# Properties

 Label 169.2.a.b.1.3 Level $169$ Weight $2$ Character 169.1 Self dual yes Analytic conductor $1.349$ Analytic rank $1$ 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] = [169,2,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$1.34947179416$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 169.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +O(q^{10})$$ $$q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +0.198062 q^{10} -4.24698 q^{11} +3.04892 q^{12} -1.89008 q^{14} -0.554958 q^{15} +0.554958 q^{16} +2.15883 q^{17} +1.64310 q^{18} -0.0881460 q^{19} -0.335126 q^{20} +5.29590 q^{21} -3.40581 q^{22} +1.49396 q^{23} +6.04892 q^{24} -4.93900 q^{25} +2.13706 q^{27} +3.19806 q^{28} +4.63102 q^{29} -0.445042 q^{30} -6.63102 q^{31} +5.82908 q^{32} +9.54288 q^{33} +1.73125 q^{34} -0.582105 q^{35} -2.78017 q^{36} +5.69202 q^{37} -0.0706876 q^{38} -0.664874 q^{40} -11.5918 q^{41} +4.24698 q^{42} -0.295897 q^{43} +5.76271 q^{44} +0.506041 q^{45} +1.19806 q^{46} -7.35690 q^{47} -1.24698 q^{48} -1.44504 q^{49} -3.96077 q^{50} -4.85086 q^{51} -10.3937 q^{53} +1.71379 q^{54} -1.04892 q^{55} +6.34481 q^{56} +0.198062 q^{57} +3.71379 q^{58} -6.78017 q^{59} +0.753020 q^{60} +3.47219 q^{61} -5.31767 q^{62} -4.82908 q^{63} +3.56465 q^{64} +7.65279 q^{66} +7.67994 q^{67} -2.92931 q^{68} -3.35690 q^{69} -0.466812 q^{70} -8.66487 q^{71} -5.51573 q^{72} +6.73556 q^{73} +4.56465 q^{74} +11.0978 q^{75} +0.119605 q^{76} +10.0097 q^{77} +9.97046 q^{79} +0.137063 q^{80} -10.9487 q^{81} -9.29590 q^{82} +1.60925 q^{83} -7.18598 q^{84} +0.533188 q^{85} -0.237291 q^{86} -10.4058 q^{87} +11.4330 q^{88} -2.88471 q^{89} +0.405813 q^{90} -2.02715 q^{92} +14.8998 q^{93} -5.89977 q^{94} -0.0217703 q^{95} -13.0978 q^{96} -8.05861 q^{97} -1.15883 q^{98} -8.70171 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 $$3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 5 q^{10} - 8 q^{11} - 5 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 9 q^{18} - 4 q^{19} + 2 q^{21} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} + q^{27} + 14 q^{28} - q^{29} - q^{30} - 5 q^{31} + 7 q^{32} + 10 q^{33} + 13 q^{34} + 4 q^{35} - 7 q^{36} + 12 q^{37} + 12 q^{38} - 3 q^{40} - 7 q^{41} + 8 q^{42} + 13 q^{43} + 11 q^{45} + 8 q^{46} - 18 q^{47} + q^{48} - 4 q^{49} + q^{50} - q^{51} + q^{53} - 3 q^{54} + 6 q^{55} - 4 q^{56} + 5 q^{57} + 3 q^{58} - 19 q^{59} + 7 q^{60} + 4 q^{61} + q^{62} - 4 q^{63} - 11 q^{64} + 5 q^{66} - q^{67} - 21 q^{68} - 6 q^{69} + 2 q^{70} - 27 q^{71} - 4 q^{72} + 9 q^{73} - 8 q^{74} + 15 q^{75} - 21 q^{76} + 8 q^{77} - 5 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 7 q^{83} - 7 q^{84} + 5 q^{85} - 18 q^{86} - 18 q^{87} + 15 q^{88} - 11 q^{89} - 12 q^{90} + 22 q^{93} + 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} + 5 q^{98} + q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 5 * q^10 - 8 * q^11 - 5 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 9 * q^18 - 4 * q^19 + 2 * q^21 + 3 * q^22 - 5 * q^23 + 9 * q^24 - 5 * q^25 + q^27 + 14 * q^28 - q^29 - q^30 - 5 * q^31 + 7 * q^32 + 10 * q^33 + 13 * q^34 + 4 * q^35 - 7 * q^36 + 12 * q^37 + 12 * q^38 - 3 * q^40 - 7 * q^41 + 8 * q^42 + 13 * q^43 + 11 * q^45 + 8 * q^46 - 18 * q^47 + q^48 - 4 * q^49 + q^50 - q^51 + q^53 - 3 * q^54 + 6 * q^55 - 4 * q^56 + 5 * q^57 + 3 * q^58 - 19 * q^59 + 7 * q^60 + 4 * q^61 + q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 - 21 * q^68 - 6 * q^69 + 2 * q^70 - 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 15 * q^75 - 21 * q^76 + 8 * q^77 - 5 * q^79 - 5 * q^80 - q^81 - 14 * q^82 - 7 * q^83 - 7 * q^84 + 5 * q^85 - 18 * q^86 - 18 * q^87 + 15 * q^88 - 11 * q^89 - 12 * q^90 + 22 * q^93 + 5 * q^94 + 3 * q^95 - 21 * q^96 + 7 * q^97 + 5 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.801938 0.567056 0.283528 0.958964i $$-0.408495\pi$$
0.283528 + 0.958964i $$0.408495\pi$$
$$3$$ −2.24698 −1.29729 −0.648647 0.761089i $$-0.724665\pi$$
−0.648647 + 0.761089i $$0.724665\pi$$
$$4$$ −1.35690 −0.678448
$$5$$ 0.246980 0.110453 0.0552263 0.998474i $$-0.482412\pi$$
0.0552263 + 0.998474i $$0.482412\pi$$
$$6$$ −1.80194 −0.735638
$$7$$ −2.35690 −0.890823 −0.445411 0.895326i $$-0.646943\pi$$
−0.445411 + 0.895326i $$0.646943\pi$$
$$8$$ −2.69202 −0.951773
$$9$$ 2.04892 0.682972
$$10$$ 0.198062 0.0626328
$$11$$ −4.24698 −1.28051 −0.640256 0.768161i $$-0.721172\pi$$
−0.640256 + 0.768161i $$0.721172\pi$$
$$12$$ 3.04892 0.880147
$$13$$ 0 0
$$14$$ −1.89008 −0.505146
$$15$$ −0.554958 −0.143290
$$16$$ 0.554958 0.138740
$$17$$ 2.15883 0.523594 0.261797 0.965123i $$-0.415685\pi$$
0.261797 + 0.965123i $$0.415685\pi$$
$$18$$ 1.64310 0.387283
$$19$$ −0.0881460 −0.0202221 −0.0101110 0.999949i $$-0.503218\pi$$
−0.0101110 + 0.999949i $$0.503218\pi$$
$$20$$ −0.335126 −0.0749364
$$21$$ 5.29590 1.15566
$$22$$ −3.40581 −0.726122
$$23$$ 1.49396 0.311512 0.155756 0.987796i $$-0.450219\pi$$
0.155756 + 0.987796i $$0.450219\pi$$
$$24$$ 6.04892 1.23473
$$25$$ −4.93900 −0.987800
$$26$$ 0 0
$$27$$ 2.13706 0.411278
$$28$$ 3.19806 0.604377
$$29$$ 4.63102 0.859959 0.429980 0.902839i $$-0.358521\pi$$
0.429980 + 0.902839i $$0.358521\pi$$
$$30$$ −0.445042 −0.0812532
$$31$$ −6.63102 −1.19097 −0.595483 0.803368i $$-0.703039\pi$$
−0.595483 + 0.803368i $$0.703039\pi$$
$$32$$ 5.82908 1.03045
$$33$$ 9.54288 1.66120
$$34$$ 1.73125 0.296907
$$35$$ −0.582105 −0.0983937
$$36$$ −2.78017 −0.463361
$$37$$ 5.69202 0.935763 0.467881 0.883791i $$-0.345017\pi$$
0.467881 + 0.883791i $$0.345017\pi$$
$$38$$ −0.0706876 −0.0114670
$$39$$ 0 0
$$40$$ −0.664874 −0.105126
$$41$$ −11.5918 −1.81033 −0.905167 0.425056i $$-0.860254\pi$$
−0.905167 + 0.425056i $$0.860254\pi$$
$$42$$ 4.24698 0.655323
$$43$$ −0.295897 −0.0451239 −0.0225619 0.999745i $$-0.507182\pi$$
−0.0225619 + 0.999745i $$0.507182\pi$$
$$44$$ 5.76271 0.868761
$$45$$ 0.506041 0.0754361
$$46$$ 1.19806 0.176645
$$47$$ −7.35690 −1.07311 −0.536557 0.843864i $$-0.680275\pi$$
−0.536557 + 0.843864i $$0.680275\pi$$
$$48$$ −1.24698 −0.179986
$$49$$ −1.44504 −0.206435
$$50$$ −3.96077 −0.560138
$$51$$ −4.85086 −0.679256
$$52$$ 0 0
$$53$$ −10.3937 −1.42769 −0.713844 0.700304i $$-0.753048\pi$$
−0.713844 + 0.700304i $$0.753048\pi$$
$$54$$ 1.71379 0.233218
$$55$$ −1.04892 −0.141436
$$56$$ 6.34481 0.847861
$$57$$ 0.198062 0.0262340
$$58$$ 3.71379 0.487645
$$59$$ −6.78017 −0.882703 −0.441351 0.897334i $$-0.645501\pi$$
−0.441351 + 0.897334i $$0.645501\pi$$
$$60$$ 0.753020 0.0972145
$$61$$ 3.47219 0.444568 0.222284 0.974982i $$-0.428649\pi$$
0.222284 + 0.974982i $$0.428649\pi$$
$$62$$ −5.31767 −0.675344
$$63$$ −4.82908 −0.608407
$$64$$ 3.56465 0.445581
$$65$$ 0 0
$$66$$ 7.65279 0.941994
$$67$$ 7.67994 0.938254 0.469127 0.883131i $$-0.344569\pi$$
0.469127 + 0.883131i $$0.344569\pi$$
$$68$$ −2.92931 −0.355231
$$69$$ −3.35690 −0.404123
$$70$$ −0.466812 −0.0557947
$$71$$ −8.66487 −1.02833 −0.514166 0.857691i $$-0.671898\pi$$
−0.514166 + 0.857691i $$0.671898\pi$$
$$72$$ −5.51573 −0.650035
$$73$$ 6.73556 0.788338 0.394169 0.919038i $$-0.371032\pi$$
0.394169 + 0.919038i $$0.371032\pi$$
$$74$$ 4.56465 0.530629
$$75$$ 11.0978 1.28147
$$76$$ 0.119605 0.0137196
$$77$$ 10.0097 1.14071
$$78$$ 0 0
$$79$$ 9.97046 1.12176 0.560882 0.827896i $$-0.310462\pi$$
0.560882 + 0.827896i $$0.310462\pi$$
$$80$$ 0.137063 0.0153241
$$81$$ −10.9487 −1.21652
$$82$$ −9.29590 −1.02656
$$83$$ 1.60925 0.176638 0.0883192 0.996092i $$-0.471850\pi$$
0.0883192 + 0.996092i $$0.471850\pi$$
$$84$$ −7.18598 −0.784055
$$85$$ 0.533188 0.0578323
$$86$$ −0.237291 −0.0255877
$$87$$ −10.4058 −1.11562
$$88$$ 11.4330 1.21876
$$89$$ −2.88471 −0.305778 −0.152889 0.988243i $$-0.548858\pi$$
−0.152889 + 0.988243i $$0.548858\pi$$
$$90$$ 0.405813 0.0427765
$$91$$ 0 0
$$92$$ −2.02715 −0.211345
$$93$$ 14.8998 1.54503
$$94$$ −5.89977 −0.608515
$$95$$ −0.0217703 −0.00223358
$$96$$ −13.0978 −1.33679
$$97$$ −8.05861 −0.818227 −0.409114 0.912483i $$-0.634162\pi$$
−0.409114 + 0.912483i $$0.634162\pi$$
$$98$$ −1.15883 −0.117060
$$99$$ −8.70171 −0.874555
$$100$$ 6.70171 0.670171
$$101$$ −13.3545 −1.32882 −0.664411 0.747367i $$-0.731318\pi$$
−0.664411 + 0.747367i $$0.731318\pi$$
$$102$$ −3.89008 −0.385176
$$103$$ 1.36227 0.134229 0.0671144 0.997745i $$-0.478621\pi$$
0.0671144 + 0.997745i $$0.478621\pi$$
$$104$$ 0 0
$$105$$ 1.30798 0.127646
$$106$$ −8.33513 −0.809579
$$107$$ 3.26875 0.316002 0.158001 0.987439i $$-0.449495\pi$$
0.158001 + 0.987439i $$0.449495\pi$$
$$108$$ −2.89977 −0.279031
$$109$$ 15.7017 1.50395 0.751976 0.659191i $$-0.229101\pi$$
0.751976 + 0.659191i $$0.229101\pi$$
$$110$$ −0.841166 −0.0802021
$$111$$ −12.7899 −1.21396
$$112$$ −1.30798 −0.123592
$$113$$ 12.0489 1.13347 0.566733 0.823901i $$-0.308207\pi$$
0.566733 + 0.823901i $$0.308207\pi$$
$$114$$ 0.158834 0.0148761
$$115$$ 0.368977 0.0344073
$$116$$ −6.28382 −0.583438
$$117$$ 0 0
$$118$$ −5.43727 −0.500541
$$119$$ −5.08815 −0.466430
$$120$$ 1.49396 0.136379
$$121$$ 7.03684 0.639712
$$122$$ 2.78448 0.252095
$$123$$ 26.0465 2.34854
$$124$$ 8.99761 0.808009
$$125$$ −2.45473 −0.219558
$$126$$ −3.87263 −0.345001
$$127$$ −9.80731 −0.870258 −0.435129 0.900368i $$-0.643297\pi$$
−0.435129 + 0.900368i $$0.643297\pi$$
$$128$$ −8.79954 −0.777777
$$129$$ 0.664874 0.0585389
$$130$$ 0 0
$$131$$ −6.57673 −0.574611 −0.287306 0.957839i $$-0.592760\pi$$
−0.287306 + 0.957839i $$0.592760\pi$$
$$132$$ −12.9487 −1.12704
$$133$$ 0.207751 0.0180143
$$134$$ 6.15883 0.532042
$$135$$ 0.527811 0.0454267
$$136$$ −5.81163 −0.498343
$$137$$ 6.21983 0.531396 0.265698 0.964056i $$-0.414398\pi$$
0.265698 + 0.964056i $$0.414398\pi$$
$$138$$ −2.69202 −0.229160
$$139$$ −14.7071 −1.24744 −0.623719 0.781648i $$-0.714379\pi$$
−0.623719 + 0.781648i $$0.714379\pi$$
$$140$$ 0.789856 0.0667550
$$141$$ 16.5308 1.39214
$$142$$ −6.94869 −0.583121
$$143$$ 0 0
$$144$$ 1.13706 0.0947553
$$145$$ 1.14377 0.0949848
$$146$$ 5.40150 0.447031
$$147$$ 3.24698 0.267806
$$148$$ −7.72348 −0.634866
$$149$$ 4.33513 0.355147 0.177574 0.984108i $$-0.443175\pi$$
0.177574 + 0.984108i $$0.443175\pi$$
$$150$$ 8.89977 0.726663
$$151$$ 3.94438 0.320989 0.160494 0.987037i $$-0.448691\pi$$
0.160494 + 0.987037i $$0.448691\pi$$
$$152$$ 0.237291 0.0192468
$$153$$ 4.42327 0.357600
$$154$$ 8.02715 0.646846
$$155$$ −1.63773 −0.131545
$$156$$ 0 0
$$157$$ 4.45473 0.355526 0.177763 0.984073i $$-0.443114\pi$$
0.177763 + 0.984073i $$0.443114\pi$$
$$158$$ 7.99569 0.636103
$$159$$ 23.3545 1.85213
$$160$$ 1.43967 0.113816
$$161$$ −3.52111 −0.277502
$$162$$ −8.78017 −0.689835
$$163$$ 16.1588 1.26566 0.632829 0.774292i $$-0.281894\pi$$
0.632829 + 0.774292i $$0.281894\pi$$
$$164$$ 15.7289 1.22822
$$165$$ 2.35690 0.183484
$$166$$ 1.29052 0.100164
$$167$$ 16.1172 1.24719 0.623594 0.781749i $$-0.285672\pi$$
0.623594 + 0.781749i $$0.285672\pi$$
$$168$$ −14.2567 −1.09993
$$169$$ 0 0
$$170$$ 0.427583 0.0327942
$$171$$ −0.180604 −0.0138111
$$172$$ 0.401501 0.0306142
$$173$$ −21.5362 −1.63736 −0.818682 0.574247i $$-0.805295\pi$$
−0.818682 + 0.574247i $$0.805295\pi$$
$$174$$ −8.34481 −0.632619
$$175$$ 11.6407 0.879955
$$176$$ −2.35690 −0.177658
$$177$$ 15.2349 1.14513
$$178$$ −2.31336 −0.173393
$$179$$ 11.4330 0.854540 0.427270 0.904124i $$-0.359475\pi$$
0.427270 + 0.904124i $$0.359475\pi$$
$$180$$ −0.686645 −0.0511795
$$181$$ 20.9705 1.55872 0.779361 0.626575i $$-0.215544\pi$$
0.779361 + 0.626575i $$0.215544\pi$$
$$182$$ 0 0
$$183$$ −7.80194 −0.576736
$$184$$ −4.02177 −0.296489
$$185$$ 1.40581 0.103357
$$186$$ 11.9487 0.876120
$$187$$ −9.16852 −0.670469
$$188$$ 9.98254 0.728052
$$189$$ −5.03684 −0.366376
$$190$$ −0.0174584 −0.00126657
$$191$$ −14.4373 −1.04464 −0.522322 0.852748i $$-0.674934\pi$$
−0.522322 + 0.852748i $$0.674934\pi$$
$$192$$ −8.00969 −0.578049
$$193$$ −13.5797 −0.977489 −0.488745 0.872427i $$-0.662545\pi$$
−0.488745 + 0.872427i $$0.662545\pi$$
$$194$$ −6.46250 −0.463980
$$195$$ 0 0
$$196$$ 1.96077 0.140055
$$197$$ −0.560335 −0.0399222 −0.0199611 0.999801i $$-0.506354\pi$$
−0.0199611 + 0.999801i $$0.506354\pi$$
$$198$$ −6.97823 −0.495921
$$199$$ 11.4916 0.814616 0.407308 0.913291i $$-0.366468\pi$$
0.407308 + 0.913291i $$0.366468\pi$$
$$200$$ 13.2959 0.940162
$$201$$ −17.2567 −1.21719
$$202$$ −10.7095 −0.753516
$$203$$ −10.9148 −0.766071
$$204$$ 6.58211 0.460840
$$205$$ −2.86294 −0.199956
$$206$$ 1.09246 0.0761151
$$207$$ 3.06100 0.212754
$$208$$ 0 0
$$209$$ 0.374354 0.0258946
$$210$$ 1.04892 0.0723822
$$211$$ 8.78448 0.604748 0.302374 0.953189i $$-0.402221\pi$$
0.302374 + 0.953189i $$0.402221\pi$$
$$212$$ 14.1032 0.968613
$$213$$ 19.4698 1.33405
$$214$$ 2.62133 0.179191
$$215$$ −0.0730805 −0.00498405
$$216$$ −5.75302 −0.391443
$$217$$ 15.6286 1.06094
$$218$$ 12.5918 0.852824
$$219$$ −15.1347 −1.02271
$$220$$ 1.42327 0.0959570
$$221$$ 0 0
$$222$$ −10.2567 −0.688383
$$223$$ −2.25906 −0.151278 −0.0756390 0.997135i $$-0.524100\pi$$
−0.0756390 + 0.997135i $$0.524100\pi$$
$$224$$ −13.7385 −0.917945
$$225$$ −10.1196 −0.674640
$$226$$ 9.66248 0.642739
$$227$$ −6.96615 −0.462359 −0.231180 0.972911i $$-0.574259\pi$$
−0.231180 + 0.972911i $$0.574259\pi$$
$$228$$ −0.268750 −0.0177984
$$229$$ −24.1739 −1.59746 −0.798728 0.601692i $$-0.794493\pi$$
−0.798728 + 0.601692i $$0.794493\pi$$
$$230$$ 0.295897 0.0195109
$$231$$ −22.4916 −1.47984
$$232$$ −12.4668 −0.818486
$$233$$ −3.06100 −0.200533 −0.100266 0.994961i $$-0.531969\pi$$
−0.100266 + 0.994961i $$0.531969\pi$$
$$234$$ 0 0
$$235$$ −1.81700 −0.118528
$$236$$ 9.19998 0.598868
$$237$$ −22.4034 −1.45526
$$238$$ −4.08038 −0.264492
$$239$$ −25.1468 −1.62661 −0.813304 0.581839i $$-0.802333\pi$$
−0.813304 + 0.581839i $$0.802333\pi$$
$$240$$ −0.307979 −0.0198799
$$241$$ −20.2664 −1.30547 −0.652735 0.757586i $$-0.726379\pi$$
−0.652735 + 0.757586i $$0.726379\pi$$
$$242$$ 5.64310 0.362752
$$243$$ 18.1903 1.16691
$$244$$ −4.71140 −0.301616
$$245$$ −0.356896 −0.0228012
$$246$$ 20.8877 1.33175
$$247$$ 0 0
$$248$$ 17.8509 1.13353
$$249$$ −3.61596 −0.229152
$$250$$ −1.96854 −0.124501
$$251$$ −23.7211 −1.49726 −0.748631 0.662987i $$-0.769288\pi$$
−0.748631 + 0.662987i $$0.769288\pi$$
$$252$$ 6.55257 0.412773
$$253$$ −6.34481 −0.398895
$$254$$ −7.86486 −0.493485
$$255$$ −1.19806 −0.0750256
$$256$$ −14.1860 −0.886624
$$257$$ 14.2241 0.887278 0.443639 0.896206i $$-0.353687\pi$$
0.443639 + 0.896206i $$0.353687\pi$$
$$258$$ 0.533188 0.0331948
$$259$$ −13.4155 −0.833599
$$260$$ 0 0
$$261$$ 9.48858 0.587329
$$262$$ −5.27413 −0.325837
$$263$$ −17.0954 −1.05415 −0.527075 0.849819i $$-0.676711\pi$$
−0.527075 + 0.849819i $$0.676711\pi$$
$$264$$ −25.6896 −1.58109
$$265$$ −2.56704 −0.157692
$$266$$ 0.166603 0.0102151
$$267$$ 6.48188 0.396684
$$268$$ −10.4209 −0.636556
$$269$$ −6.46681 −0.394288 −0.197144 0.980374i $$-0.563167\pi$$
−0.197144 + 0.980374i $$0.563167\pi$$
$$270$$ 0.423272 0.0257595
$$271$$ 6.44803 0.391690 0.195845 0.980635i $$-0.437255\pi$$
0.195845 + 0.980635i $$0.437255\pi$$
$$272$$ 1.19806 0.0726432
$$273$$ 0 0
$$274$$ 4.98792 0.301331
$$275$$ 20.9758 1.26489
$$276$$ 4.55496 0.274176
$$277$$ 13.4601 0.808739 0.404370 0.914596i $$-0.367491\pi$$
0.404370 + 0.914596i $$0.367491\pi$$
$$278$$ −11.7942 −0.707367
$$279$$ −13.5864 −0.813398
$$280$$ 1.56704 0.0936485
$$281$$ 5.03684 0.300472 0.150236 0.988650i $$-0.451997\pi$$
0.150236 + 0.988650i $$0.451997\pi$$
$$282$$ 13.2567 0.789423
$$283$$ 22.1280 1.31537 0.657686 0.753293i $$-0.271536\pi$$
0.657686 + 0.753293i $$0.271536\pi$$
$$284$$ 11.7573 0.697669
$$285$$ 0.0489173 0.00289761
$$286$$ 0 0
$$287$$ 27.3207 1.61269
$$288$$ 11.9433 0.703766
$$289$$ −12.3394 −0.725849
$$290$$ 0.917231 0.0538616
$$291$$ 18.1075 1.06148
$$292$$ −9.13946 −0.534846
$$293$$ 14.9463 0.873172 0.436586 0.899663i $$-0.356187\pi$$
0.436586 + 0.899663i $$0.356187\pi$$
$$294$$ 2.60388 0.151861
$$295$$ −1.67456 −0.0974968
$$296$$ −15.3230 −0.890634
$$297$$ −9.07606 −0.526647
$$298$$ 3.47650 0.201388
$$299$$ 0 0
$$300$$ −15.0586 −0.869409
$$301$$ 0.697398 0.0401974
$$302$$ 3.16315 0.182019
$$303$$ 30.0073 1.72387
$$304$$ −0.0489173 −0.00280560
$$305$$ 0.857560 0.0491037
$$306$$ 3.54719 0.202779
$$307$$ 19.1293 1.09177 0.545883 0.837861i $$-0.316194\pi$$
0.545883 + 0.837861i $$0.316194\pi$$
$$308$$ −13.5821 −0.773912
$$309$$ −3.06100 −0.174134
$$310$$ −1.31336 −0.0745936
$$311$$ −0.269815 −0.0152998 −0.00764990 0.999971i $$-0.502435\pi$$
−0.00764990 + 0.999971i $$0.502435\pi$$
$$312$$ 0 0
$$313$$ −23.3937 −1.32229 −0.661146 0.750257i $$-0.729930\pi$$
−0.661146 + 0.750257i $$0.729930\pi$$
$$314$$ 3.57242 0.201603
$$315$$ −1.19269 −0.0672002
$$316$$ −13.5289 −0.761059
$$317$$ −13.9952 −0.786050 −0.393025 0.919528i $$-0.628571\pi$$
−0.393025 + 0.919528i $$0.628571\pi$$
$$318$$ 18.7289 1.05026
$$319$$ −19.6679 −1.10119
$$320$$ 0.880395 0.0492156
$$321$$ −7.34481 −0.409948
$$322$$ −2.82371 −0.157359
$$323$$ −0.190293 −0.0105882
$$324$$ 14.8562 0.825346
$$325$$ 0 0
$$326$$ 12.9584 0.717698
$$327$$ −35.2814 −1.95107
$$328$$ 31.2054 1.72303
$$329$$ 17.3394 0.955954
$$330$$ 1.89008 0.104046
$$331$$ 17.8213 0.979548 0.489774 0.871849i $$-0.337079\pi$$
0.489774 + 0.871849i $$0.337079\pi$$
$$332$$ −2.18359 −0.119840
$$333$$ 11.6625 0.639100
$$334$$ 12.9250 0.707225
$$335$$ 1.89679 0.103633
$$336$$ 2.93900 0.160336
$$337$$ −27.8485 −1.51700 −0.758501 0.651672i $$-0.774068\pi$$
−0.758501 + 0.651672i $$0.774068\pi$$
$$338$$ 0 0
$$339$$ −27.0737 −1.47044
$$340$$ −0.723480 −0.0392362
$$341$$ 28.1618 1.52505
$$342$$ −0.144833 −0.00783167
$$343$$ 19.9041 1.07472
$$344$$ 0.796561 0.0429477
$$345$$ −0.829085 −0.0446364
$$346$$ −17.2707 −0.928477
$$347$$ 1.50365 0.0807200 0.0403600 0.999185i $$-0.487150\pi$$
0.0403600 + 0.999185i $$0.487150\pi$$
$$348$$ 14.1196 0.756890
$$349$$ −14.1860 −0.759358 −0.379679 0.925118i $$-0.623966\pi$$
−0.379679 + 0.925118i $$0.623966\pi$$
$$350$$ 9.33513 0.498983
$$351$$ 0 0
$$352$$ −24.7560 −1.31950
$$353$$ −7.16852 −0.381542 −0.190771 0.981635i $$-0.561099\pi$$
−0.190771 + 0.981635i $$0.561099\pi$$
$$354$$ 12.2174 0.649350
$$355$$ −2.14005 −0.113582
$$356$$ 3.91425 0.207455
$$357$$ 11.4330 0.605096
$$358$$ 9.16852 0.484571
$$359$$ 19.8853 1.04951 0.524753 0.851255i $$-0.324158\pi$$
0.524753 + 0.851255i $$0.324158\pi$$
$$360$$ −1.36227 −0.0717981
$$361$$ −18.9922 −0.999591
$$362$$ 16.8170 0.883882
$$363$$ −15.8116 −0.829895
$$364$$ 0 0
$$365$$ 1.66355 0.0870740
$$366$$ −6.25667 −0.327041
$$367$$ 1.08383 0.0565757 0.0282878 0.999600i $$-0.490994\pi$$
0.0282878 + 0.999600i $$0.490994\pi$$
$$368$$ 0.829085 0.0432190
$$369$$ −23.7506 −1.23641
$$370$$ 1.12737 0.0586094
$$371$$ 24.4969 1.27182
$$372$$ −20.2174 −1.04823
$$373$$ −6.13036 −0.317418 −0.158709 0.987325i $$-0.550733\pi$$
−0.158709 + 0.987325i $$0.550733\pi$$
$$374$$ −7.35258 −0.380193
$$375$$ 5.51573 0.284831
$$376$$ 19.8049 1.02136
$$377$$ 0 0
$$378$$ −4.03923 −0.207756
$$379$$ −2.40880 −0.123732 −0.0618658 0.998084i $$-0.519705\pi$$
−0.0618658 + 0.998084i $$0.519705\pi$$
$$380$$ 0.0295400 0.00151537
$$381$$ 22.0368 1.12898
$$382$$ −11.5778 −0.592371
$$383$$ −30.3913 −1.55292 −0.776462 0.630164i $$-0.782988\pi$$
−0.776462 + 0.630164i $$0.782988\pi$$
$$384$$ 19.7724 1.00901
$$385$$ 2.47219 0.125994
$$386$$ −10.8901 −0.554291
$$387$$ −0.606268 −0.0308184
$$388$$ 10.9347 0.555125
$$389$$ −15.9409 −0.808237 −0.404118 0.914707i $$-0.632422\pi$$
−0.404118 + 0.914707i $$0.632422\pi$$
$$390$$ 0 0
$$391$$ 3.22521 0.163106
$$392$$ 3.89008 0.196479
$$393$$ 14.7778 0.745440
$$394$$ −0.449354 −0.0226381
$$395$$ 2.46250 0.123902
$$396$$ 11.8073 0.593340
$$397$$ 16.9148 0.848931 0.424466 0.905444i $$-0.360462\pi$$
0.424466 + 0.905444i $$0.360462\pi$$
$$398$$ 9.21552 0.461932
$$399$$ −0.466812 −0.0233698
$$400$$ −2.74094 −0.137047
$$401$$ 26.6625 1.33146 0.665730 0.746192i $$-0.268120\pi$$
0.665730 + 0.746192i $$0.268120\pi$$
$$402$$ −13.8388 −0.690215
$$403$$ 0 0
$$404$$ 18.1207 0.901537
$$405$$ −2.70410 −0.134368
$$406$$ −8.75302 −0.434405
$$407$$ −24.1739 −1.19826
$$408$$ 13.0586 0.646497
$$409$$ 28.5163 1.41004 0.705021 0.709187i $$-0.250938\pi$$
0.705021 + 0.709187i $$0.250938\pi$$
$$410$$ −2.29590 −0.113386
$$411$$ −13.9758 −0.689377
$$412$$ −1.84846 −0.0910672
$$413$$ 15.9801 0.786332
$$414$$ 2.45473 0.120643
$$415$$ 0.397452 0.0195102
$$416$$ 0 0
$$417$$ 33.0465 1.61830
$$418$$ 0.300209 0.0146837
$$419$$ −29.6093 −1.44651 −0.723253 0.690583i $$-0.757354\pi$$
−0.723253 + 0.690583i $$0.757354\pi$$
$$420$$ −1.77479 −0.0866009
$$421$$ −11.6606 −0.568301 −0.284151 0.958780i $$-0.591712\pi$$
−0.284151 + 0.958780i $$0.591712\pi$$
$$422$$ 7.04461 0.342926
$$423$$ −15.0737 −0.732907
$$424$$ 27.9801 1.35884
$$425$$ −10.6625 −0.517206
$$426$$ 15.6136 0.756480
$$427$$ −8.18359 −0.396032
$$428$$ −4.43535 −0.214391
$$429$$ 0 0
$$430$$ −0.0586060 −0.00282623
$$431$$ 4.34913 0.209490 0.104745 0.994499i $$-0.466597\pi$$
0.104745 + 0.994499i $$0.466597\pi$$
$$432$$ 1.18598 0.0570605
$$433$$ −14.3884 −0.691460 −0.345730 0.938334i $$-0.612369\pi$$
−0.345730 + 0.938334i $$0.612369\pi$$
$$434$$ 12.5332 0.601612
$$435$$ −2.57002 −0.123223
$$436$$ −21.3056 −1.02035
$$437$$ −0.131687 −0.00629942
$$438$$ −12.1371 −0.579931
$$439$$ −20.2325 −0.965645 −0.482822 0.875718i $$-0.660388\pi$$
−0.482822 + 0.875718i $$0.660388\pi$$
$$440$$ 2.82371 0.134615
$$441$$ −2.96077 −0.140989
$$442$$ 0 0
$$443$$ 8.12200 0.385888 0.192944 0.981210i $$-0.438196\pi$$
0.192944 + 0.981210i $$0.438196\pi$$
$$444$$ 17.3545 0.823608
$$445$$ −0.712464 −0.0337740
$$446$$ −1.81163 −0.0857830
$$447$$ −9.74094 −0.460731
$$448$$ −8.40150 −0.396934
$$449$$ 12.4916 0.589513 0.294757 0.955572i $$-0.404761\pi$$
0.294757 + 0.955572i $$0.404761\pi$$
$$450$$ −8.11529 −0.382559
$$451$$ 49.2301 2.31816
$$452$$ −16.3491 −0.768998
$$453$$ −8.86294 −0.416417
$$454$$ −5.58642 −0.262184
$$455$$ 0 0
$$456$$ −0.533188 −0.0249688
$$457$$ 5.98121 0.279789 0.139895 0.990166i $$-0.455324\pi$$
0.139895 + 0.990166i $$0.455324\pi$$
$$458$$ −19.3860 −0.905847
$$459$$ 4.61356 0.215343
$$460$$ −0.500664 −0.0233436
$$461$$ −2.05669 −0.0957895 −0.0478947 0.998852i $$-0.515251\pi$$
−0.0478947 + 0.998852i $$0.515251\pi$$
$$462$$ −18.0368 −0.839150
$$463$$ −8.44935 −0.392675 −0.196337 0.980536i $$-0.562905\pi$$
−0.196337 + 0.980536i $$0.562905\pi$$
$$464$$ 2.57002 0.119310
$$465$$ 3.67994 0.170653
$$466$$ −2.45473 −0.113713
$$467$$ 33.5139 1.55084 0.775420 0.631446i $$-0.217538\pi$$
0.775420 + 0.631446i $$0.217538\pi$$
$$468$$ 0 0
$$469$$ −18.1008 −0.835818
$$470$$ −1.45712 −0.0672121
$$471$$ −10.0097 −0.461222
$$472$$ 18.2524 0.840133
$$473$$ 1.25667 0.0577817
$$474$$ −17.9661 −0.825213
$$475$$ 0.435353 0.0199754
$$476$$ 6.90408 0.316448
$$477$$ −21.2959 −0.975072
$$478$$ −20.1661 −0.922377
$$479$$ −24.7313 −1.13000 −0.565000 0.825091i $$-0.691124\pi$$
−0.565000 + 0.825091i $$0.691124\pi$$
$$480$$ −3.23490 −0.147652
$$481$$ 0 0
$$482$$ −16.2524 −0.740275
$$483$$ 7.91185 0.360002
$$484$$ −9.54825 −0.434012
$$485$$ −1.99031 −0.0903754
$$486$$ 14.5875 0.661702
$$487$$ −37.7555 −1.71087 −0.855433 0.517913i $$-0.826709\pi$$
−0.855433 + 0.517913i $$0.826709\pi$$
$$488$$ −9.34721 −0.423128
$$489$$ −36.3086 −1.64193
$$490$$ −0.286208 −0.0129296
$$491$$ 31.3110 1.41304 0.706522 0.707691i $$-0.250263\pi$$
0.706522 + 0.707691i $$0.250263\pi$$
$$492$$ −35.3424 −1.59336
$$493$$ 9.99761 0.450270
$$494$$ 0 0
$$495$$ −2.14914 −0.0965969
$$496$$ −3.67994 −0.165234
$$497$$ 20.4222 0.916061
$$498$$ −2.89977 −0.129942
$$499$$ 21.4873 0.961902 0.480951 0.876748i $$-0.340292\pi$$
0.480951 + 0.876748i $$0.340292\pi$$
$$500$$ 3.33081 0.148959
$$501$$ −36.2150 −1.61797
$$502$$ −19.0228 −0.849031
$$503$$ 37.5924 1.67616 0.838081 0.545546i $$-0.183678\pi$$
0.838081 + 0.545546i $$0.183678\pi$$
$$504$$ 13.0000 0.579066
$$505$$ −3.29829 −0.146772
$$506$$ −5.08815 −0.226196
$$507$$ 0 0
$$508$$ 13.3075 0.590425
$$509$$ −17.1075 −0.758278 −0.379139 0.925340i $$-0.623780\pi$$
−0.379139 + 0.925340i $$0.623780\pi$$
$$510$$ −0.960771 −0.0425437
$$511$$ −15.8750 −0.702269
$$512$$ 6.22282 0.275012
$$513$$ −0.188374 −0.00831690
$$514$$ 11.4069 0.503136
$$515$$ 0.336454 0.0148259
$$516$$ −0.902165 −0.0397156
$$517$$ 31.2446 1.37414
$$518$$ −10.7584 −0.472697
$$519$$ 48.3913 2.12414
$$520$$ 0 0
$$521$$ −19.8465 −0.869493 −0.434746 0.900553i $$-0.643162\pi$$
−0.434746 + 0.900553i $$0.643162\pi$$
$$522$$ 7.60925 0.333048
$$523$$ −11.4300 −0.499798 −0.249899 0.968272i $$-0.580397\pi$$
−0.249899 + 0.968272i $$0.580397\pi$$
$$524$$ 8.92394 0.389844
$$525$$ −26.1564 −1.14156
$$526$$ −13.7095 −0.597762
$$527$$ −14.3153 −0.623583
$$528$$ 5.29590 0.230474
$$529$$ −20.7681 −0.902960
$$530$$ −2.05861 −0.0894201
$$531$$ −13.8920 −0.602862
$$532$$ −0.281896 −0.0122218
$$533$$ 0 0
$$534$$ 5.19806 0.224942
$$535$$ 0.807315 0.0349033
$$536$$ −20.6746 −0.893005
$$537$$ −25.6896 −1.10859
$$538$$ −5.18598 −0.223584
$$539$$ 6.13706 0.264342
$$540$$ −0.716185 −0.0308197
$$541$$ 16.1884 0.695993 0.347996 0.937496i $$-0.386862\pi$$
0.347996 + 0.937496i $$0.386862\pi$$
$$542$$ 5.17092 0.222110
$$543$$ −47.1202 −2.02212
$$544$$ 12.5840 0.539536
$$545$$ 3.87800 0.166115
$$546$$ 0 0
$$547$$ 5.33081 0.227929 0.113965 0.993485i $$-0.463645\pi$$
0.113965 + 0.993485i $$0.463645\pi$$
$$548$$ −8.43967 −0.360525
$$549$$ 7.11423 0.303628
$$550$$ 16.8213 0.717263
$$551$$ −0.408206 −0.0173902
$$552$$ 9.03684 0.384633
$$553$$ −23.4993 −0.999293
$$554$$ 10.7942 0.458600
$$555$$ −3.15883 −0.134085
$$556$$ 19.9560 0.846322
$$557$$ 7.39075 0.313156 0.156578 0.987666i $$-0.449954\pi$$
0.156578 + 0.987666i $$0.449954\pi$$
$$558$$ −10.8955 −0.461242
$$559$$ 0 0
$$560$$ −0.323044 −0.0136511
$$561$$ 20.6015 0.869795
$$562$$ 4.03923 0.170385
$$563$$ −9.47889 −0.399488 −0.199744 0.979848i $$-0.564011\pi$$
−0.199744 + 0.979848i $$0.564011\pi$$
$$564$$ −22.4306 −0.944497
$$565$$ 2.97584 0.125194
$$566$$ 17.7453 0.745889
$$567$$ 25.8049 1.08370
$$568$$ 23.3260 0.978738
$$569$$ −10.1438 −0.425249 −0.212624 0.977134i $$-0.568201\pi$$
−0.212624 + 0.977134i $$0.568201\pi$$
$$570$$ 0.0392287 0.00164311
$$571$$ −14.0925 −0.589751 −0.294876 0.955536i $$-0.595278\pi$$
−0.294876 + 0.955536i $$0.595278\pi$$
$$572$$ 0 0
$$573$$ 32.4403 1.35521
$$574$$ 21.9095 0.914483
$$575$$ −7.37867 −0.307712
$$576$$ 7.30367 0.304319
$$577$$ 25.1545 1.04720 0.523598 0.851965i $$-0.324589\pi$$
0.523598 + 0.851965i $$0.324589\pi$$
$$578$$ −9.89546 −0.411597
$$579$$ 30.5133 1.26809
$$580$$ −1.55197 −0.0644422
$$581$$ −3.79284 −0.157354
$$582$$ 14.5211 0.601919
$$583$$ 44.1420 1.82817
$$584$$ −18.1323 −0.750319
$$585$$ 0 0
$$586$$ 11.9860 0.495137
$$587$$ 43.8353 1.80928 0.904639 0.426180i $$-0.140141\pi$$
0.904639 + 0.426180i $$0.140141\pi$$
$$588$$ −4.40581 −0.181693
$$589$$ 0.584498 0.0240838
$$590$$ −1.34290 −0.0552861
$$591$$ 1.25906 0.0517909
$$592$$ 3.15883 0.129827
$$593$$ −24.9965 −1.02648 −0.513242 0.858244i $$-0.671556\pi$$
−0.513242 + 0.858244i $$0.671556\pi$$
$$594$$ −7.27844 −0.298638
$$595$$ −1.25667 −0.0515184
$$596$$ −5.88231 −0.240949
$$597$$ −25.8213 −1.05680
$$598$$ 0 0
$$599$$ −6.24027 −0.254971 −0.127485 0.991840i $$-0.540691\pi$$
−0.127485 + 0.991840i $$0.540691\pi$$
$$600$$ −29.8756 −1.21967
$$601$$ 6.32975 0.258196 0.129098 0.991632i $$-0.458792\pi$$
0.129098 + 0.991632i $$0.458792\pi$$
$$602$$ 0.559270 0.0227941
$$603$$ 15.7356 0.640802
$$604$$ −5.35211 −0.217774
$$605$$ 1.73795 0.0706579
$$606$$ 24.0640 0.977532
$$607$$ −43.6480 −1.77162 −0.885809 0.464050i $$-0.846396\pi$$
−0.885809 + 0.464050i $$0.846396\pi$$
$$608$$ −0.513811 −0.0208378
$$609$$ 24.5254 0.993820
$$610$$ 0.687710 0.0278445
$$611$$ 0 0
$$612$$ −6.00192 −0.242613
$$613$$ −25.9541 −1.04827 −0.524137 0.851634i $$-0.675612\pi$$
−0.524137 + 0.851634i $$0.675612\pi$$
$$614$$ 15.3405 0.619092
$$615$$ 6.43296 0.259402
$$616$$ −26.9463 −1.08570
$$617$$ 45.9396 1.84946 0.924729 0.380626i $$-0.124291\pi$$
0.924729 + 0.380626i $$0.124291\pi$$
$$618$$ −2.45473 −0.0987437
$$619$$ 6.73556 0.270725 0.135363 0.990796i $$-0.456780\pi$$
0.135363 + 0.990796i $$0.456780\pi$$
$$620$$ 2.22223 0.0892467
$$621$$ 3.19269 0.128118
$$622$$ −0.216375 −0.00867583
$$623$$ 6.79895 0.272394
$$624$$ 0 0
$$625$$ 24.0887 0.963549
$$626$$ −18.7603 −0.749813
$$627$$ −0.841166 −0.0335930
$$628$$ −6.04461 −0.241206
$$629$$ 12.2881 0.489960
$$630$$ −0.956459 −0.0381063
$$631$$ 45.0998 1.79539 0.897696 0.440614i $$-0.145239\pi$$
0.897696 + 0.440614i $$0.145239\pi$$
$$632$$ −26.8407 −1.06767
$$633$$ −19.7385 −0.784537
$$634$$ −11.2233 −0.445734
$$635$$ −2.42221 −0.0961223
$$636$$ −31.6896 −1.25658
$$637$$ 0 0
$$638$$ −15.7724 −0.624435
$$639$$ −17.7536 −0.702322
$$640$$ −2.17331 −0.0859075
$$641$$ 32.5821 1.28692 0.643458 0.765482i $$-0.277499\pi$$
0.643458 + 0.765482i $$0.277499\pi$$
$$642$$ −5.89008 −0.232463
$$643$$ 25.5754 1.00860 0.504298 0.863530i $$-0.331751\pi$$
0.504298 + 0.863530i $$0.331751\pi$$
$$644$$ 4.77777 0.188271
$$645$$ 0.164210 0.00646578
$$646$$ −0.152603 −0.00600408
$$647$$ −30.1715 −1.18616 −0.593082 0.805142i $$-0.702089\pi$$
−0.593082 + 0.805142i $$0.702089\pi$$
$$648$$ 29.4741 1.15785
$$649$$ 28.7952 1.13031
$$650$$ 0 0
$$651$$ −35.1172 −1.37635
$$652$$ −21.9259 −0.858683
$$653$$ 36.9028 1.44412 0.722058 0.691832i $$-0.243196\pi$$
0.722058 + 0.691832i $$0.243196\pi$$
$$654$$ −28.2935 −1.10636
$$655$$ −1.62432 −0.0634673
$$656$$ −6.43296 −0.251165
$$657$$ 13.8006 0.538413
$$658$$ 13.9051 0.542079
$$659$$ 23.6866 0.922701 0.461350 0.887218i $$-0.347365\pi$$
0.461350 + 0.887218i $$0.347365\pi$$
$$660$$ −3.19806 −0.124484
$$661$$ −31.7590 −1.23528 −0.617641 0.786460i $$-0.711911\pi$$
−0.617641 + 0.786460i $$0.711911\pi$$
$$662$$ 14.2916 0.555458
$$663$$ 0 0
$$664$$ −4.33214 −0.168120
$$665$$ 0.0513102 0.00198973
$$666$$ 9.35258 0.362405
$$667$$ 6.91856 0.267888
$$668$$ −21.8694 −0.846152
$$669$$ 5.07606 0.196252
$$670$$ 1.52111 0.0587655
$$671$$ −14.7463 −0.569275
$$672$$ 30.8702 1.19085
$$673$$ −7.50232 −0.289193 −0.144597 0.989491i $$-0.546188\pi$$
−0.144597 + 0.989491i $$0.546188\pi$$
$$674$$ −22.3327 −0.860225
$$675$$ −10.5550 −0.406261
$$676$$ 0 0
$$677$$ −35.0315 −1.34637 −0.673184 0.739475i $$-0.735074\pi$$
−0.673184 + 0.739475i $$0.735074\pi$$
$$678$$ −21.7114 −0.833821
$$679$$ 18.9933 0.728896
$$680$$ −1.43535 −0.0550433
$$681$$ 15.6528 0.599816
$$682$$ 22.5840 0.864787
$$683$$ −24.0834 −0.921524 −0.460762 0.887524i $$-0.652424\pi$$
−0.460762 + 0.887524i $$0.652424\pi$$
$$684$$ 0.245061 0.00937013
$$685$$ 1.53617 0.0586941
$$686$$ 15.9618 0.609426
$$687$$ 54.3183 2.07237
$$688$$ −0.164210 −0.00626046
$$689$$ 0 0
$$690$$ −0.664874 −0.0253113
$$691$$ 2.01447 0.0766342 0.0383171 0.999266i $$-0.487800\pi$$
0.0383171 + 0.999266i $$0.487800\pi$$
$$692$$ 29.2223 1.11087
$$693$$ 20.5090 0.779073
$$694$$ 1.20583 0.0457728
$$695$$ −3.63235 −0.137783
$$696$$ 28.0127 1.06182
$$697$$ −25.0248 −0.947880
$$698$$ −11.3763 −0.430598
$$699$$ 6.87800 0.260150
$$700$$ −15.7952 −0.597004
$$701$$ −48.8189 −1.84387 −0.921933 0.387350i $$-0.873390\pi$$
−0.921933 + 0.387350i $$0.873390\pi$$
$$702$$ 0 0
$$703$$ −0.501729 −0.0189231
$$704$$ −15.1390 −0.570572
$$705$$ 4.08277 0.153766
$$706$$ −5.74871 −0.216355
$$707$$ 31.4752 1.18375
$$708$$ −20.6722 −0.776908
$$709$$ −20.8060 −0.781385 −0.390693 0.920521i $$-0.627764\pi$$
−0.390693 + 0.920521i $$0.627764\pi$$
$$710$$ −1.71618 −0.0644073
$$711$$ 20.4286 0.766134
$$712$$ 7.76569 0.291032
$$713$$ −9.90648 −0.371000
$$714$$ 9.16852 0.343123
$$715$$ 0 0
$$716$$ −15.5133 −0.579761
$$717$$ 56.5042 2.11019
$$718$$ 15.9468 0.595128
$$719$$ 21.4306 0.799225 0.399613 0.916684i $$-0.369145\pi$$
0.399613 + 0.916684i $$0.369145\pi$$
$$720$$ 0.280831 0.0104660
$$721$$ −3.21073 −0.119574
$$722$$ −15.2306 −0.566824
$$723$$ 45.5381 1.69358
$$724$$ −28.4547 −1.05751
$$725$$ −22.8726 −0.849468
$$726$$ −12.6799 −0.470597
$$727$$ 13.4862 0.500175 0.250088 0.968223i $$-0.419541\pi$$
0.250088 + 0.968223i $$0.419541\pi$$
$$728$$ 0 0
$$729$$ −8.02715 −0.297302
$$730$$ 1.33406 0.0493758
$$731$$ −0.638792 −0.0236266
$$732$$ 10.5864 0.391285
$$733$$ −43.5424 −1.60828 −0.804138 0.594443i $$-0.797373\pi$$
−0.804138 + 0.594443i $$0.797373\pi$$
$$734$$ 0.869167 0.0320816
$$735$$ 0.801938 0.0295799
$$736$$ 8.70841 0.320996
$$737$$ −32.6165 −1.20145
$$738$$ −19.0465 −0.701112
$$739$$ 20.0543 0.737709 0.368855 0.929487i $$-0.379750\pi$$
0.368855 + 0.929487i $$0.379750\pi$$
$$740$$ −1.90754 −0.0701226
$$741$$ 0 0
$$742$$ 19.6450 0.721191
$$743$$ −33.1685 −1.21684 −0.608418 0.793617i $$-0.708195\pi$$
−0.608418 + 0.793617i $$0.708195\pi$$
$$744$$ −40.1105 −1.47052
$$745$$ 1.07069 0.0392270
$$746$$ −4.91617 −0.179994
$$747$$ 3.29722 0.120639
$$748$$ 12.4407 0.454878
$$749$$ −7.70410 −0.281502
$$750$$ 4.42327 0.161515
$$751$$ 39.2814 1.43340 0.716700 0.697382i $$-0.245652\pi$$
0.716700 + 0.697382i $$0.245652\pi$$
$$752$$ −4.08277 −0.148883
$$753$$ 53.3008 1.94239
$$754$$ 0 0
$$755$$ 0.974181 0.0354541
$$756$$ 6.83446 0.248567
$$757$$ −46.6426 −1.69526 −0.847628 0.530592i $$-0.821970\pi$$
−0.847628 + 0.530592i $$0.821970\pi$$
$$758$$ −1.93171 −0.0701627
$$759$$ 14.2567 0.517484
$$760$$ 0.0586060 0.00212586
$$761$$ −21.8984 −0.793818 −0.396909 0.917858i $$-0.629917\pi$$
−0.396909 + 0.917858i $$0.629917\pi$$
$$762$$ 17.6722 0.640195
$$763$$ −37.0073 −1.33975
$$764$$ 19.5899 0.708737
$$765$$ 1.09246 0.0394979
$$766$$ −24.3720 −0.880595
$$767$$ 0 0
$$768$$ 31.8756 1.15021
$$769$$ 46.7096 1.68439 0.842196 0.539172i $$-0.181263\pi$$
0.842196 + 0.539172i $$0.181263\pi$$
$$770$$ 1.98254 0.0714458
$$771$$ −31.9614 −1.15106
$$772$$ 18.4263 0.663175
$$773$$ −30.2416 −1.08771 −0.543857 0.839178i $$-0.683037\pi$$
−0.543857 + 0.839178i $$0.683037\pi$$
$$774$$ −0.486189 −0.0174757
$$775$$ 32.7506 1.17644
$$776$$ 21.6939 0.778767
$$777$$ 30.1444 1.08142
$$778$$ −12.7836 −0.458315
$$779$$ 1.02177 0.0366087
$$780$$ 0 0
$$781$$ 36.7995 1.31679
$$782$$ 2.58642 0.0924901
$$783$$ 9.89679 0.353682
$$784$$ −0.801938 −0.0286406
$$785$$ 1.10023 0.0392688
$$786$$ 11.8509 0.422706
$$787$$ 28.7023 1.02313 0.511563 0.859246i $$-0.329067\pi$$
0.511563 + 0.859246i $$0.329067\pi$$
$$788$$ 0.760316 0.0270851
$$789$$ 38.4131 1.36754
$$790$$ 1.97477 0.0702592
$$791$$ −28.3980 −1.00972
$$792$$ 23.4252 0.832378
$$793$$ 0 0
$$794$$ 13.5646 0.481391
$$795$$ 5.76809 0.204573
$$796$$ −15.5929 −0.552674
$$797$$ −18.5418 −0.656785 −0.328392 0.944541i $$-0.606507\pi$$
−0.328392 + 0.944541i $$0.606507\pi$$
$$798$$ −0.374354 −0.0132520
$$799$$ −15.8823 −0.561876
$$800$$ −28.7899 −1.01788
$$801$$ −5.91053 −0.208838
$$802$$ 21.3817 0.755012
$$803$$ −28.6058 −1.00948
$$804$$ 23.4155 0.825801
$$805$$ −0.869641 −0.0306508
$$806$$ 0 0
$$807$$ 14.5308 0.511508
$$808$$ 35.9506 1.26474
$$809$$ −10.0677 −0.353962 −0.176981 0.984214i $$-0.556633\pi$$
−0.176981 + 0.984214i $$0.556633\pi$$
$$810$$ −2.16852 −0.0761941
$$811$$ −10.0285 −0.352147 −0.176074 0.984377i $$-0.556340\pi$$
−0.176074 + 0.984377i $$0.556340\pi$$
$$812$$ 14.8103 0.519740
$$813$$ −14.4886 −0.508137
$$814$$ −19.3860 −0.679478
$$815$$ 3.99090 0.139795
$$816$$ −2.69202 −0.0942396
$$817$$ 0.0260821 0.000912498 0
$$818$$ 22.8683 0.799572
$$819$$ 0 0
$$820$$ 3.88471 0.135660
$$821$$ −26.1704 −0.913355 −0.456677 0.889632i $$-0.650961\pi$$
−0.456677 + 0.889632i $$0.650961\pi$$
$$822$$ −11.2078 −0.390915
$$823$$ 1.82238 0.0635242 0.0317621 0.999495i $$-0.489888\pi$$
0.0317621 + 0.999495i $$0.489888\pi$$
$$824$$ −3.66727 −0.127755
$$825$$ −47.1323 −1.64094
$$826$$ 12.8151 0.445894
$$827$$ −32.2941 −1.12298 −0.561488 0.827485i $$-0.689771\pi$$
−0.561488 + 0.827485i $$0.689771\pi$$
$$828$$ −4.15346 −0.144343
$$829$$ 15.1002 0.524453 0.262226 0.965006i $$-0.415543\pi$$
0.262226 + 0.965006i $$0.415543\pi$$
$$830$$ 0.318732 0.0110634
$$831$$ −30.2446 −1.04917
$$832$$ 0 0
$$833$$ −3.11960 −0.108088
$$834$$ 26.5013 0.917663
$$835$$ 3.98062 0.137755
$$836$$ −0.507960 −0.0175682
$$837$$ −14.1709 −0.489818
$$838$$ −23.7448 −0.820250
$$839$$ −32.9965 −1.13917 −0.569584 0.821933i $$-0.692896\pi$$
−0.569584 + 0.821933i $$0.692896\pi$$
$$840$$ −3.52111 −0.121490
$$841$$ −7.55363 −0.260470
$$842$$ −9.35105 −0.322258
$$843$$ −11.3177 −0.389801
$$844$$ −11.9196 −0.410290
$$845$$ 0 0
$$846$$ −12.0881 −0.415599
$$847$$ −16.5851 −0.569870
$$848$$ −5.76809 −0.198077
$$849$$ −49.7211 −1.70642
$$850$$ −8.55065 −0.293285
$$851$$ 8.50365 0.291501
$$852$$ −26.4185 −0.905082
$$853$$ −37.7802 −1.29357 −0.646784 0.762673i $$-0.723887\pi$$
−0.646784 + 0.762673i $$0.723887\pi$$
$$854$$ −6.56273 −0.224572
$$855$$ −0.0446055 −0.00152547
$$856$$ −8.79954 −0.300762
$$857$$ 27.3623 0.934677 0.467339 0.884078i $$-0.345213\pi$$
0.467339 + 0.884078i $$0.345213\pi$$
$$858$$ 0 0
$$859$$ −20.0629 −0.684538 −0.342269 0.939602i $$-0.611195\pi$$
−0.342269 + 0.939602i $$0.611195\pi$$
$$860$$ 0.0991626 0.00338142
$$861$$ −61.3889 −2.09213
$$862$$ 3.48773 0.118792
$$863$$ −6.14483 −0.209173 −0.104586 0.994516i $$-0.533352\pi$$
−0.104586 + 0.994516i $$0.533352\pi$$
$$864$$ 12.4571 0.423800
$$865$$ −5.31900 −0.180851
$$866$$ −11.5386 −0.392096
$$867$$ 27.7265 0.941640
$$868$$ −21.2064 −0.719793
$$869$$ −42.3443 −1.43643
$$870$$ −2.06100 −0.0698744
$$871$$ 0 0
$$872$$ −42.2693 −1.43142
$$873$$ −16.5114 −0.558827
$$874$$ −0.105604 −0.00357212
$$875$$ 5.78554 0.195587
$$876$$ 20.5362 0.693853
$$877$$ −13.5077 −0.456123 −0.228061 0.973647i $$-0.573239\pi$$
−0.228061 + 0.973647i $$0.573239\pi$$
$$878$$ −16.2252 −0.547574
$$879$$ −33.5840 −1.13276
$$880$$ −0.582105 −0.0196228
$$881$$ −5.23431 −0.176348 −0.0881741 0.996105i $$-0.528103\pi$$
−0.0881741 + 0.996105i $$0.528103\pi$$
$$882$$ −2.37435 −0.0799487
$$883$$ −4.57301 −0.153894 −0.0769470 0.997035i $$-0.524517\pi$$
−0.0769470 + 0.997035i $$0.524517\pi$$
$$884$$ 0 0
$$885$$ 3.76271 0.126482
$$886$$ 6.51334 0.218820
$$887$$ −1.64071 −0.0550897 −0.0275448 0.999621i $$-0.508769\pi$$
−0.0275448 + 0.999621i $$0.508769\pi$$
$$888$$ 34.4306 1.15541
$$889$$ 23.1148 0.775246
$$890$$ −0.571352 −0.0191517
$$891$$ 46.4989 1.55777
$$892$$ 3.06531 0.102634
$$893$$ 0.648481 0.0217006
$$894$$ −7.81163 −0.261260
$$895$$ 2.82371 0.0943861
$$896$$ 20.7396 0.692862
$$897$$ 0 0
$$898$$ 10.0175 0.334287
$$899$$ −30.7084 −1.02418
$$900$$ 13.7313 0.457708
$$901$$ −22.4383 −0.747529
$$902$$ 39.4795 1.31452
$$903$$ −1.56704 −0.0521478
$$904$$ −32.4359 −1.07880
$$905$$ 5.17928 0.172165
$$906$$ −7.10752 −0.236132
$$907$$ 8.10215 0.269027 0.134514 0.990912i $$-0.457053\pi$$
0.134514 + 0.990912i $$0.457053\pi$$
$$908$$ 9.45234 0.313687
$$909$$ −27.3623 −0.907549
$$910$$ 0 0
$$911$$ −9.18119 −0.304187 −0.152093 0.988366i $$-0.548601\pi$$
−0.152093 + 0.988366i $$0.548601\pi$$
$$912$$ 0.109916 0.00363969
$$913$$ −6.83446 −0.226188
$$914$$ 4.79656 0.158656
$$915$$ −1.92692 −0.0637020
$$916$$ 32.8015 1.08379
$$917$$ 15.5007 0.511877
$$918$$ 3.69979 0.122111
$$919$$ 27.5036 0.907262 0.453631 0.891190i $$-0.350128\pi$$
0.453631 + 0.891190i $$0.350128\pi$$
$$920$$ −0.993295 −0.0327480
$$921$$ −42.9831 −1.41634
$$922$$ −1.64933 −0.0543180
$$923$$ 0 0
$$924$$ 30.5187 1.00399
$$925$$ −28.1129 −0.924346
$$926$$ −6.77586 −0.222668
$$927$$ 2.79118 0.0916745
$$928$$ 26.9946 0.886142
$$929$$ 24.2131 0.794407 0.397203 0.917731i $$-0.369981\pi$$
0.397203 + 0.917731i $$0.369981\pi$$
$$930$$ 2.95108 0.0967698
$$931$$ 0.127375 0.00417454
$$932$$ 4.15346 0.136051
$$933$$ 0.606268 0.0198483
$$934$$ 26.8761 0.879412
$$935$$ −2.26444 −0.0740550
$$936$$ 0 0
$$937$$ 11.1830 0.365333 0.182666 0.983175i $$-0.441527\pi$$
0.182666 + 0.983175i $$0.441527\pi$$
$$938$$ −14.5157 −0.473955
$$939$$ 52.5652 1.71540
$$940$$ 2.46548 0.0804152
$$941$$ −15.9638 −0.520404 −0.260202 0.965554i $$-0.583789\pi$$
−0.260202 + 0.965554i $$0.583789\pi$$
$$942$$ −8.02715 −0.261539
$$943$$ −17.3177 −0.563941
$$944$$ −3.76271 −0.122466
$$945$$ −1.24400 −0.0404672
$$946$$ 1.00777 0.0327654
$$947$$ 6.51466 0.211698 0.105849 0.994382i $$-0.466244\pi$$
0.105849 + 0.994382i $$0.466244\pi$$
$$948$$ 30.3991 0.987317
$$949$$ 0 0
$$950$$ 0.349126 0.0113271
$$951$$ 31.4470 1.01974
$$952$$ 13.6974 0.443935
$$953$$ 47.6469 1.54344 0.771718 0.635965i $$-0.219398\pi$$
0.771718 + 0.635965i $$0.219398\pi$$
$$954$$ −17.0780 −0.552920
$$955$$ −3.56571 −0.115384
$$956$$ 34.1215 1.10357
$$957$$ 44.1933 1.42857
$$958$$ −19.8329 −0.640773
$$959$$ −14.6595 −0.473380
$$960$$ −1.97823 −0.0638471
$$961$$ 12.9705 0.418402
$$962$$ 0 0
$$963$$ 6.69740 0.215821
$$964$$ 27.4993 0.885694
$$965$$ −3.35391 −0.107966
$$966$$ 6.34481 0.204141
$$967$$ −43.8122 −1.40891 −0.704453 0.709751i $$-0.748808\pi$$
−0.704453 + 0.709751i $$0.748808\pi$$
$$968$$ −18.9433 −0.608861
$$969$$ 0.427583 0.0137360
$$970$$ −1.59611 −0.0512479
$$971$$ 4.29483 0.137828 0.0689139 0.997623i $$-0.478047\pi$$
0.0689139 + 0.997623i $$0.478047\pi$$
$$972$$ −24.6823 −0.791686
$$973$$ 34.6631 1.11125
$$974$$ −30.2776 −0.970156
$$975$$ 0 0
$$976$$ 1.92692 0.0616792
$$977$$ 26.8019 0.857470 0.428735 0.903430i $$-0.358959\pi$$
0.428735 + 0.903430i $$0.358959\pi$$
$$978$$ −29.1172 −0.931066
$$979$$ 12.2513 0.391553
$$980$$ 0.484271 0.0154695
$$981$$ 32.1715 1.02716
$$982$$ 25.1094 0.801275
$$983$$ 27.2495 0.869124 0.434562 0.900642i $$-0.356903\pi$$
0.434562 + 0.900642i $$0.356903\pi$$
$$984$$ −70.1178 −2.23527
$$985$$ −0.138391 −0.00440951
$$986$$ 8.01746 0.255328
$$987$$ −38.9614 −1.24015
$$988$$ 0 0
$$989$$ −0.442058 −0.0140566
$$990$$ −1.72348 −0.0547758
$$991$$ 24.3889 0.774740 0.387370 0.921924i $$-0.373384\pi$$
0.387370 + 0.921924i $$0.373384\pi$$
$$992$$ −38.6528 −1.22723
$$993$$ −40.0441 −1.27076
$$994$$ 16.3773 0.519458
$$995$$ 2.83818 0.0899764
$$996$$ 4.90648 0.155468
$$997$$ 31.3207 0.991935 0.495967 0.868341i $$-0.334814\pi$$
0.495967 + 0.868341i $$0.334814\pi$$
$$998$$ 17.2314 0.545452
$$999$$ 12.1642 0.384859
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 169.2.a.b.1.3 3
3.2 odd 2 1521.2.a.r.1.1 3
4.3 odd 2 2704.2.a.z.1.3 3
5.4 even 2 4225.2.a.bg.1.1 3
7.6 odd 2 8281.2.a.bf.1.3 3
13.2 odd 12 169.2.e.b.147.5 12
13.3 even 3 169.2.c.c.22.1 6
13.4 even 6 169.2.c.b.146.3 6
13.5 odd 4 169.2.b.b.168.2 6
13.6 odd 12 169.2.e.b.23.2 12
13.7 odd 12 169.2.e.b.23.5 12
13.8 odd 4 169.2.b.b.168.5 6
13.9 even 3 169.2.c.c.146.1 6
13.10 even 6 169.2.c.b.22.3 6
13.11 odd 12 169.2.e.b.147.2 12
13.12 even 2 169.2.a.c.1.1 yes 3
39.5 even 4 1521.2.b.l.1351.5 6
39.8 even 4 1521.2.b.l.1351.2 6
39.38 odd 2 1521.2.a.o.1.3 3
52.31 even 4 2704.2.f.o.337.5 6
52.47 even 4 2704.2.f.o.337.6 6
52.51 odd 2 2704.2.a.ba.1.3 3
65.64 even 2 4225.2.a.bb.1.3 3
91.90 odd 2 8281.2.a.bj.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 1.1 even 1 trivial
169.2.a.c.1.1 yes 3 13.12 even 2
169.2.b.b.168.2 6 13.5 odd 4
169.2.b.b.168.5 6 13.8 odd 4
169.2.c.b.22.3 6 13.10 even 6
169.2.c.b.146.3 6 13.4 even 6
169.2.c.c.22.1 6 13.3 even 3
169.2.c.c.146.1 6 13.9 even 3
169.2.e.b.23.2 12 13.6 odd 12
169.2.e.b.23.5 12 13.7 odd 12
169.2.e.b.147.2 12 13.11 odd 12
169.2.e.b.147.5 12 13.2 odd 12
1521.2.a.o.1.3 3 39.38 odd 2
1521.2.a.r.1.1 3 3.2 odd 2
1521.2.b.l.1351.2 6 39.8 even 4
1521.2.b.l.1351.5 6 39.5 even 4
2704.2.a.z.1.3 3 4.3 odd 2
2704.2.a.ba.1.3 3 52.51 odd 2
2704.2.f.o.337.5 6 52.31 even 4
2704.2.f.o.337.6 6 52.47 even 4
4225.2.a.bb.1.3 3 65.64 even 2
4225.2.a.bg.1.1 3 5.4 even 2
8281.2.a.bf.1.3 3 7.6 odd 2
8281.2.a.bj.1.1 3 91.90 odd 2