$$963$$ −39.5513 −1.27452
$$964$$ −46.1820 −1.48742
$$965$$ −16.7814 −0.540212
$$966$$ −39.2313 −1.26225
$$967$$ 51.5268 1.65699 0.828495 0.559996i $$-0.189197\pi$$
0.828495 + 0.559996i $$0.189197\pi$$
$$968$$ −19.5348 −0.627873
$$969$$ −7.68842 −0.246987
$$970$$ −10.4115 −0.334292
$$971$$ 9.69729 0.311201 0.155600 0.987820i $$-0.450269\pi$$
0.155600 + 0.987820i $$0.450269\pi$$
$$972$$ 59.7626 1.91689
$$973$$ 10.1190 0.324399
$$974$$ 19.9526 0.639322
$$975$$ 12.1147 0.387980
$$976$$ −6.74579 −0.215927
$$977$$ −9.97213 −0.319037 −0.159518 0.987195i $$-0.550994\pi$$
−0.159518 + 0.987195i $$0.550994\pi$$
$$978$$ 83.3541 2.66537
$$979$$ 3.17216 0.101383
$$980$$ 9.51742 0.304023
$$981$$ 13.3975 0.427749
$$982$$ 33.3943 1.06565
$$983$$ −57.0771 −1.82048 −0.910239 0.414084i $$-0.864102\pi$$
−0.910239 + 0.414084i $$0.864102\pi$$
$$984$$ 52.6269 1.67768
$$985$$ 11.6929 0.372568
$$986$$ 0 0
$$987$$ −18.9540 −0.603312
$$988$$ −8.78291 −0.279422
$$989$$ −19.2871 −0.613295
$$990$$ 1.64466 0.0522707
$$991$$ −30.9565 −0.983366 −0.491683 0.870774i $$-0.663618\pi$$
−0.491683 + 0.870774i $$0.663618\pi$$
$$992$$ −43.0634 −1.36726
$$993$$ −16.6752 −0.529173
$$994$$ −9.79133 −0.310562
$$995$$ 1.69617 0.0537721
$$996$$ −45.3608 −1.43731
$$997$$ 27.5065 0.871140 0.435570 0.900155i $$-0.356547\pi$$
0.435570 + 0.900155i $$0.356547\pi$$
$$998$$ −85.3980 −2.70323
$$999$$ −8.01653 −0.253632
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.r.1.11 12
29.9 even 14 145.2.k.a.81.1 24
29.13 even 14 145.2.k.a.111.1 yes 24
29.28 even 2 4205.2.a.q.1.2 12
145.9 even 14 725.2.l.d.226.4 24
145.13 odd 28 725.2.r.c.24.1 48
145.38 odd 28 725.2.r.c.574.8 48
145.42 odd 28 725.2.r.c.24.8 48
145.67 odd 28 725.2.r.c.574.1 48
145.129 even 14 725.2.l.d.401.4 24

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.k.a.81.1 24 29.9 even 14
145.2.k.a.111.1 yes 24 29.13 even 14
725.2.l.d.226.4 24 145.9 even 14
725.2.l.d.401.4 24 145.129 even 14
725.2.r.c.24.1 48 145.13 odd 28
725.2.r.c.24.8 48 145.42 odd 28
725.2.r.c.574.1 48 145.67 odd 28
725.2.r.c.574.8 48 145.38 odd 28
4205.2.a.q.1.2 12 29.28 even 2
4205.2.a.r.1.11 12 1.1 even 1 trivial