Properties

 Label 1045.2.a.f.1.5 Level $1045$ Weight $2$ Character 1045.1 Self dual yes Analytic conductor $8.344$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.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: $$8.34436701122$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.7281497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ x^6 - 2*x^5 - 5*x^4 + 7*x^3 + 6*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 1.5 Root $$2.59744$$ of defining polynomial Character $$\chi$$ $$=$$ 1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.21244 q^{2} +1.77266 q^{3} -0.529980 q^{4} -1.00000 q^{5} +2.14925 q^{6} -3.37010 q^{7} -3.06746 q^{8} +0.142317 q^{9} +O(q^{10})$$ $$q+1.21244 q^{2} +1.77266 q^{3} -0.529980 q^{4} -1.00000 q^{5} +2.14925 q^{6} -3.37010 q^{7} -3.06746 q^{8} +0.142317 q^{9} -1.21244 q^{10} -1.00000 q^{11} -0.939473 q^{12} -0.439786 q^{13} -4.08605 q^{14} -1.77266 q^{15} -2.65916 q^{16} -3.08872 q^{17} +0.172551 q^{18} -1.00000 q^{19} +0.529980 q^{20} -5.97403 q^{21} -1.21244 q^{22} +1.45571 q^{23} -5.43756 q^{24} +1.00000 q^{25} -0.533216 q^{26} -5.06570 q^{27} +1.78608 q^{28} +5.85137 q^{29} -2.14925 q^{30} -6.80721 q^{31} +2.91083 q^{32} -1.77266 q^{33} -3.74490 q^{34} +3.37010 q^{35} -0.0754249 q^{36} -2.73402 q^{37} -1.21244 q^{38} -0.779590 q^{39} +3.06746 q^{40} -3.67980 q^{41} -7.24317 q^{42} +4.59330 q^{43} +0.529980 q^{44} -0.142317 q^{45} +1.76497 q^{46} +0.210971 q^{47} -4.71378 q^{48} +4.35755 q^{49} +1.21244 q^{50} -5.47525 q^{51} +0.233078 q^{52} +5.17998 q^{53} -6.14187 q^{54} +1.00000 q^{55} +10.3376 q^{56} -1.77266 q^{57} +7.09446 q^{58} -10.8732 q^{59} +0.939473 q^{60} -5.66286 q^{61} -8.25336 q^{62} -0.479621 q^{63} +8.84754 q^{64} +0.439786 q^{65} -2.14925 q^{66} +6.14764 q^{67} +1.63696 q^{68} +2.58048 q^{69} +4.08605 q^{70} -7.15175 q^{71} -0.436550 q^{72} +4.15351 q^{73} -3.31484 q^{74} +1.77266 q^{75} +0.529980 q^{76} +3.37010 q^{77} -0.945209 q^{78} +15.4881 q^{79} +2.65916 q^{80} -9.40670 q^{81} -4.46155 q^{82} +3.35537 q^{83} +3.16612 q^{84} +3.08872 q^{85} +5.56911 q^{86} +10.3725 q^{87} +3.06746 q^{88} -6.26557 q^{89} -0.172551 q^{90} +1.48212 q^{91} -0.771498 q^{92} -12.0669 q^{93} +0.255791 q^{94} +1.00000 q^{95} +5.15991 q^{96} -6.70633 q^{97} +5.28329 q^{98} -0.142317 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 $$6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.21244 0.857327 0.428664 0.903464i $$-0.358985\pi$$
0.428664 + 0.903464i $$0.358985\pi$$
$$3$$ 1.77266 1.02344 0.511722 0.859151i $$-0.329008\pi$$
0.511722 + 0.859151i $$0.329008\pi$$
$$4$$ −0.529980 −0.264990
$$5$$ −1.00000 −0.447214
$$6$$ 2.14925 0.877427
$$7$$ −3.37010 −1.27378 −0.636888 0.770956i $$-0.719779\pi$$
−0.636888 + 0.770956i $$0.719779\pi$$
$$8$$ −3.06746 −1.08451
$$9$$ 0.142317 0.0474389
$$10$$ −1.21244 −0.383408
$$11$$ −1.00000 −0.301511
$$12$$ −0.939473 −0.271203
$$13$$ −0.439786 −0.121975 −0.0609873 0.998139i $$-0.519425\pi$$
−0.0609873 + 0.998139i $$0.519425\pi$$
$$14$$ −4.08605 −1.09204
$$15$$ −1.77266 −0.457698
$$16$$ −2.65916 −0.664790
$$17$$ −3.08872 −0.749125 −0.374563 0.927202i $$-0.622207\pi$$
−0.374563 + 0.927202i $$0.622207\pi$$
$$18$$ 0.172551 0.0406706
$$19$$ −1.00000 −0.229416
$$20$$ 0.529980 0.118507
$$21$$ −5.97403 −1.30364
$$22$$ −1.21244 −0.258494
$$23$$ 1.45571 0.303537 0.151768 0.988416i $$-0.451503\pi$$
0.151768 + 0.988416i $$0.451503\pi$$
$$24$$ −5.43756 −1.10994
$$25$$ 1.00000 0.200000
$$26$$ −0.533216 −0.104572
$$27$$ −5.06570 −0.974894
$$28$$ 1.78608 0.337538
$$29$$ 5.85137 1.08657 0.543286 0.839548i $$-0.317180\pi$$
0.543286 + 0.839548i $$0.317180\pi$$
$$30$$ −2.14925 −0.392397
$$31$$ −6.80721 −1.22261 −0.611306 0.791394i $$-0.709355\pi$$
−0.611306 + 0.791394i $$0.709355\pi$$
$$32$$ 2.91083 0.514567
$$33$$ −1.77266 −0.308580
$$34$$ −3.74490 −0.642245
$$35$$ 3.37010 0.569650
$$36$$ −0.0754249 −0.0125708
$$37$$ −2.73402 −0.449470 −0.224735 0.974420i $$-0.572152\pi$$
−0.224735 + 0.974420i $$0.572152\pi$$
$$38$$ −1.21244 −0.196684
$$39$$ −0.779590 −0.124834
$$40$$ 3.06746 0.485008
$$41$$ −3.67980 −0.574687 −0.287344 0.957828i $$-0.592772\pi$$
−0.287344 + 0.957828i $$0.592772\pi$$
$$42$$ −7.24317 −1.11765
$$43$$ 4.59330 0.700471 0.350236 0.936662i $$-0.386102\pi$$
0.350236 + 0.936662i $$0.386102\pi$$
$$44$$ 0.529980 0.0798975
$$45$$ −0.142317 −0.0212153
$$46$$ 1.76497 0.260230
$$47$$ 0.210971 0.0307733 0.0153867 0.999882i $$-0.495102\pi$$
0.0153867 + 0.999882i $$0.495102\pi$$
$$48$$ −4.71378 −0.680376
$$49$$ 4.35755 0.622507
$$50$$ 1.21244 0.171465
$$51$$ −5.47525 −0.766688
$$52$$ 0.233078 0.0323221
$$53$$ 5.17998 0.711525 0.355762 0.934576i $$-0.384221\pi$$
0.355762 + 0.934576i $$0.384221\pi$$
$$54$$ −6.14187 −0.835803
$$55$$ 1.00000 0.134840
$$56$$ 10.3376 1.38142
$$57$$ −1.77266 −0.234794
$$58$$ 7.09446 0.931548
$$59$$ −10.8732 −1.41557 −0.707785 0.706428i $$-0.750305\pi$$
−0.707785 + 0.706428i $$0.750305\pi$$
$$60$$ 0.939473 0.121285
$$61$$ −5.66286 −0.725055 −0.362527 0.931973i $$-0.618086\pi$$
−0.362527 + 0.931973i $$0.618086\pi$$
$$62$$ −8.25336 −1.04818
$$63$$ −0.479621 −0.0604265
$$64$$ 8.84754 1.10594
$$65$$ 0.439786 0.0545487
$$66$$ −2.14925 −0.264554
$$67$$ 6.14764 0.751054 0.375527 0.926811i $$-0.377462\pi$$
0.375527 + 0.926811i $$0.377462\pi$$
$$68$$ 1.63696 0.198511
$$69$$ 2.58048 0.310653
$$70$$ 4.08605 0.488377
$$71$$ −7.15175 −0.848756 −0.424378 0.905485i $$-0.639507\pi$$
−0.424378 + 0.905485i $$0.639507\pi$$
$$72$$ −0.436550 −0.0514479
$$73$$ 4.15351 0.486132 0.243066 0.970010i $$-0.421847\pi$$
0.243066 + 0.970010i $$0.421847\pi$$
$$74$$ −3.31484 −0.385343
$$75$$ 1.77266 0.204689
$$76$$ 0.529980 0.0607929
$$77$$ 3.37010 0.384058
$$78$$ −0.945209 −0.107024
$$79$$ 15.4881 1.74254 0.871272 0.490800i $$-0.163295\pi$$
0.871272 + 0.490800i $$0.163295\pi$$
$$80$$ 2.65916 0.297303
$$81$$ −9.40670 −1.04519
$$82$$ −4.46155 −0.492695
$$83$$ 3.35537 0.368299 0.184150 0.982898i $$-0.441047\pi$$
0.184150 + 0.982898i $$0.441047\pi$$
$$84$$ 3.16612 0.345452
$$85$$ 3.08872 0.335019
$$86$$ 5.56911 0.600533
$$87$$ 10.3725 1.11205
$$88$$ 3.06746 0.326992
$$89$$ −6.26557 −0.664149 −0.332075 0.943253i $$-0.607749\pi$$
−0.332075 + 0.943253i $$0.607749\pi$$
$$90$$ −0.172551 −0.0181885
$$91$$ 1.48212 0.155368
$$92$$ −0.771498 −0.0804342
$$93$$ −12.0669 −1.25128
$$94$$ 0.255791 0.0263828
$$95$$ 1.00000 0.102598
$$96$$ 5.15991 0.526631
$$97$$ −6.70633 −0.680924 −0.340462 0.940258i $$-0.610584\pi$$
−0.340462 + 0.940258i $$0.610584\pi$$
$$98$$ 5.28329 0.533692
$$99$$ −0.142317 −0.0143034
$$100$$ −0.529980 −0.0529980
$$101$$ 19.2123 1.91169 0.955846 0.293868i $$-0.0949426\pi$$
0.955846 + 0.293868i $$0.0949426\pi$$
$$102$$ −6.63843 −0.657302
$$103$$ −15.0086 −1.47884 −0.739418 0.673246i $$-0.764899\pi$$
−0.739418 + 0.673246i $$0.764899\pi$$
$$104$$ 1.34902 0.132283
$$105$$ 5.97403 0.583006
$$106$$ 6.28043 0.610010
$$107$$ −7.63909 −0.738499 −0.369250 0.929330i $$-0.620385\pi$$
−0.369250 + 0.929330i $$0.620385\pi$$
$$108$$ 2.68472 0.258337
$$109$$ 0.0458556 0.00439217 0.00219609 0.999998i $$-0.499301\pi$$
0.00219609 + 0.999998i $$0.499301\pi$$
$$110$$ 1.21244 0.115602
$$111$$ −4.84648 −0.460008
$$112$$ 8.96163 0.846795
$$113$$ 8.43432 0.793434 0.396717 0.917941i $$-0.370149\pi$$
0.396717 + 0.917941i $$0.370149\pi$$
$$114$$ −2.14925 −0.201296
$$115$$ −1.45571 −0.135746
$$116$$ −3.10111 −0.287931
$$117$$ −0.0625888 −0.00578634
$$118$$ −13.1831 −1.21361
$$119$$ 10.4093 0.954218
$$120$$ 5.43756 0.496379
$$121$$ 1.00000 0.0909091
$$122$$ −6.86590 −0.621609
$$123$$ −6.52302 −0.588161
$$124$$ 3.60769 0.323980
$$125$$ −1.00000 −0.0894427
$$126$$ −0.581513 −0.0518053
$$127$$ 1.50239 0.133316 0.0666579 0.997776i $$-0.478766\pi$$
0.0666579 + 0.997776i $$0.478766\pi$$
$$128$$ 4.90548 0.433588
$$129$$ 8.14235 0.716894
$$130$$ 0.533216 0.0467661
$$131$$ −0.608483 −0.0531634 −0.0265817 0.999647i $$-0.508462\pi$$
−0.0265817 + 0.999647i $$0.508462\pi$$
$$132$$ 0.939473 0.0817706
$$133$$ 3.37010 0.292224
$$134$$ 7.45367 0.643899
$$135$$ 5.06570 0.435986
$$136$$ 9.47453 0.812434
$$137$$ −21.4534 −1.83289 −0.916443 0.400166i $$-0.868953\pi$$
−0.916443 + 0.400166i $$0.868953\pi$$
$$138$$ 3.12869 0.266331
$$139$$ −8.85639 −0.751189 −0.375595 0.926784i $$-0.622561\pi$$
−0.375595 + 0.926784i $$0.622561\pi$$
$$140$$ −1.78608 −0.150952
$$141$$ 0.373980 0.0314948
$$142$$ −8.67109 −0.727662
$$143$$ 0.439786 0.0367767
$$144$$ −0.378443 −0.0315369
$$145$$ −5.85137 −0.485930
$$146$$ 5.03590 0.416774
$$147$$ 7.72445 0.637102
$$148$$ 1.44898 0.119105
$$149$$ −2.25215 −0.184503 −0.0922515 0.995736i $$-0.529406\pi$$
−0.0922515 + 0.995736i $$0.529406\pi$$
$$150$$ 2.14925 0.175485
$$151$$ −17.3616 −1.41286 −0.706432 0.707781i $$-0.749696\pi$$
−0.706432 + 0.707781i $$0.749696\pi$$
$$152$$ 3.06746 0.248804
$$153$$ −0.439576 −0.0355376
$$154$$ 4.08605 0.329264
$$155$$ 6.80721 0.546769
$$156$$ 0.413167 0.0330798
$$157$$ −3.75886 −0.299990 −0.149995 0.988687i $$-0.547926\pi$$
−0.149995 + 0.988687i $$0.547926\pi$$
$$158$$ 18.7784 1.49393
$$159$$ 9.18233 0.728206
$$160$$ −2.91083 −0.230122
$$161$$ −4.90589 −0.386638
$$162$$ −11.4051 −0.896069
$$163$$ −11.8435 −0.927655 −0.463828 0.885925i $$-0.653524\pi$$
−0.463828 + 0.885925i $$0.653524\pi$$
$$164$$ 1.95022 0.152286
$$165$$ 1.77266 0.138001
$$166$$ 4.06819 0.315753
$$167$$ 6.33806 0.490454 0.245227 0.969466i $$-0.421137\pi$$
0.245227 + 0.969466i $$0.421137\pi$$
$$168$$ 18.3251 1.41381
$$169$$ −12.8066 −0.985122
$$170$$ 3.74490 0.287221
$$171$$ −0.142317 −0.0108832
$$172$$ −2.43436 −0.185618
$$173$$ −8.21796 −0.624800 −0.312400 0.949951i $$-0.601133\pi$$
−0.312400 + 0.949951i $$0.601133\pi$$
$$174$$ 12.5760 0.953388
$$175$$ −3.37010 −0.254755
$$176$$ 2.65916 0.200442
$$177$$ −19.2745 −1.44876
$$178$$ −7.59665 −0.569393
$$179$$ 11.2448 0.840476 0.420238 0.907414i $$-0.361947\pi$$
0.420238 + 0.907414i $$0.361947\pi$$
$$180$$ 0.0754249 0.00562184
$$181$$ 8.27670 0.615202 0.307601 0.951515i $$-0.400474\pi$$
0.307601 + 0.951515i $$0.400474\pi$$
$$182$$ 1.79699 0.133202
$$183$$ −10.0383 −0.742053
$$184$$ −4.46533 −0.329189
$$185$$ 2.73402 0.201009
$$186$$ −14.6304 −1.07275
$$187$$ 3.08872 0.225870
$$188$$ −0.111811 −0.00815462
$$189$$ 17.0719 1.24180
$$190$$ 1.21244 0.0879599
$$191$$ 3.21777 0.232830 0.116415 0.993201i $$-0.462860\pi$$
0.116415 + 0.993201i $$0.462860\pi$$
$$192$$ 15.6837 1.13187
$$193$$ 19.0447 1.37087 0.685433 0.728136i $$-0.259613\pi$$
0.685433 + 0.728136i $$0.259613\pi$$
$$194$$ −8.13104 −0.583775
$$195$$ 0.779590 0.0558276
$$196$$ −2.30941 −0.164958
$$197$$ 2.04266 0.145534 0.0727668 0.997349i $$-0.476817\pi$$
0.0727668 + 0.997349i $$0.476817\pi$$
$$198$$ −0.172551 −0.0122627
$$199$$ 4.61202 0.326937 0.163469 0.986549i $$-0.447732\pi$$
0.163469 + 0.986549i $$0.447732\pi$$
$$200$$ −3.06746 −0.216902
$$201$$ 10.8977 0.768662
$$202$$ 23.2938 1.63895
$$203$$ −19.7197 −1.38405
$$204$$ 2.90177 0.203165
$$205$$ 3.67980 0.257008
$$206$$ −18.1970 −1.26785
$$207$$ 0.207172 0.0143994
$$208$$ 1.16946 0.0810876
$$209$$ 1.00000 0.0691714
$$210$$ 7.24317 0.499827
$$211$$ −16.5102 −1.13661 −0.568303 0.822819i $$-0.692400\pi$$
−0.568303 + 0.822819i $$0.692400\pi$$
$$212$$ −2.74528 −0.188547
$$213$$ −12.6776 −0.868655
$$214$$ −9.26197 −0.633135
$$215$$ −4.59330 −0.313260
$$216$$ 15.5388 1.05728
$$217$$ 22.9410 1.55733
$$218$$ 0.0555973 0.00376553
$$219$$ 7.36276 0.497529
$$220$$ −0.529980 −0.0357312
$$221$$ 1.35838 0.0913742
$$222$$ −5.87609 −0.394377
$$223$$ 3.43650 0.230125 0.115062 0.993358i $$-0.463293\pi$$
0.115062 + 0.993358i $$0.463293\pi$$
$$224$$ −9.80979 −0.655444
$$225$$ 0.142317 0.00948777
$$226$$ 10.2261 0.680233
$$227$$ −9.66762 −0.641663 −0.320831 0.947136i $$-0.603962\pi$$
−0.320831 + 0.947136i $$0.603962\pi$$
$$228$$ 0.939473 0.0622181
$$229$$ 11.7897 0.779084 0.389542 0.921009i $$-0.372633\pi$$
0.389542 + 0.921009i $$0.372633\pi$$
$$230$$ −1.76497 −0.116379
$$231$$ 5.97403 0.393062
$$232$$ −17.9488 −1.17840
$$233$$ 2.00908 0.131619 0.0658097 0.997832i $$-0.479037\pi$$
0.0658097 + 0.997832i $$0.479037\pi$$
$$234$$ −0.0758854 −0.00496078
$$235$$ −0.210971 −0.0137622
$$236$$ 5.76258 0.375112
$$237$$ 27.4551 1.78340
$$238$$ 12.6207 0.818077
$$239$$ −6.36104 −0.411461 −0.205731 0.978609i $$-0.565957\pi$$
−0.205731 + 0.978609i $$0.565957\pi$$
$$240$$ 4.71378 0.304273
$$241$$ −28.3775 −1.82796 −0.913978 0.405763i $$-0.867006\pi$$
−0.913978 + 0.405763i $$0.867006\pi$$
$$242$$ 1.21244 0.0779388
$$243$$ −1.47777 −0.0947989
$$244$$ 3.00120 0.192132
$$245$$ −4.35755 −0.278394
$$246$$ −7.90879 −0.504246
$$247$$ 0.439786 0.0279829
$$248$$ 20.8808 1.32594
$$249$$ 5.94792 0.376934
$$250$$ −1.21244 −0.0766817
$$251$$ −17.7932 −1.12310 −0.561550 0.827443i $$-0.689795\pi$$
−0.561550 + 0.827443i $$0.689795\pi$$
$$252$$ 0.254189 0.0160124
$$253$$ −1.45571 −0.0915198
$$254$$ 1.82157 0.114295
$$255$$ 5.47525 0.342873
$$256$$ −11.7475 −0.734217
$$257$$ −22.5168 −1.40456 −0.702279 0.711902i $$-0.747834\pi$$
−0.702279 + 0.711902i $$0.747834\pi$$
$$258$$ 9.87214 0.614612
$$259$$ 9.21391 0.572524
$$260$$ −0.233078 −0.0144549
$$261$$ 0.832747 0.0515457
$$262$$ −0.737751 −0.0455784
$$263$$ −5.74826 −0.354453 −0.177226 0.984170i $$-0.556712\pi$$
−0.177226 + 0.984170i $$0.556712\pi$$
$$264$$ 5.43756 0.334658
$$265$$ −5.17998 −0.318204
$$266$$ 4.08605 0.250532
$$267$$ −11.1067 −0.679720
$$268$$ −3.25813 −0.199022
$$269$$ −5.69265 −0.347087 −0.173544 0.984826i $$-0.555522\pi$$
−0.173544 + 0.984826i $$0.555522\pi$$
$$270$$ 6.14187 0.373782
$$271$$ −16.0881 −0.977282 −0.488641 0.872485i $$-0.662507\pi$$
−0.488641 + 0.872485i $$0.662507\pi$$
$$272$$ 8.21341 0.498011
$$273$$ 2.62729 0.159011
$$274$$ −26.0110 −1.57138
$$275$$ −1.00000 −0.0603023
$$276$$ −1.36760 −0.0823200
$$277$$ −3.44717 −0.207120 −0.103560 0.994623i $$-0.533023\pi$$
−0.103560 + 0.994623i $$0.533023\pi$$
$$278$$ −10.7379 −0.644015
$$279$$ −0.968779 −0.0579993
$$280$$ −10.3376 −0.617792
$$281$$ −8.74423 −0.521637 −0.260819 0.965388i $$-0.583992\pi$$
−0.260819 + 0.965388i $$0.583992\pi$$
$$282$$ 0.453430 0.0270013
$$283$$ 30.1987 1.79513 0.897564 0.440885i $$-0.145335\pi$$
0.897564 + 0.440885i $$0.145335\pi$$
$$284$$ 3.79028 0.224912
$$285$$ 1.77266 0.105003
$$286$$ 0.533216 0.0315297
$$287$$ 12.4013 0.732024
$$288$$ 0.414260 0.0244105
$$289$$ −7.45980 −0.438812
$$290$$ −7.09446 −0.416601
$$291$$ −11.8880 −0.696888
$$292$$ −2.20128 −0.128820
$$293$$ −11.1213 −0.649714 −0.324857 0.945763i $$-0.605316\pi$$
−0.324857 + 0.945763i $$0.605316\pi$$
$$294$$ 9.36546 0.546205
$$295$$ 10.8732 0.633062
$$296$$ 8.38649 0.487455
$$297$$ 5.06570 0.293941
$$298$$ −2.73060 −0.158179
$$299$$ −0.640201 −0.0370238
$$300$$ −0.939473 −0.0542405
$$301$$ −15.4799 −0.892244
$$302$$ −21.0499 −1.21129
$$303$$ 34.0568 1.95651
$$304$$ 2.65916 0.152513
$$305$$ 5.66286 0.324254
$$306$$ −0.532961 −0.0304674
$$307$$ 24.8969 1.42094 0.710469 0.703728i $$-0.248483\pi$$
0.710469 + 0.703728i $$0.248483\pi$$
$$308$$ −1.78608 −0.101772
$$309$$ −26.6050 −1.51351
$$310$$ 8.25336 0.468760
$$311$$ 9.24020 0.523963 0.261982 0.965073i $$-0.415624\pi$$
0.261982 + 0.965073i $$0.415624\pi$$
$$312$$ 2.39136 0.135384
$$313$$ −14.8438 −0.839021 −0.419511 0.907750i $$-0.637798\pi$$
−0.419511 + 0.907750i $$0.637798\pi$$
$$314$$ −4.55740 −0.257189
$$315$$ 0.479621 0.0270236
$$316$$ −8.20837 −0.461757
$$317$$ 33.8110 1.89902 0.949509 0.313740i $$-0.101582\pi$$
0.949509 + 0.313740i $$0.101582\pi$$
$$318$$ 11.1331 0.624311
$$319$$ −5.85137 −0.327614
$$320$$ −8.84754 −0.494593
$$321$$ −13.5415 −0.755813
$$322$$ −5.94811 −0.331475
$$323$$ 3.08872 0.171861
$$324$$ 4.98536 0.276964
$$325$$ −0.439786 −0.0243949
$$326$$ −14.3596 −0.795304
$$327$$ 0.0812863 0.00449514
$$328$$ 11.2876 0.623255
$$329$$ −0.710993 −0.0391983
$$330$$ 2.14925 0.118312
$$331$$ 11.5116 0.632737 0.316369 0.948636i $$-0.397536\pi$$
0.316369 + 0.948636i $$0.397536\pi$$
$$332$$ −1.77828 −0.0975956
$$333$$ −0.389096 −0.0213223
$$334$$ 7.68455 0.420480
$$335$$ −6.14764 −0.335882
$$336$$ 15.8859 0.866647
$$337$$ 25.3591 1.38140 0.690698 0.723143i $$-0.257303\pi$$
0.690698 + 0.723143i $$0.257303\pi$$
$$338$$ −15.5273 −0.844572
$$339$$ 14.9512 0.812036
$$340$$ −1.63696 −0.0887766
$$341$$ 6.80721 0.368631
$$342$$ −0.172551 −0.00933048
$$343$$ 8.90531 0.480841
$$344$$ −14.0897 −0.759668
$$345$$ −2.58048 −0.138928
$$346$$ −9.96382 −0.535658
$$347$$ −9.66755 −0.518982 −0.259491 0.965746i $$-0.583555\pi$$
−0.259491 + 0.965746i $$0.583555\pi$$
$$348$$ −5.49721 −0.294681
$$349$$ 30.9349 1.65591 0.827953 0.560797i $$-0.189505\pi$$
0.827953 + 0.560797i $$0.189505\pi$$
$$350$$ −4.08605 −0.218409
$$351$$ 2.22782 0.118912
$$352$$ −2.91083 −0.155148
$$353$$ 10.9128 0.580830 0.290415 0.956901i $$-0.406207\pi$$
0.290415 + 0.956901i $$0.406207\pi$$
$$354$$ −23.3692 −1.24206
$$355$$ 7.15175 0.379575
$$356$$ 3.32063 0.175993
$$357$$ 18.4521 0.976589
$$358$$ 13.6337 0.720563
$$359$$ −29.0146 −1.53133 −0.765667 0.643237i $$-0.777591\pi$$
−0.765667 + 0.643237i $$0.777591\pi$$
$$360$$ 0.436550 0.0230082
$$361$$ 1.00000 0.0526316
$$362$$ 10.0350 0.527429
$$363$$ 1.77266 0.0930404
$$364$$ −0.785494 −0.0411711
$$365$$ −4.15351 −0.217405
$$366$$ −12.1709 −0.636182
$$367$$ 21.1172 1.10231 0.551154 0.834404i $$-0.314188\pi$$
0.551154 + 0.834404i $$0.314188\pi$$
$$368$$ −3.87097 −0.201788
$$369$$ −0.523696 −0.0272625
$$370$$ 3.31484 0.172331
$$371$$ −17.4570 −0.906324
$$372$$ 6.39519 0.331575
$$373$$ 24.3430 1.26043 0.630215 0.776421i $$-0.282967\pi$$
0.630215 + 0.776421i $$0.282967\pi$$
$$374$$ 3.74490 0.193644
$$375$$ −1.77266 −0.0915397
$$376$$ −0.647146 −0.0333740
$$377$$ −2.57335 −0.132534
$$378$$ 20.6987 1.06463
$$379$$ 12.5458 0.644433 0.322216 0.946666i $$-0.395572\pi$$
0.322216 + 0.946666i $$0.395572\pi$$
$$380$$ −0.529980 −0.0271874
$$381$$ 2.66323 0.136441
$$382$$ 3.90137 0.199611
$$383$$ 8.27908 0.423041 0.211521 0.977374i $$-0.432158\pi$$
0.211521 + 0.977374i $$0.432158\pi$$
$$384$$ 8.69575 0.443753
$$385$$ −3.37010 −0.171756
$$386$$ 23.0906 1.17528
$$387$$ 0.653702 0.0332296
$$388$$ 3.55422 0.180438
$$389$$ −7.91740 −0.401428 −0.200714 0.979650i $$-0.564326\pi$$
−0.200714 + 0.979650i $$0.564326\pi$$
$$390$$ 0.945209 0.0478625
$$391$$ −4.49629 −0.227387
$$392$$ −13.3666 −0.675116
$$393$$ −1.07863 −0.0544098
$$394$$ 2.47661 0.124770
$$395$$ −15.4881 −0.779290
$$396$$ 0.0754249 0.00379024
$$397$$ −24.3775 −1.22347 −0.611737 0.791062i $$-0.709529\pi$$
−0.611737 + 0.791062i $$0.709529\pi$$
$$398$$ 5.59181 0.280292
$$399$$ 5.97403 0.299076
$$400$$ −2.65916 −0.132958
$$401$$ 32.6120 1.62857 0.814283 0.580469i $$-0.197131\pi$$
0.814283 + 0.580469i $$0.197131\pi$$
$$402$$ 13.2128 0.658995
$$403$$ 2.99372 0.149128
$$404$$ −10.1821 −0.506579
$$405$$ 9.40670 0.467422
$$406$$ −23.9090 −1.18658
$$407$$ 2.73402 0.135520
$$408$$ 16.7951 0.831481
$$409$$ −36.7429 −1.81682 −0.908409 0.418083i $$-0.862702\pi$$
−0.908409 + 0.418083i $$0.862702\pi$$
$$410$$ 4.46155 0.220340
$$411$$ −38.0295 −1.87586
$$412$$ 7.95423 0.391877
$$413$$ 36.6437 1.80312
$$414$$ 0.251184 0.0123450
$$415$$ −3.35537 −0.164708
$$416$$ −1.28014 −0.0627642
$$417$$ −15.6994 −0.768801
$$418$$ 1.21244 0.0593026
$$419$$ −20.7229 −1.01238 −0.506191 0.862422i $$-0.668947\pi$$
−0.506191 + 0.862422i $$0.668947\pi$$
$$420$$ −3.16612 −0.154491
$$421$$ −23.3391 −1.13748 −0.568738 0.822518i $$-0.692568\pi$$
−0.568738 + 0.822518i $$0.692568\pi$$
$$422$$ −20.0176 −0.974443
$$423$$ 0.0300247 0.00145985
$$424$$ −15.8894 −0.771656
$$425$$ −3.08872 −0.149825
$$426$$ −15.3709 −0.744722
$$427$$ 19.0844 0.923558
$$428$$ 4.04857 0.195695
$$429$$ 0.779590 0.0376390
$$430$$ −5.56911 −0.268567
$$431$$ 2.35724 0.113544 0.0567721 0.998387i $$-0.481919\pi$$
0.0567721 + 0.998387i $$0.481919\pi$$
$$432$$ 13.4705 0.648100
$$433$$ 37.3199 1.79348 0.896741 0.442555i $$-0.145928\pi$$
0.896741 + 0.442555i $$0.145928\pi$$
$$434$$ 27.8146 1.33515
$$435$$ −10.3725 −0.497322
$$436$$ −0.0243026 −0.00116388
$$437$$ −1.45571 −0.0696361
$$438$$ 8.92693 0.426545
$$439$$ 26.3625 1.25821 0.629106 0.777320i $$-0.283421\pi$$
0.629106 + 0.777320i $$0.283421\pi$$
$$440$$ −3.06746 −0.146235
$$441$$ 0.620152 0.0295310
$$442$$ 1.64695 0.0783376
$$443$$ −7.31633 −0.347609 −0.173805 0.984780i $$-0.555606\pi$$
−0.173805 + 0.984780i $$0.555606\pi$$
$$444$$ 2.56854 0.121897
$$445$$ 6.26557 0.297017
$$446$$ 4.16656 0.197292
$$447$$ −3.99228 −0.188829
$$448$$ −29.8171 −1.40872
$$449$$ −0.872582 −0.0411797 −0.0205898 0.999788i $$-0.506554\pi$$
−0.0205898 + 0.999788i $$0.506554\pi$$
$$450$$ 0.172551 0.00813412
$$451$$ 3.67980 0.173275
$$452$$ −4.47002 −0.210252
$$453$$ −30.7761 −1.44599
$$454$$ −11.7215 −0.550115
$$455$$ −1.48212 −0.0694829
$$456$$ 5.43756 0.254637
$$457$$ −2.04749 −0.0957774 −0.0478887 0.998853i $$-0.515249\pi$$
−0.0478887 + 0.998853i $$0.515249\pi$$
$$458$$ 14.2943 0.667930
$$459$$ 15.6465 0.730317
$$460$$ 0.771498 0.0359713
$$461$$ 26.6538 1.24139 0.620695 0.784052i $$-0.286851\pi$$
0.620695 + 0.784052i $$0.286851\pi$$
$$462$$ 7.24317 0.336983
$$463$$ −19.5335 −0.907798 −0.453899 0.891053i $$-0.649967\pi$$
−0.453899 + 0.891053i $$0.649967\pi$$
$$464$$ −15.5597 −0.722343
$$465$$ 12.0669 0.559587
$$466$$ 2.43590 0.112841
$$467$$ −29.3029 −1.35598 −0.677988 0.735073i $$-0.737148\pi$$
−0.677988 + 0.735073i $$0.737148\pi$$
$$468$$ 0.0331708 0.00153332
$$469$$ −20.7182 −0.956675
$$470$$ −0.255791 −0.0117987
$$471$$ −6.66317 −0.307023
$$472$$ 33.3531 1.53520
$$473$$ −4.59330 −0.211200
$$474$$ 33.2877 1.52896
$$475$$ −1.00000 −0.0458831
$$476$$ −5.51671 −0.252858
$$477$$ 0.737197 0.0337539
$$478$$ −7.71240 −0.352757
$$479$$ −0.965639 −0.0441212 −0.0220606 0.999757i $$-0.507023\pi$$
−0.0220606 + 0.999757i $$0.507023\pi$$
$$480$$ −5.15991 −0.235517
$$481$$ 1.20238 0.0548239
$$482$$ −34.4061 −1.56716
$$483$$ −8.69646 −0.395703
$$484$$ −0.529980 −0.0240900
$$485$$ 6.70633 0.304519
$$486$$ −1.79171 −0.0812737
$$487$$ −3.36125 −0.152313 −0.0761564 0.997096i $$-0.524265\pi$$
−0.0761564 + 0.997096i $$0.524265\pi$$
$$488$$ 17.3706 0.786329
$$489$$ −20.9945 −0.949404
$$490$$ −5.28329 −0.238675
$$491$$ −32.5832 −1.47046 −0.735230 0.677818i $$-0.762926\pi$$
−0.735230 + 0.677818i $$0.762926\pi$$
$$492$$ 3.45707 0.155857
$$493$$ −18.0733 −0.813978
$$494$$ 0.533216 0.0239905
$$495$$ 0.142317 0.00639665
$$496$$ 18.1015 0.812780
$$497$$ 24.1021 1.08113
$$498$$ 7.21152 0.323156
$$499$$ 7.53868 0.337477 0.168739 0.985661i $$-0.446031\pi$$
0.168739 + 0.985661i $$0.446031\pi$$
$$500$$ 0.529980 0.0237014
$$501$$ 11.2352 0.501953
$$502$$ −21.5733 −0.962864
$$503$$ −9.02729 −0.402507 −0.201254 0.979539i $$-0.564502\pi$$
−0.201254 + 0.979539i $$0.564502\pi$$
$$504$$ 1.47122 0.0655332
$$505$$ −19.2123 −0.854935
$$506$$ −1.76497 −0.0784624
$$507$$ −22.7017 −1.00822
$$508$$ −0.796238 −0.0353273
$$509$$ −20.8922 −0.926031 −0.463015 0.886350i $$-0.653233\pi$$
−0.463015 + 0.886350i $$0.653233\pi$$
$$510$$ 6.63843 0.293955
$$511$$ −13.9977 −0.619223
$$512$$ −24.0541 −1.06305
$$513$$ 5.06570 0.223656
$$514$$ −27.3003 −1.20417
$$515$$ 15.0086 0.661356
$$516$$ −4.31528 −0.189970
$$517$$ −0.210971 −0.00927850
$$518$$ 11.1713 0.490841
$$519$$ −14.5676 −0.639448
$$520$$ −1.34902 −0.0591586
$$521$$ −43.6640 −1.91296 −0.956478 0.291803i $$-0.905745\pi$$
−0.956478 + 0.291803i $$0.905745\pi$$
$$522$$ 1.00966 0.0441916
$$523$$ 26.4677 1.15735 0.578675 0.815558i $$-0.303570\pi$$
0.578675 + 0.815558i $$0.303570\pi$$
$$524$$ 0.322484 0.0140878
$$525$$ −5.97403 −0.260728
$$526$$ −6.96944 −0.303882
$$527$$ 21.0256 0.915889
$$528$$ 4.71378 0.205141
$$529$$ −20.8809 −0.907865
$$530$$ −6.28043 −0.272805
$$531$$ −1.54744 −0.0671530
$$532$$ −1.78608 −0.0774365
$$533$$ 1.61832 0.0700973
$$534$$ −13.4663 −0.582742
$$535$$ 7.63909 0.330267
$$536$$ −18.8576 −0.814526
$$537$$ 19.9332 0.860180
$$538$$ −6.90202 −0.297567
$$539$$ −4.35755 −0.187693
$$540$$ −2.68472 −0.115532
$$541$$ −1.04162 −0.0447829 −0.0223915 0.999749i $$-0.507128\pi$$
−0.0223915 + 0.999749i $$0.507128\pi$$
$$542$$ −19.5059 −0.837851
$$543$$ 14.6718 0.629625
$$544$$ −8.99075 −0.385475
$$545$$ −0.0458556 −0.00196424
$$546$$ 3.18545 0.136324
$$547$$ −5.27609 −0.225589 −0.112795 0.993618i $$-0.535980\pi$$
−0.112795 + 0.993618i $$0.535980\pi$$
$$548$$ 11.3699 0.485696
$$549$$ −0.805918 −0.0343958
$$550$$ −1.21244 −0.0516988
$$551$$ −5.85137 −0.249277
$$552$$ −7.91551 −0.336907
$$553$$ −52.1963 −2.21961
$$554$$ −4.17949 −0.177570
$$555$$ 4.84648 0.205722
$$556$$ 4.69371 0.199058
$$557$$ −12.9219 −0.547517 −0.273759 0.961798i $$-0.588267\pi$$
−0.273759 + 0.961798i $$0.588267\pi$$
$$558$$ −1.17459 −0.0497244
$$559$$ −2.02007 −0.0854397
$$560$$ −8.96163 −0.378698
$$561$$ 5.47525 0.231165
$$562$$ −10.6019 −0.447214
$$563$$ −8.76657 −0.369467 −0.184733 0.982789i $$-0.559142\pi$$
−0.184733 + 0.982789i $$0.559142\pi$$
$$564$$ −0.198202 −0.00834580
$$565$$ −8.43432 −0.354834
$$566$$ 36.6142 1.53901
$$567$$ 31.7015 1.33134
$$568$$ 21.9377 0.920485
$$569$$ −14.7828 −0.619726 −0.309863 0.950781i $$-0.600283\pi$$
−0.309863 + 0.950781i $$0.600283\pi$$
$$570$$ 2.14925 0.0900221
$$571$$ 22.6278 0.946943 0.473472 0.880809i $$-0.343001\pi$$
0.473472 + 0.880809i $$0.343001\pi$$
$$572$$ −0.233078 −0.00974547
$$573$$ 5.70401 0.238288
$$574$$ 15.0358 0.627584
$$575$$ 1.45571 0.0607074
$$576$$ 1.25915 0.0524647
$$577$$ 3.73371 0.155436 0.0777182 0.996975i $$-0.475237\pi$$
0.0777182 + 0.996975i $$0.475237\pi$$
$$578$$ −9.04459 −0.376205
$$579$$ 33.7597 1.40301
$$580$$ 3.10111 0.128767
$$581$$ −11.3079 −0.469131
$$582$$ −14.4136 −0.597461
$$583$$ −5.17998 −0.214533
$$584$$ −12.7407 −0.527215
$$585$$ 0.0625888 0.00258773
$$586$$ −13.4840 −0.557018
$$587$$ −19.6389 −0.810584 −0.405292 0.914187i $$-0.632830\pi$$
−0.405292 + 0.914187i $$0.632830\pi$$
$$588$$ −4.09380 −0.168826
$$589$$ 6.80721 0.280486
$$590$$ 13.1831 0.542741
$$591$$ 3.62094 0.148946
$$592$$ 7.27020 0.298803
$$593$$ −32.8110 −1.34739 −0.673693 0.739011i $$-0.735293\pi$$
−0.673693 + 0.739011i $$0.735293\pi$$
$$594$$ 6.14187 0.252004
$$595$$ −10.4093 −0.426739
$$596$$ 1.19359 0.0488914
$$597$$ 8.17553 0.334602
$$598$$ −0.776208 −0.0317415
$$599$$ −41.5943 −1.69950 −0.849750 0.527187i $$-0.823247\pi$$
−0.849750 + 0.527187i $$0.823247\pi$$
$$600$$ −5.43756 −0.221987
$$601$$ −16.1727 −0.659698 −0.329849 0.944034i $$-0.606998\pi$$
−0.329849 + 0.944034i $$0.606998\pi$$
$$602$$ −18.7685 −0.764945
$$603$$ 0.874911 0.0356291
$$604$$ 9.20127 0.374395
$$605$$ −1.00000 −0.0406558
$$606$$ 41.2919 1.67737
$$607$$ −14.9877 −0.608330 −0.304165 0.952619i $$-0.598377\pi$$
−0.304165 + 0.952619i $$0.598377\pi$$
$$608$$ −2.91083 −0.118050
$$609$$ −34.9563 −1.41650
$$610$$ 6.86590 0.277992
$$611$$ −0.0927821 −0.00375356
$$612$$ 0.232967 0.00941712
$$613$$ 19.1145 0.772027 0.386014 0.922493i $$-0.373852\pi$$
0.386014 + 0.922493i $$0.373852\pi$$
$$614$$ 30.1860 1.21821
$$615$$ 6.52302 0.263033
$$616$$ −10.3376 −0.416515
$$617$$ −1.20030 −0.0483223 −0.0241612 0.999708i $$-0.507691\pi$$
−0.0241612 + 0.999708i $$0.507691\pi$$
$$618$$ −32.2571 −1.29757
$$619$$ −44.2783 −1.77969 −0.889847 0.456258i $$-0.849189\pi$$
−0.889847 + 0.456258i $$0.849189\pi$$
$$620$$ −3.60769 −0.144888
$$621$$ −7.37419 −0.295916
$$622$$ 11.2032 0.449208
$$623$$ 21.1156 0.845978
$$624$$ 2.07306 0.0829886
$$625$$ 1.00000 0.0400000
$$626$$ −17.9973 −0.719316
$$627$$ 1.77266 0.0707931
$$628$$ 1.99212 0.0794942
$$629$$ 8.44462 0.336709
$$630$$ 0.581513 0.0231680
$$631$$ 7.76811 0.309244 0.154622 0.987974i $$-0.450584\pi$$
0.154622 + 0.987974i $$0.450584\pi$$
$$632$$ −47.5090 −1.88981
$$633$$ −29.2669 −1.16325
$$634$$ 40.9940 1.62808
$$635$$ −1.50239 −0.0596206
$$636$$ −4.86645 −0.192967
$$637$$ −1.91639 −0.0759301
$$638$$ −7.09446 −0.280872
$$639$$ −1.01781 −0.0402640
$$640$$ −4.90548 −0.193906
$$641$$ 27.0513 1.06846 0.534231 0.845339i $$-0.320601\pi$$
0.534231 + 0.845339i $$0.320601\pi$$
$$642$$ −16.4183 −0.647979
$$643$$ −15.1507 −0.597485 −0.298742 0.954334i $$-0.596567\pi$$
−0.298742 + 0.954334i $$0.596567\pi$$
$$644$$ 2.60002 0.102455
$$645$$ −8.14235 −0.320605
$$646$$ 3.74490 0.147341
$$647$$ −22.7874 −0.895866 −0.447933 0.894067i $$-0.647840\pi$$
−0.447933 + 0.894067i $$0.647840\pi$$
$$648$$ 28.8546 1.13352
$$649$$ 10.8732 0.426810
$$650$$ −0.533216 −0.0209144
$$651$$ 40.6665 1.59385
$$652$$ 6.27682 0.245819
$$653$$ 36.1144 1.41327 0.706633 0.707580i $$-0.250213\pi$$
0.706633 + 0.707580i $$0.250213\pi$$
$$654$$ 0.0985551 0.00385381
$$655$$ 0.608483 0.0237754
$$656$$ 9.78517 0.382047
$$657$$ 0.591113 0.0230615
$$658$$ −0.862040 −0.0336058
$$659$$ −5.96009 −0.232172 −0.116086 0.993239i $$-0.537035\pi$$
−0.116086 + 0.993239i $$0.537035\pi$$
$$660$$ −0.939473 −0.0365689
$$661$$ 47.5667 1.85013 0.925065 0.379808i $$-0.124010\pi$$
0.925065 + 0.379808i $$0.124010\pi$$
$$662$$ 13.9572 0.542463
$$663$$ 2.40794 0.0935165
$$664$$ −10.2924 −0.399425
$$665$$ −3.37010 −0.130687
$$666$$ −0.471757 −0.0182802
$$667$$ 8.51791 0.329815
$$668$$ −3.35905 −0.129965
$$669$$ 6.09174 0.235520
$$670$$ −7.45367 −0.287960
$$671$$ 5.66286 0.218612
$$672$$ −17.3894 −0.670811
$$673$$ 23.3646 0.900639 0.450319 0.892868i $$-0.351310\pi$$
0.450319 + 0.892868i $$0.351310\pi$$
$$674$$ 30.7465 1.18431
$$675$$ −5.06570 −0.194979
$$676$$ 6.78723 0.261047
$$677$$ −22.7258 −0.873423 −0.436712 0.899602i $$-0.643857\pi$$
−0.436712 + 0.899602i $$0.643857\pi$$
$$678$$ 18.1274 0.696180
$$679$$ 22.6010 0.867346
$$680$$ −9.47453 −0.363331
$$681$$ −17.1374 −0.656706
$$682$$ 8.25336 0.316038
$$683$$ −6.08405 −0.232800 −0.116400 0.993202i $$-0.537135\pi$$
−0.116400 + 0.993202i $$0.537135\pi$$
$$684$$ 0.0754249 0.00288394
$$685$$ 21.4534 0.819691
$$686$$ 10.7972 0.412239
$$687$$ 20.8991 0.797350
$$688$$ −12.2143 −0.465667
$$689$$ −2.27808 −0.0867880
$$690$$ −3.12869 −0.119107
$$691$$ 10.2313 0.389215 0.194608 0.980881i $$-0.437657\pi$$
0.194608 + 0.980881i $$0.437657\pi$$
$$692$$ 4.35535 0.165566
$$693$$ 0.479621 0.0182193
$$694$$ −11.7214 −0.444937
$$695$$ 8.85639 0.335942
$$696$$ −31.8171 −1.20603
$$697$$ 11.3659 0.430513
$$698$$ 37.5068 1.41965
$$699$$ 3.56141 0.134705
$$700$$ 1.78608 0.0675076
$$701$$ 20.1154 0.759749 0.379875 0.925038i $$-0.375967\pi$$
0.379875 + 0.925038i $$0.375967\pi$$
$$702$$ 2.70111 0.101947
$$703$$ 2.73402 0.103115
$$704$$ −8.84754 −0.333454
$$705$$ −0.373980 −0.0140849
$$706$$ 13.2312 0.497962
$$707$$ −64.7472 −2.43507
$$708$$ 10.2151 0.383906
$$709$$ −18.6469 −0.700301 −0.350150 0.936694i $$-0.613870\pi$$
−0.350150 + 0.936694i $$0.613870\pi$$
$$710$$ 8.67109 0.325420
$$711$$ 2.20421 0.0826643
$$712$$ 19.2194 0.720277
$$713$$ −9.90934 −0.371108
$$714$$ 22.3721 0.837257
$$715$$ −0.439786 −0.0164471
$$716$$ −5.95952 −0.222718
$$717$$ −11.2759 −0.421108
$$718$$ −35.1786 −1.31285
$$719$$ −25.8776 −0.965073 −0.482537 0.875876i $$-0.660284\pi$$
−0.482537 + 0.875876i $$0.660284\pi$$
$$720$$ 0.378443 0.0141037
$$721$$ 50.5803 1.88371
$$722$$ 1.21244 0.0451225
$$723$$ −50.3036 −1.87081
$$724$$ −4.38648 −0.163022
$$725$$ 5.85137 0.217314
$$726$$ 2.14925 0.0797661
$$727$$ 48.1800 1.78690 0.893450 0.449164i $$-0.148278\pi$$
0.893450 + 0.449164i $$0.148278\pi$$
$$728$$ −4.54634 −0.168499
$$729$$ 25.6005 0.948167
$$730$$ −5.03590 −0.186387
$$731$$ −14.1874 −0.524741
$$732$$ 5.32010 0.196637
$$733$$ 5.47157 0.202097 0.101049 0.994881i $$-0.467780\pi$$
0.101049 + 0.994881i $$0.467780\pi$$
$$734$$ 25.6034 0.945038
$$735$$ −7.72445 −0.284921
$$736$$ 4.23733 0.156190
$$737$$ −6.14764 −0.226451
$$738$$ −0.634952 −0.0233729
$$739$$ 11.0454 0.406311 0.203156 0.979146i $$-0.434880\pi$$
0.203156 + 0.979146i $$0.434880\pi$$
$$740$$ −1.44898 −0.0532654
$$741$$ 0.779590 0.0286389
$$742$$ −21.1657 −0.777016
$$743$$ −16.8303 −0.617443 −0.308721 0.951153i $$-0.599901\pi$$
−0.308721 + 0.951153i $$0.599901\pi$$
$$744$$ 37.0146 1.35702
$$745$$ 2.25215 0.0825122
$$746$$ 29.5145 1.08060
$$747$$ 0.477524 0.0174717
$$748$$ −1.63696 −0.0598532
$$749$$ 25.7445 0.940683
$$750$$ −2.14925 −0.0784795
$$751$$ −8.34710 −0.304590 −0.152295 0.988335i $$-0.548666\pi$$
−0.152295 + 0.988335i $$0.548666\pi$$
$$752$$ −0.561007 −0.0204578
$$753$$ −31.5413 −1.14943
$$754$$ −3.12004 −0.113625
$$755$$ 17.3616 0.631852
$$756$$ −9.04776 −0.329064
$$757$$ 2.18852 0.0795431 0.0397716 0.999209i $$-0.487337\pi$$
0.0397716 + 0.999209i $$0.487337\pi$$
$$758$$ 15.2110 0.552490
$$759$$ −2.58048 −0.0936654
$$760$$ −3.06746 −0.111268
$$761$$ 33.4322 1.21192 0.605958 0.795496i $$-0.292790\pi$$
0.605958 + 0.795496i $$0.292790\pi$$
$$762$$ 3.22901 0.116975
$$763$$ −0.154538 −0.00559464
$$764$$ −1.70535 −0.0616975
$$765$$ 0.439576 0.0158929
$$766$$ 10.0379 0.362685
$$767$$ 4.78188 0.172664
$$768$$ −20.8242 −0.751430
$$769$$ 35.7102 1.28774 0.643871 0.765134i $$-0.277327\pi$$
0.643871 + 0.765134i $$0.277327\pi$$
$$770$$ −4.08605 −0.147251
$$771$$ −39.9145 −1.43749
$$772$$ −10.0933 −0.363266
$$773$$ 12.1707 0.437749 0.218875 0.975753i $$-0.429761\pi$$
0.218875 + 0.975753i $$0.429761\pi$$
$$774$$ 0.792577 0.0284886
$$775$$ −6.80721 −0.244522
$$776$$ 20.5714 0.738469
$$777$$ 16.3331 0.585947
$$778$$ −9.59940 −0.344155
$$779$$ 3.67980 0.131842
$$780$$ −0.413167 −0.0147938
$$781$$ 7.15175 0.255910
$$782$$ −5.45150 −0.194945
$$783$$ −29.6413 −1.05929
$$784$$ −11.5874 −0.413837
$$785$$ 3.75886 0.134159
$$786$$ −1.30778 −0.0466470
$$787$$ 47.9347 1.70869 0.854344 0.519707i $$-0.173959\pi$$
0.854344 + 0.519707i $$0.173959\pi$$
$$788$$ −1.08257 −0.0385649
$$789$$ −10.1897 −0.362763
$$790$$ −18.7784 −0.668106
$$791$$ −28.4245 −1.01066
$$792$$ 0.436550 0.0155121
$$793$$ 2.49044 0.0884383
$$794$$ −29.5564 −1.04892
$$795$$ −9.18233 −0.325664
$$796$$ −2.44428 −0.0866351
$$797$$ 50.1321 1.77577 0.887885 0.460066i $$-0.152174\pi$$
0.887885 + 0.460066i $$0.152174\pi$$
$$798$$ 7.24317 0.256406
$$799$$ −0.651631 −0.0230531
$$800$$ 2.91083 0.102913
$$801$$ −0.891695 −0.0315065
$$802$$ 39.5402 1.39621
$$803$$ −4.15351 −0.146574
$$804$$ −5.77555 −0.203688
$$805$$ 4.90589 0.172910
$$806$$ 3.62971 0.127851
$$807$$ −10.0911 −0.355225
$$808$$ −58.9328 −2.07325
$$809$$ −39.8704 −1.40177 −0.700884 0.713275i $$-0.747211\pi$$
−0.700884 + 0.713275i $$0.747211\pi$$
$$810$$ 11.4051 0.400734
$$811$$ −54.0597 −1.89829 −0.949147 0.314835i $$-0.898051\pi$$
−0.949147 + 0.314835i $$0.898051\pi$$
$$812$$ 10.4510 0.366759
$$813$$ −28.5187 −1.00019
$$814$$ 3.31484 0.116185
$$815$$ 11.8435 0.414860
$$816$$ 14.5596 0.509687
$$817$$ −4.59330 −0.160699
$$818$$ −44.5487 −1.55761
$$819$$ 0.210930 0.00737050
$$820$$ −1.95022 −0.0681046
$$821$$ 28.2714 0.986677 0.493339 0.869837i $$-0.335776\pi$$
0.493339 + 0.869837i $$0.335776\pi$$
$$822$$ −46.1086 −1.60822
$$823$$ −28.9439 −1.00892 −0.504461 0.863434i $$-0.668309\pi$$
−0.504461 + 0.863434i $$0.668309\pi$$
$$824$$ 46.0381 1.60381
$$825$$ −1.77266 −0.0617160
$$826$$ 44.4285 1.54586
$$827$$ −6.79361 −0.236237 −0.118118 0.993000i $$-0.537686\pi$$
−0.118118 + 0.993000i $$0.537686\pi$$
$$828$$ −0.109797 −0.00381571
$$829$$ 4.79089 0.166394 0.0831972 0.996533i $$-0.473487\pi$$
0.0831972 + 0.996533i $$0.473487\pi$$
$$830$$ −4.06819 −0.141209
$$831$$ −6.11065 −0.211976
$$832$$ −3.89102 −0.134897
$$833$$ −13.4593 −0.466336
$$834$$ −19.0346 −0.659114
$$835$$ −6.33806 −0.219338
$$836$$ −0.529980 −0.0183297
$$837$$ 34.4833 1.19192
$$838$$ −25.1254 −0.867942
$$839$$ −46.8222 −1.61648 −0.808241 0.588852i $$-0.799580\pi$$
−0.808241 + 0.588852i $$0.799580\pi$$
$$840$$ −18.3251 −0.632276
$$841$$ 5.23853 0.180639
$$842$$ −28.2973 −0.975190
$$843$$ −15.5005 −0.533867
$$844$$ 8.75005 0.301189
$$845$$ 12.8066 0.440560
$$846$$ 0.0364033 0.00125157
$$847$$ −3.37010 −0.115798
$$848$$ −13.7744 −0.473015
$$849$$ 53.5320 1.83721
$$850$$ −3.74490 −0.128449
$$851$$ −3.97994 −0.136431
$$852$$ 6.71888 0.230185
$$853$$ 4.99555 0.171044 0.0855222 0.996336i $$-0.472744\pi$$
0.0855222 + 0.996336i $$0.472744\pi$$
$$854$$ 23.1387 0.791791
$$855$$ 0.142317 0.00486712
$$856$$ 23.4326 0.800910
$$857$$ 3.32386 0.113541 0.0567705 0.998387i $$-0.481920\pi$$
0.0567705 + 0.998387i $$0.481920\pi$$
$$858$$ 0.945209 0.0322689
$$859$$ −32.5496 −1.11058 −0.555288 0.831658i $$-0.687392\pi$$
−0.555288 + 0.831658i $$0.687392\pi$$
$$860$$ 2.43436 0.0830108
$$861$$ 21.9832 0.749186
$$862$$ 2.85802 0.0973446
$$863$$ 7.39803 0.251832 0.125916 0.992041i $$-0.459813\pi$$
0.125916 + 0.992041i $$0.459813\pi$$
$$864$$ −14.7454 −0.501649
$$865$$ 8.21796 0.279419
$$866$$ 45.2483 1.53760
$$867$$ −13.2237 −0.449099
$$868$$ −12.1583 −0.412678
$$869$$ −15.4881 −0.525397
$$870$$ −12.5760 −0.426368
$$871$$ −2.70365 −0.0916096
$$872$$ −0.140660 −0.00476335
$$873$$ −0.954421 −0.0323023
$$874$$ −1.76497 −0.0597009
$$875$$ 3.37010 0.113930
$$876$$ −3.90211 −0.131840
$$877$$ 40.5912 1.37067 0.685333 0.728230i $$-0.259657\pi$$
0.685333 + 0.728230i $$0.259657\pi$$
$$878$$ 31.9630 1.07870
$$879$$ −19.7143 −0.664946
$$880$$ −2.65916 −0.0896403
$$881$$ 16.4133 0.552978 0.276489 0.961017i $$-0.410829\pi$$
0.276489 + 0.961017i $$0.410829\pi$$
$$882$$ 0.751899 0.0253178
$$883$$ 41.3719 1.39227 0.696137 0.717909i $$-0.254900\pi$$
0.696137 + 0.717909i $$0.254900\pi$$
$$884$$ −0.719912 −0.0242133
$$885$$ 19.2745 0.647904
$$886$$ −8.87064 −0.298015
$$887$$ 11.9068 0.399791 0.199896 0.979817i $$-0.435940\pi$$
0.199896 + 0.979817i $$0.435940\pi$$
$$888$$ 14.8664 0.498883
$$889$$ −5.06321 −0.169814
$$890$$ 7.59665 0.254640
$$891$$ 9.40670 0.315136
$$892$$ −1.82128 −0.0609808
$$893$$ −0.210971 −0.00705988
$$894$$ −4.84042 −0.161888
$$895$$ −11.2448 −0.375872
$$896$$ −16.5320 −0.552294
$$897$$ −1.13486 −0.0378918
$$898$$ −1.05796 −0.0353045
$$899$$ −39.8315 −1.32846
$$900$$ −0.0754249 −0.00251416
$$901$$ −15.9995 −0.533021
$$902$$ 4.46155 0.148553
$$903$$ −27.4405 −0.913162
$$904$$ −25.8719 −0.860487
$$905$$ −8.27670 −0.275127
$$906$$ −37.3143 −1.23968
$$907$$ −18.1787 −0.603614 −0.301807 0.953369i $$-0.597590\pi$$
−0.301807 + 0.953369i $$0.597590\pi$$
$$908$$ 5.12365 0.170034
$$909$$ 2.73422 0.0906885
$$910$$ −1.79699 −0.0595696
$$911$$ 32.5463 1.07831 0.539154 0.842207i $$-0.318744\pi$$
0.539154 + 0.842207i $$0.318744\pi$$
$$912$$ 4.71378 0.156089
$$913$$ −3.35537 −0.111046
$$914$$ −2.48246 −0.0821125
$$915$$ 10.0383 0.331856
$$916$$ −6.24830 −0.206450
$$917$$ 2.05065 0.0677183
$$918$$ 18.9705 0.626121
$$919$$ −22.8963 −0.755280 −0.377640 0.925952i $$-0.623264\pi$$
−0.377640 + 0.925952i $$0.623264\pi$$
$$920$$ 4.46533 0.147218
$$921$$ 44.1336 1.45425
$$922$$ 32.3162 1.06428
$$923$$ 3.14524 0.103527
$$924$$ −3.16612 −0.104158
$$925$$ −2.73402 −0.0898940
$$926$$ −23.6832 −0.778280
$$927$$ −2.13597 −0.0701543
$$928$$ 17.0324 0.559115
$$929$$ −39.9363 −1.31027 −0.655135 0.755512i $$-0.727388\pi$$
−0.655135 + 0.755512i $$0.727388\pi$$
$$930$$ 14.6304 0.479749
$$931$$ −4.35755 −0.142813
$$932$$ −1.06477 −0.0348778
$$933$$ 16.3797 0.536248
$$934$$ −35.5281 −1.16252
$$935$$ −3.08872 −0.101012
$$936$$ 0.191989 0.00627534
$$937$$ 51.4680 1.68139 0.840693 0.541511i $$-0.182148\pi$$
0.840693 + 0.541511i $$0.182148\pi$$
$$938$$ −25.1196 −0.820184
$$939$$ −26.3130 −0.858692
$$940$$ 0.111811 0.00364686
$$941$$ 56.3979 1.83852 0.919259 0.393652i $$-0.128789\pi$$
0.919259 + 0.393652i $$0.128789\pi$$
$$942$$ −8.07872 −0.263219
$$943$$ −5.35672 −0.174439
$$944$$ 28.9136 0.941057
$$945$$ −17.0719 −0.555348
$$946$$ −5.56911 −0.181068
$$947$$ 26.6837 0.867103 0.433552 0.901129i $$-0.357260\pi$$
0.433552 + 0.901129i $$0.357260\pi$$
$$948$$ −14.5506 −0.472583
$$949$$ −1.82666 −0.0592957
$$950$$ −1.21244 −0.0393369
$$951$$ 59.9354 1.94354
$$952$$ −31.9301 −1.03486
$$953$$ −14.9629 −0.484695 −0.242348 0.970190i $$-0.577917\pi$$
−0.242348 + 0.970190i $$0.577917\pi$$
$$954$$ 0.893810 0.0289382
$$955$$ −3.21777 −0.104125
$$956$$ 3.37122 0.109033
$$957$$ −10.3725 −0.335295
$$958$$ −1.17078 −0.0378263
$$959$$ 72.2999 2.33469
$$960$$ −15.6837 −0.506188
$$961$$ 15.3382 0.494779
$$962$$ 1.45782 0.0470021
$$963$$ −1.08717 −0.0350336
$$964$$ 15.0395 0.484390
$$965$$ −19.0447 −0.613070
$$966$$ −10.5440 −0.339247
$$967$$ 19.7493 0.635094 0.317547 0.948242i $$-0.397141\pi$$
0.317547 + 0.948242i $$0.397141\pi$$
$$968$$ −3.06746 −0.0985919
$$969$$ 5.47525 0.175890
$$970$$ 8.13104 0.261072
$$971$$ −53.2159 −1.70778 −0.853889 0.520455i $$-0.825762\pi$$
−0.853889 + 0.520455i $$0.825762\pi$$
$$972$$ 0.783187 0.0251207
$$973$$ 29.8469 0.956847
$$974$$ −4.07533 −0.130582
$$975$$ −0.779590 −0.0249669
$$976$$ 15.0585 0.482009
$$977$$ −49.9728 −1.59877 −0.799385 0.600819i $$-0.794841\pi$$
−0.799385 + 0.600819i $$0.794841\pi$$
$$978$$ −25.4546 −0.813950
$$979$$ 6.26557 0.200249
$$980$$ 2.30941 0.0737715
$$981$$ 0.00652601 0.000208360 0
$$982$$ −39.5053 −1.26067
$$983$$ 29.2283 0.932239 0.466119 0.884722i $$-0.345652\pi$$
0.466119 + 0.884722i $$0.345652\pi$$
$$984$$ 20.0091 0.637866
$$985$$ −2.04266 −0.0650846
$$986$$ −21.9128 −0.697846
$$987$$ −1.26035 −0.0401173
$$988$$ −0.233078 −0.00741519
$$989$$ 6.68651 0.212619
$$990$$ 0.172551 0.00548403
$$991$$ −7.92503 −0.251747 −0.125873 0.992046i $$-0.540173\pi$$
−0.125873 + 0.992046i $$0.540173\pi$$
$$992$$ −19.8147 −0.629116
$$993$$ 20.4062 0.647571
$$994$$ 29.2224 0.926879
$$995$$ −4.61202 −0.146211
$$996$$ −3.15228 −0.0998837
$$997$$ 11.9229 0.377601 0.188800 0.982015i $$-0.439540\pi$$
0.188800 + 0.982015i $$0.439540\pi$$
$$998$$ 9.14022 0.289329
$$999$$ 13.8497 0.438185
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 1045.2.a.f.1.5 6
3.2 odd 2 9405.2.a.z.1.2 6
5.4 even 2 5225.2.a.l.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.f.1.5 6 1.1 even 1 trivial
5225.2.a.l.1.2 6 5.4 even 2
9405.2.a.z.1.2 6 3.2 odd 2