# Properties

 Label 966.2.a.l.1.1 Level $966$ Weight $2$ Character 966.1 Self dual yes Analytic conductor $7.714$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, 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("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.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: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 966.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.56155 q^{10} +5.12311 q^{11} -1.00000 q^{12} -3.56155 q^{13} +1.00000 q^{14} +1.56155 q^{15} +1.00000 q^{16} -1.12311 q^{17} -1.00000 q^{18} -5.12311 q^{19} -1.56155 q^{20} +1.00000 q^{21} -5.12311 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.56155 q^{25} +3.56155 q^{26} -1.00000 q^{27} -1.00000 q^{28} +7.56155 q^{29} -1.56155 q^{30} +3.12311 q^{31} -1.00000 q^{32} -5.12311 q^{33} +1.12311 q^{34} +1.56155 q^{35} +1.00000 q^{36} -1.56155 q^{37} +5.12311 q^{38} +3.56155 q^{39} +1.56155 q^{40} +3.56155 q^{41} -1.00000 q^{42} +6.68466 q^{43} +5.12311 q^{44} -1.56155 q^{45} -1.00000 q^{46} +2.43845 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.56155 q^{50} +1.12311 q^{51} -3.56155 q^{52} +14.2462 q^{53} +1.00000 q^{54} -8.00000 q^{55} +1.00000 q^{56} +5.12311 q^{57} -7.56155 q^{58} +4.87689 q^{59} +1.56155 q^{60} +0.876894 q^{61} -3.12311 q^{62} -1.00000 q^{63} +1.00000 q^{64} +5.56155 q^{65} +5.12311 q^{66} -1.12311 q^{67} -1.12311 q^{68} -1.00000 q^{69} -1.56155 q^{70} -9.36932 q^{71} -1.00000 q^{72} +9.12311 q^{73} +1.56155 q^{74} +2.56155 q^{75} -5.12311 q^{76} -5.12311 q^{77} -3.56155 q^{78} +14.2462 q^{79} -1.56155 q^{80} +1.00000 q^{81} -3.56155 q^{82} -9.12311 q^{83} +1.00000 q^{84} +1.75379 q^{85} -6.68466 q^{86} -7.56155 q^{87} -5.12311 q^{88} +14.0000 q^{89} +1.56155 q^{90} +3.56155 q^{91} +1.00000 q^{92} -3.12311 q^{93} -2.43845 q^{94} +8.00000 q^{95} +1.00000 q^{96} -12.4384 q^{97} -1.00000 q^{98} +5.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} - 2 q^{12} - 3 q^{13} + 2 q^{14} - q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 2 q^{19} + q^{20} + 2 q^{21} - 2 q^{22} + 2 q^{23} + 2 q^{24} - q^{25} + 3 q^{26} - 2 q^{27} - 2 q^{28} + 11 q^{29} + q^{30} - 2 q^{31} - 2 q^{32} - 2 q^{33} - 6 q^{34} - q^{35} + 2 q^{36} + q^{37} + 2 q^{38} + 3 q^{39} - q^{40} + 3 q^{41} - 2 q^{42} + q^{43} + 2 q^{44} + q^{45} - 2 q^{46} + 9 q^{47} - 2 q^{48} + 2 q^{49} + q^{50} - 6 q^{51} - 3 q^{52} + 12 q^{53} + 2 q^{54} - 16 q^{55} + 2 q^{56} + 2 q^{57} - 11 q^{58} + 18 q^{59} - q^{60} + 10 q^{61} + 2 q^{62} - 2 q^{63} + 2 q^{64} + 7 q^{65} + 2 q^{66} + 6 q^{67} + 6 q^{68} - 2 q^{69} + q^{70} + 6 q^{71} - 2 q^{72} + 10 q^{73} - q^{74} + q^{75} - 2 q^{76} - 2 q^{77} - 3 q^{78} + 12 q^{79} + q^{80} + 2 q^{81} - 3 q^{82} - 10 q^{83} + 2 q^{84} + 20 q^{85} - q^{86} - 11 q^{87} - 2 q^{88} + 28 q^{89} - q^{90} + 3 q^{91} + 2 q^{92} + 2 q^{93} - 9 q^{94} + 16 q^{95} + 2 q^{96} - 29 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - q^10 + 2 * q^11 - 2 * q^12 - 3 * q^13 + 2 * q^14 - q^15 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 2 * q^19 + q^20 + 2 * q^21 - 2 * q^22 + 2 * q^23 + 2 * q^24 - q^25 + 3 * q^26 - 2 * q^27 - 2 * q^28 + 11 * q^29 + q^30 - 2 * q^31 - 2 * q^32 - 2 * q^33 - 6 * q^34 - q^35 + 2 * q^36 + q^37 + 2 * q^38 + 3 * q^39 - q^40 + 3 * q^41 - 2 * q^42 + q^43 + 2 * q^44 + q^45 - 2 * q^46 + 9 * q^47 - 2 * q^48 + 2 * q^49 + q^50 - 6 * q^51 - 3 * q^52 + 12 * q^53 + 2 * q^54 - 16 * q^55 + 2 * q^56 + 2 * q^57 - 11 * q^58 + 18 * q^59 - q^60 + 10 * q^61 + 2 * q^62 - 2 * q^63 + 2 * q^64 + 7 * q^65 + 2 * q^66 + 6 * q^67 + 6 * q^68 - 2 * q^69 + q^70 + 6 * q^71 - 2 * q^72 + 10 * q^73 - q^74 + q^75 - 2 * q^76 - 2 * q^77 - 3 * q^78 + 12 * q^79 + q^80 + 2 * q^81 - 3 * q^82 - 10 * q^83 + 2 * q^84 + 20 * q^85 - q^86 - 11 * q^87 - 2 * q^88 + 28 * q^89 - q^90 + 3 * q^91 + 2 * q^92 + 2 * q^93 - 9 * q^94 + 16 * q^95 + 2 * q^96 - 29 * q^97 - 2 * q^98 + 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.56155 −0.698348 −0.349174 0.937058i $$-0.613538\pi$$
−0.349174 + 0.937058i $$0.613538\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.56155 0.493806
$$11$$ 5.12311 1.54467 0.772337 0.635213i $$-0.219088\pi$$
0.772337 + 0.635213i $$0.219088\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −3.56155 −0.987797 −0.493899 0.869520i $$-0.664429\pi$$
−0.493899 + 0.869520i $$0.664429\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 1.56155 0.403191
$$16$$ 1.00000 0.250000
$$17$$ −1.12311 −0.272393 −0.136197 0.990682i $$-0.543488\pi$$
−0.136197 + 0.990682i $$0.543488\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −5.12311 −1.17532 −0.587661 0.809108i $$-0.699951\pi$$
−0.587661 + 0.809108i $$0.699951\pi$$
$$20$$ −1.56155 −0.349174
$$21$$ 1.00000 0.218218
$$22$$ −5.12311 −1.09225
$$23$$ 1.00000 0.208514
$$24$$ 1.00000 0.204124
$$25$$ −2.56155 −0.512311
$$26$$ 3.56155 0.698478
$$27$$ −1.00000 −0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 7.56155 1.40415 0.702073 0.712105i $$-0.252258\pi$$
0.702073 + 0.712105i $$0.252258\pi$$
$$30$$ −1.56155 −0.285099
$$31$$ 3.12311 0.560926 0.280463 0.959865i $$-0.409512\pi$$
0.280463 + 0.959865i $$0.409512\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −5.12311 −0.891818
$$34$$ 1.12311 0.192611
$$35$$ 1.56155 0.263951
$$36$$ 1.00000 0.166667
$$37$$ −1.56155 −0.256718 −0.128359 0.991728i $$-0.540971\pi$$
−0.128359 + 0.991728i $$0.540971\pi$$
$$38$$ 5.12311 0.831077
$$39$$ 3.56155 0.570305
$$40$$ 1.56155 0.246903
$$41$$ 3.56155 0.556221 0.278111 0.960549i $$-0.410292\pi$$
0.278111 + 0.960549i $$0.410292\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 6.68466 1.01940 0.509700 0.860352i $$-0.329756\pi$$
0.509700 + 0.860352i $$0.329756\pi$$
$$44$$ 5.12311 0.772337
$$45$$ −1.56155 −0.232783
$$46$$ −1.00000 −0.147442
$$47$$ 2.43845 0.355684 0.177842 0.984059i $$-0.443088\pi$$
0.177842 + 0.984059i $$0.443088\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 2.56155 0.362258
$$51$$ 1.12311 0.157266
$$52$$ −3.56155 −0.493899
$$53$$ 14.2462 1.95687 0.978434 0.206561i $$-0.0662271\pi$$
0.978434 + 0.206561i $$0.0662271\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −8.00000 −1.07872
$$56$$ 1.00000 0.133631
$$57$$ 5.12311 0.678572
$$58$$ −7.56155 −0.992881
$$59$$ 4.87689 0.634918 0.317459 0.948272i $$-0.397170\pi$$
0.317459 + 0.948272i $$0.397170\pi$$
$$60$$ 1.56155 0.201596
$$61$$ 0.876894 0.112275 0.0561374 0.998423i $$-0.482122\pi$$
0.0561374 + 0.998423i $$0.482122\pi$$
$$62$$ −3.12311 −0.396635
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 5.56155 0.689826
$$66$$ 5.12311 0.630611
$$67$$ −1.12311 −0.137209 −0.0686046 0.997644i $$-0.521855\pi$$
−0.0686046 + 0.997644i $$0.521855\pi$$
$$68$$ −1.12311 −0.136197
$$69$$ −1.00000 −0.120386
$$70$$ −1.56155 −0.186641
$$71$$ −9.36932 −1.11193 −0.555967 0.831205i $$-0.687652\pi$$
−0.555967 + 0.831205i $$0.687652\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 9.12311 1.06778 0.533889 0.845554i $$-0.320730\pi$$
0.533889 + 0.845554i $$0.320730\pi$$
$$74$$ 1.56155 0.181527
$$75$$ 2.56155 0.295783
$$76$$ −5.12311 −0.587661
$$77$$ −5.12311 −0.583832
$$78$$ −3.56155 −0.403266
$$79$$ 14.2462 1.60282 0.801412 0.598113i $$-0.204082\pi$$
0.801412 + 0.598113i $$0.204082\pi$$
$$80$$ −1.56155 −0.174587
$$81$$ 1.00000 0.111111
$$82$$ −3.56155 −0.393308
$$83$$ −9.12311 −1.00139 −0.500695 0.865624i $$-0.666922\pi$$
−0.500695 + 0.865624i $$0.666922\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 1.75379 0.190225
$$86$$ −6.68466 −0.720825
$$87$$ −7.56155 −0.810684
$$88$$ −5.12311 −0.546125
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 1.56155 0.164602
$$91$$ 3.56155 0.373352
$$92$$ 1.00000 0.104257
$$93$$ −3.12311 −0.323851
$$94$$ −2.43845 −0.251507
$$95$$ 8.00000 0.820783
$$96$$ 1.00000 0.102062
$$97$$ −12.4384 −1.26293 −0.631466 0.775403i $$-0.717547\pi$$
−0.631466 + 0.775403i $$0.717547\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 5.12311 0.514891
$$100$$ −2.56155 −0.256155
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ −1.12311 −0.111204
$$103$$ 18.9309 1.86531 0.932657 0.360764i $$-0.117484\pi$$
0.932657 + 0.360764i $$0.117484\pi$$
$$104$$ 3.56155 0.349239
$$105$$ −1.56155 −0.152392
$$106$$ −14.2462 −1.38371
$$107$$ −5.12311 −0.495269 −0.247635 0.968853i $$-0.579653\pi$$
−0.247635 + 0.968853i $$0.579653\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.4384 0.999822 0.499911 0.866077i $$-0.333366\pi$$
0.499911 + 0.866077i $$0.333366\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 1.56155 0.148216
$$112$$ −1.00000 −0.0944911
$$113$$ −2.68466 −0.252551 −0.126276 0.991995i $$-0.540302\pi$$
−0.126276 + 0.991995i $$0.540302\pi$$
$$114$$ −5.12311 −0.479823
$$115$$ −1.56155 −0.145616
$$116$$ 7.56155 0.702073
$$117$$ −3.56155 −0.329266
$$118$$ −4.87689 −0.448955
$$119$$ 1.12311 0.102955
$$120$$ −1.56155 −0.142550
$$121$$ 15.2462 1.38602
$$122$$ −0.876894 −0.0793903
$$123$$ −3.56155 −0.321134
$$124$$ 3.12311 0.280463
$$125$$ 11.8078 1.05612
$$126$$ 1.00000 0.0890871
$$127$$ −16.6847 −1.48052 −0.740262 0.672318i $$-0.765299\pi$$
−0.740262 + 0.672318i $$0.765299\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −6.68466 −0.588551
$$130$$ −5.56155 −0.487780
$$131$$ 4.87689 0.426096 0.213048 0.977042i $$-0.431661\pi$$
0.213048 + 0.977042i $$0.431661\pi$$
$$132$$ −5.12311 −0.445909
$$133$$ 5.12311 0.444230
$$134$$ 1.12311 0.0970215
$$135$$ 1.56155 0.134397
$$136$$ 1.12311 0.0963055
$$137$$ −0.438447 −0.0374591 −0.0187295 0.999825i $$-0.505962\pi$$
−0.0187295 + 0.999825i $$0.505962\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ 3.80776 0.322970 0.161485 0.986875i $$-0.448372\pi$$
0.161485 + 0.986875i $$0.448372\pi$$
$$140$$ 1.56155 0.131975
$$141$$ −2.43845 −0.205354
$$142$$ 9.36932 0.786256
$$143$$ −18.2462 −1.52582
$$144$$ 1.00000 0.0833333
$$145$$ −11.8078 −0.980581
$$146$$ −9.12311 −0.755034
$$147$$ −1.00000 −0.0824786
$$148$$ −1.56155 −0.128359
$$149$$ −6.24621 −0.511710 −0.255855 0.966715i $$-0.582357\pi$$
−0.255855 + 0.966715i $$0.582357\pi$$
$$150$$ −2.56155 −0.209150
$$151$$ 7.31534 0.595314 0.297657 0.954673i $$-0.403795\pi$$
0.297657 + 0.954673i $$0.403795\pi$$
$$152$$ 5.12311 0.415539
$$153$$ −1.12311 −0.0907977
$$154$$ 5.12311 0.412832
$$155$$ −4.87689 −0.391722
$$156$$ 3.56155 0.285152
$$157$$ 14.2462 1.13697 0.568486 0.822693i $$-0.307529\pi$$
0.568486 + 0.822693i $$0.307529\pi$$
$$158$$ −14.2462 −1.13337
$$159$$ −14.2462 −1.12980
$$160$$ 1.56155 0.123452
$$161$$ −1.00000 −0.0788110
$$162$$ −1.00000 −0.0785674
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ 3.56155 0.278111
$$165$$ 8.00000 0.622799
$$166$$ 9.12311 0.708090
$$167$$ −14.2462 −1.10240 −0.551202 0.834372i $$-0.685831\pi$$
−0.551202 + 0.834372i $$0.685831\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −0.315342 −0.0242570
$$170$$ −1.75379 −0.134509
$$171$$ −5.12311 −0.391774
$$172$$ 6.68466 0.509700
$$173$$ 18.4924 1.40595 0.702976 0.711213i $$-0.251854\pi$$
0.702976 + 0.711213i $$0.251854\pi$$
$$174$$ 7.56155 0.573240
$$175$$ 2.56155 0.193635
$$176$$ 5.12311 0.386169
$$177$$ −4.87689 −0.366570
$$178$$ −14.0000 −1.04934
$$179$$ −3.80776 −0.284606 −0.142303 0.989823i $$-0.545451\pi$$
−0.142303 + 0.989823i $$0.545451\pi$$
$$180$$ −1.56155 −0.116391
$$181$$ −15.1231 −1.12409 −0.562046 0.827106i $$-0.689986\pi$$
−0.562046 + 0.827106i $$0.689986\pi$$
$$182$$ −3.56155 −0.264000
$$183$$ −0.876894 −0.0648219
$$184$$ −1.00000 −0.0737210
$$185$$ 2.43845 0.179278
$$186$$ 3.12311 0.228997
$$187$$ −5.75379 −0.420759
$$188$$ 2.43845 0.177842
$$189$$ 1.00000 0.0727393
$$190$$ −8.00000 −0.580381
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 2.68466 0.193246 0.0966230 0.995321i $$-0.469196\pi$$
0.0966230 + 0.995321i $$0.469196\pi$$
$$194$$ 12.4384 0.893028
$$195$$ −5.56155 −0.398271
$$196$$ 1.00000 0.0714286
$$197$$ 24.9309 1.77625 0.888125 0.459601i $$-0.152008\pi$$
0.888125 + 0.459601i $$0.152008\pi$$
$$198$$ −5.12311 −0.364083
$$199$$ −21.1771 −1.50120 −0.750602 0.660755i $$-0.770236\pi$$
−0.750602 + 0.660755i $$0.770236\pi$$
$$200$$ 2.56155 0.181129
$$201$$ 1.12311 0.0792178
$$202$$ −16.2462 −1.14308
$$203$$ −7.56155 −0.530717
$$204$$ 1.12311 0.0786331
$$205$$ −5.56155 −0.388436
$$206$$ −18.9309 −1.31898
$$207$$ 1.00000 0.0695048
$$208$$ −3.56155 −0.246949
$$209$$ −26.2462 −1.81549
$$210$$ 1.56155 0.107757
$$211$$ −16.4924 −1.13539 −0.567693 0.823241i $$-0.692164\pi$$
−0.567693 + 0.823241i $$0.692164\pi$$
$$212$$ 14.2462 0.978434
$$213$$ 9.36932 0.641975
$$214$$ 5.12311 0.350208
$$215$$ −10.4384 −0.711896
$$216$$ 1.00000 0.0680414
$$217$$ −3.12311 −0.212010
$$218$$ −10.4384 −0.706981
$$219$$ −9.12311 −0.616482
$$220$$ −8.00000 −0.539360
$$221$$ 4.00000 0.269069
$$222$$ −1.56155 −0.104805
$$223$$ 3.12311 0.209139 0.104569 0.994518i $$-0.466654\pi$$
0.104569 + 0.994518i $$0.466654\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −2.56155 −0.170770
$$226$$ 2.68466 0.178581
$$227$$ −6.19224 −0.410993 −0.205497 0.978658i $$-0.565881\pi$$
−0.205497 + 0.978658i $$0.565881\pi$$
$$228$$ 5.12311 0.339286
$$229$$ 2.24621 0.148434 0.0742169 0.997242i $$-0.476354\pi$$
0.0742169 + 0.997242i $$0.476354\pi$$
$$230$$ 1.56155 0.102966
$$231$$ 5.12311 0.337076
$$232$$ −7.56155 −0.496440
$$233$$ −18.4924 −1.21148 −0.605739 0.795663i $$-0.707123\pi$$
−0.605739 + 0.795663i $$0.707123\pi$$
$$234$$ 3.56155 0.232826
$$235$$ −3.80776 −0.248391
$$236$$ 4.87689 0.317459
$$237$$ −14.2462 −0.925391
$$238$$ −1.12311 −0.0728001
$$239$$ 4.49242 0.290591 0.145295 0.989388i $$-0.453587\pi$$
0.145295 + 0.989388i $$0.453587\pi$$
$$240$$ 1.56155 0.100798
$$241$$ −20.0540 −1.29179 −0.645895 0.763426i $$-0.723516\pi$$
−0.645895 + 0.763426i $$0.723516\pi$$
$$242$$ −15.2462 −0.980064
$$243$$ −1.00000 −0.0641500
$$244$$ 0.876894 0.0561374
$$245$$ −1.56155 −0.0997639
$$246$$ 3.56155 0.227076
$$247$$ 18.2462 1.16098
$$248$$ −3.12311 −0.198317
$$249$$ 9.12311 0.578153
$$250$$ −11.8078 −0.746789
$$251$$ −0.438447 −0.0276745 −0.0138373 0.999904i $$-0.504405\pi$$
−0.0138373 + 0.999904i $$0.504405\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 5.12311 0.322087
$$254$$ 16.6847 1.04689
$$255$$ −1.75379 −0.109827
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 6.68466 0.416169
$$259$$ 1.56155 0.0970302
$$260$$ 5.56155 0.344913
$$261$$ 7.56155 0.468048
$$262$$ −4.87689 −0.301296
$$263$$ 22.9309 1.41398 0.706989 0.707225i $$-0.250053\pi$$
0.706989 + 0.707225i $$0.250053\pi$$
$$264$$ 5.12311 0.315305
$$265$$ −22.2462 −1.36657
$$266$$ −5.12311 −0.314118
$$267$$ −14.0000 −0.856786
$$268$$ −1.12311 −0.0686046
$$269$$ 12.2462 0.746665 0.373332 0.927698i $$-0.378215\pi$$
0.373332 + 0.927698i $$0.378215\pi$$
$$270$$ −1.56155 −0.0950331
$$271$$ 14.2462 0.865396 0.432698 0.901539i $$-0.357562\pi$$
0.432698 + 0.901539i $$0.357562\pi$$
$$272$$ −1.12311 −0.0680983
$$273$$ −3.56155 −0.215555
$$274$$ 0.438447 0.0264876
$$275$$ −13.1231 −0.791353
$$276$$ −1.00000 −0.0601929
$$277$$ 13.1231 0.788491 0.394245 0.919005i $$-0.371006\pi$$
0.394245 + 0.919005i $$0.371006\pi$$
$$278$$ −3.80776 −0.228375
$$279$$ 3.12311 0.186975
$$280$$ −1.56155 −0.0933206
$$281$$ 4.93087 0.294151 0.147076 0.989125i $$-0.453014\pi$$
0.147076 + 0.989125i $$0.453014\pi$$
$$282$$ 2.43845 0.145207
$$283$$ −10.4924 −0.623710 −0.311855 0.950130i $$-0.600950\pi$$
−0.311855 + 0.950130i $$0.600950\pi$$
$$284$$ −9.36932 −0.555967
$$285$$ −8.00000 −0.473879
$$286$$ 18.2462 1.07892
$$287$$ −3.56155 −0.210232
$$288$$ −1.00000 −0.0589256
$$289$$ −15.7386 −0.925802
$$290$$ 11.8078 0.693376
$$291$$ 12.4384 0.729155
$$292$$ 9.12311 0.533889
$$293$$ 27.6155 1.61332 0.806658 0.591018i $$-0.201274\pi$$
0.806658 + 0.591018i $$0.201274\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −7.61553 −0.443393
$$296$$ 1.56155 0.0907634
$$297$$ −5.12311 −0.297273
$$298$$ 6.24621 0.361833
$$299$$ −3.56155 −0.205970
$$300$$ 2.56155 0.147891
$$301$$ −6.68466 −0.385297
$$302$$ −7.31534 −0.420951
$$303$$ −16.2462 −0.933320
$$304$$ −5.12311 −0.293830
$$305$$ −1.36932 −0.0784069
$$306$$ 1.12311 0.0642037
$$307$$ 2.93087 0.167274 0.0836368 0.996496i $$-0.473346\pi$$
0.0836368 + 0.996496i $$0.473346\pi$$
$$308$$ −5.12311 −0.291916
$$309$$ −18.9309 −1.07694
$$310$$ 4.87689 0.276989
$$311$$ 18.7386 1.06257 0.531285 0.847193i $$-0.321709\pi$$
0.531285 + 0.847193i $$0.321709\pi$$
$$312$$ −3.56155 −0.201633
$$313$$ 4.24621 0.240010 0.120005 0.992773i $$-0.461709\pi$$
0.120005 + 0.992773i $$0.461709\pi$$
$$314$$ −14.2462 −0.803960
$$315$$ 1.56155 0.0879835
$$316$$ 14.2462 0.801412
$$317$$ −17.8078 −1.00018 −0.500092 0.865972i $$-0.666700\pi$$
−0.500092 + 0.865972i $$0.666700\pi$$
$$318$$ 14.2462 0.798888
$$319$$ 38.7386 2.16895
$$320$$ −1.56155 −0.0872935
$$321$$ 5.12311 0.285944
$$322$$ 1.00000 0.0557278
$$323$$ 5.75379 0.320149
$$324$$ 1.00000 0.0555556
$$325$$ 9.12311 0.506059
$$326$$ 8.00000 0.443079
$$327$$ −10.4384 −0.577247
$$328$$ −3.56155 −0.196654
$$329$$ −2.43845 −0.134436
$$330$$ −8.00000 −0.440386
$$331$$ −24.4924 −1.34623 −0.673113 0.739540i $$-0.735043\pi$$
−0.673113 + 0.739540i $$0.735043\pi$$
$$332$$ −9.12311 −0.500695
$$333$$ −1.56155 −0.0855726
$$334$$ 14.2462 0.779518
$$335$$ 1.75379 0.0958197
$$336$$ 1.00000 0.0545545
$$337$$ 27.3693 1.49090 0.745451 0.666561i $$-0.232234\pi$$
0.745451 + 0.666561i $$0.232234\pi$$
$$338$$ 0.315342 0.0171523
$$339$$ 2.68466 0.145811
$$340$$ 1.75379 0.0951125
$$341$$ 16.0000 0.866449
$$342$$ 5.12311 0.277026
$$343$$ −1.00000 −0.0539949
$$344$$ −6.68466 −0.360413
$$345$$ 1.56155 0.0840712
$$346$$ −18.4924 −0.994159
$$347$$ 14.9309 0.801531 0.400766 0.916181i $$-0.368744\pi$$
0.400766 + 0.916181i $$0.368744\pi$$
$$348$$ −7.56155 −0.405342
$$349$$ −16.2462 −0.869640 −0.434820 0.900517i $$-0.643188\pi$$
−0.434820 + 0.900517i $$0.643188\pi$$
$$350$$ −2.56155 −0.136921
$$351$$ 3.56155 0.190102
$$352$$ −5.12311 −0.273062
$$353$$ −33.8078 −1.79941 −0.899703 0.436503i $$-0.856217\pi$$
−0.899703 + 0.436503i $$0.856217\pi$$
$$354$$ 4.87689 0.259204
$$355$$ 14.6307 0.776516
$$356$$ 14.0000 0.741999
$$357$$ −1.12311 −0.0594411
$$358$$ 3.80776 0.201247
$$359$$ −6.43845 −0.339808 −0.169904 0.985461i $$-0.554346\pi$$
−0.169904 + 0.985461i $$0.554346\pi$$
$$360$$ 1.56155 0.0823011
$$361$$ 7.24621 0.381380
$$362$$ 15.1231 0.794853
$$363$$ −15.2462 −0.800219
$$364$$ 3.56155 0.186676
$$365$$ −14.2462 −0.745681
$$366$$ 0.876894 0.0458360
$$367$$ −18.4384 −0.962479 −0.481240 0.876589i $$-0.659813\pi$$
−0.481240 + 0.876589i $$0.659813\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 3.56155 0.185407
$$370$$ −2.43845 −0.126769
$$371$$ −14.2462 −0.739626
$$372$$ −3.12311 −0.161925
$$373$$ 15.1231 0.783045 0.391522 0.920169i $$-0.371949\pi$$
0.391522 + 0.920169i $$0.371949\pi$$
$$374$$ 5.75379 0.297521
$$375$$ −11.8078 −0.609750
$$376$$ −2.43845 −0.125753
$$377$$ −26.9309 −1.38701
$$378$$ −1.00000 −0.0514344
$$379$$ 20.0540 1.03010 0.515052 0.857159i $$-0.327773\pi$$
0.515052 + 0.857159i $$0.327773\pi$$
$$380$$ 8.00000 0.410391
$$381$$ 16.6847 0.854781
$$382$$ 20.0000 1.02329
$$383$$ −6.24621 −0.319166 −0.159583 0.987184i $$-0.551015\pi$$
−0.159583 + 0.987184i $$0.551015\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 8.00000 0.407718
$$386$$ −2.68466 −0.136646
$$387$$ 6.68466 0.339800
$$388$$ −12.4384 −0.631466
$$389$$ −29.3693 −1.48908 −0.744542 0.667576i $$-0.767332\pi$$
−0.744542 + 0.667576i $$0.767332\pi$$
$$390$$ 5.56155 0.281620
$$391$$ −1.12311 −0.0567979
$$392$$ −1.00000 −0.0505076
$$393$$ −4.87689 −0.246007
$$394$$ −24.9309 −1.25600
$$395$$ −22.2462 −1.11933
$$396$$ 5.12311 0.257446
$$397$$ −7.75379 −0.389152 −0.194576 0.980887i $$-0.562333\pi$$
−0.194576 + 0.980887i $$0.562333\pi$$
$$398$$ 21.1771 1.06151
$$399$$ −5.12311 −0.256476
$$400$$ −2.56155 −0.128078
$$401$$ 16.2462 0.811297 0.405649 0.914029i $$-0.367046\pi$$
0.405649 + 0.914029i $$0.367046\pi$$
$$402$$ −1.12311 −0.0560154
$$403$$ −11.1231 −0.554081
$$404$$ 16.2462 0.808279
$$405$$ −1.56155 −0.0775942
$$406$$ 7.56155 0.375274
$$407$$ −8.00000 −0.396545
$$408$$ −1.12311 −0.0556020
$$409$$ 33.6155 1.66218 0.831090 0.556137i $$-0.187717\pi$$
0.831090 + 0.556137i $$0.187717\pi$$
$$410$$ 5.56155 0.274666
$$411$$ 0.438447 0.0216270
$$412$$ 18.9309 0.932657
$$413$$ −4.87689 −0.239976
$$414$$ −1.00000 −0.0491473
$$415$$ 14.2462 0.699319
$$416$$ 3.56155 0.174619
$$417$$ −3.80776 −0.186467
$$418$$ 26.2462 1.28374
$$419$$ 23.8617 1.16572 0.582861 0.812572i $$-0.301933\pi$$
0.582861 + 0.812572i $$0.301933\pi$$
$$420$$ −1.56155 −0.0761960
$$421$$ 33.1771 1.61695 0.808476 0.588529i $$-0.200293\pi$$
0.808476 + 0.588529i $$0.200293\pi$$
$$422$$ 16.4924 0.802839
$$423$$ 2.43845 0.118561
$$424$$ −14.2462 −0.691857
$$425$$ 2.87689 0.139550
$$426$$ −9.36932 −0.453945
$$427$$ −0.876894 −0.0424359
$$428$$ −5.12311 −0.247635
$$429$$ 18.2462 0.880935
$$430$$ 10.4384 0.503387
$$431$$ −3.31534 −0.159694 −0.0798472 0.996807i $$-0.525443\pi$$
−0.0798472 + 0.996807i $$0.525443\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 14.6847 0.705700 0.352850 0.935680i $$-0.385213\pi$$
0.352850 + 0.935680i $$0.385213\pi$$
$$434$$ 3.12311 0.149914
$$435$$ 11.8078 0.566139
$$436$$ 10.4384 0.499911
$$437$$ −5.12311 −0.245071
$$438$$ 9.12311 0.435919
$$439$$ −23.6155 −1.12711 −0.563554 0.826079i $$-0.690566\pi$$
−0.563554 + 0.826079i $$0.690566\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 1.00000 0.0476190
$$442$$ −4.00000 −0.190261
$$443$$ 9.06913 0.430887 0.215444 0.976516i $$-0.430880\pi$$
0.215444 + 0.976516i $$0.430880\pi$$
$$444$$ 1.56155 0.0741080
$$445$$ −21.8617 −1.03635
$$446$$ −3.12311 −0.147883
$$447$$ 6.24621 0.295436
$$448$$ −1.00000 −0.0472456
$$449$$ −0.246211 −0.0116194 −0.00580971 0.999983i $$-0.501849\pi$$
−0.00580971 + 0.999983i $$0.501849\pi$$
$$450$$ 2.56155 0.120753
$$451$$ 18.2462 0.859181
$$452$$ −2.68466 −0.126276
$$453$$ −7.31534 −0.343705
$$454$$ 6.19224 0.290616
$$455$$ −5.56155 −0.260730
$$456$$ −5.12311 −0.239911
$$457$$ 7.36932 0.344722 0.172361 0.985034i $$-0.444860\pi$$
0.172361 + 0.985034i $$0.444860\pi$$
$$458$$ −2.24621 −0.104959
$$459$$ 1.12311 0.0524221
$$460$$ −1.56155 −0.0728078
$$461$$ 40.7386 1.89739 0.948694 0.316197i $$-0.102406\pi$$
0.948694 + 0.316197i $$0.102406\pi$$
$$462$$ −5.12311 −0.238348
$$463$$ −13.5616 −0.630259 −0.315129 0.949049i $$-0.602048\pi$$
−0.315129 + 0.949049i $$0.602048\pi$$
$$464$$ 7.56155 0.351036
$$465$$ 4.87689 0.226161
$$466$$ 18.4924 0.856645
$$467$$ 2.68466 0.124231 0.0621156 0.998069i $$-0.480215\pi$$
0.0621156 + 0.998069i $$0.480215\pi$$
$$468$$ −3.56155 −0.164633
$$469$$ 1.12311 0.0518602
$$470$$ 3.80776 0.175639
$$471$$ −14.2462 −0.656431
$$472$$ −4.87689 −0.224477
$$473$$ 34.2462 1.57464
$$474$$ 14.2462 0.654350
$$475$$ 13.1231 0.602129
$$476$$ 1.12311 0.0514775
$$477$$ 14.2462 0.652289
$$478$$ −4.49242 −0.205479
$$479$$ 31.6155 1.44455 0.722275 0.691606i $$-0.243096\pi$$
0.722275 + 0.691606i $$0.243096\pi$$
$$480$$ −1.56155 −0.0712748
$$481$$ 5.56155 0.253585
$$482$$ 20.0540 0.913434
$$483$$ 1.00000 0.0455016
$$484$$ 15.2462 0.693010
$$485$$ 19.4233 0.881966
$$486$$ 1.00000 0.0453609
$$487$$ −5.56155 −0.252018 −0.126009 0.992029i $$-0.540217\pi$$
−0.126009 + 0.992029i $$0.540217\pi$$
$$488$$ −0.876894 −0.0396951
$$489$$ 8.00000 0.361773
$$490$$ 1.56155 0.0705438
$$491$$ −26.7386 −1.20670 −0.603349 0.797477i $$-0.706167\pi$$
−0.603349 + 0.797477i $$0.706167\pi$$
$$492$$ −3.56155 −0.160567
$$493$$ −8.49242 −0.382479
$$494$$ −18.2462 −0.820936
$$495$$ −8.00000 −0.359573
$$496$$ 3.12311 0.140232
$$497$$ 9.36932 0.420271
$$498$$ −9.12311 −0.408816
$$499$$ 41.3693 1.85194 0.925972 0.377591i $$-0.123247\pi$$
0.925972 + 0.377591i $$0.123247\pi$$
$$500$$ 11.8078 0.528059
$$501$$ 14.2462 0.636474
$$502$$ 0.438447 0.0195689
$$503$$ −32.1080 −1.43162 −0.715811 0.698294i $$-0.753943\pi$$
−0.715811 + 0.698294i $$0.753943\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ −25.3693 −1.12892
$$506$$ −5.12311 −0.227750
$$507$$ 0.315342 0.0140048
$$508$$ −16.6847 −0.740262
$$509$$ −3.36932 −0.149342 −0.0746712 0.997208i $$-0.523791\pi$$
−0.0746712 + 0.997208i $$0.523791\pi$$
$$510$$ 1.75379 0.0776591
$$511$$ −9.12311 −0.403582
$$512$$ −1.00000 −0.0441942
$$513$$ 5.12311 0.226191
$$514$$ 2.00000 0.0882162
$$515$$ −29.5616 −1.30264
$$516$$ −6.68466 −0.294276
$$517$$ 12.4924 0.549416
$$518$$ −1.56155 −0.0686107
$$519$$ −18.4924 −0.811727
$$520$$ −5.56155 −0.243890
$$521$$ 0.246211 0.0107867 0.00539336 0.999985i $$-0.498283\pi$$
0.00539336 + 0.999985i $$0.498283\pi$$
$$522$$ −7.56155 −0.330960
$$523$$ 14.0000 0.612177 0.306089 0.952003i $$-0.400980\pi$$
0.306089 + 0.952003i $$0.400980\pi$$
$$524$$ 4.87689 0.213048
$$525$$ −2.56155 −0.111795
$$526$$ −22.9309 −0.999833
$$527$$ −3.50758 −0.152792
$$528$$ −5.12311 −0.222955
$$529$$ 1.00000 0.0434783
$$530$$ 22.2462 0.966314
$$531$$ 4.87689 0.211639
$$532$$ 5.12311 0.222115
$$533$$ −12.6847 −0.549434
$$534$$ 14.0000 0.605839
$$535$$ 8.00000 0.345870
$$536$$ 1.12311 0.0485108
$$537$$ 3.80776 0.164317
$$538$$ −12.2462 −0.527972
$$539$$ 5.12311 0.220668
$$540$$ 1.56155 0.0671985
$$541$$ −15.3693 −0.660779 −0.330389 0.943845i $$-0.607180\pi$$
−0.330389 + 0.943845i $$0.607180\pi$$
$$542$$ −14.2462 −0.611927
$$543$$ 15.1231 0.648995
$$544$$ 1.12311 0.0481528
$$545$$ −16.3002 −0.698223
$$546$$ 3.56155 0.152420
$$547$$ −3.61553 −0.154589 −0.0772944 0.997008i $$-0.524628\pi$$
−0.0772944 + 0.997008i $$0.524628\pi$$
$$548$$ −0.438447 −0.0187295
$$549$$ 0.876894 0.0374249
$$550$$ 13.1231 0.559571
$$551$$ −38.7386 −1.65032
$$552$$ 1.00000 0.0425628
$$553$$ −14.2462 −0.605811
$$554$$ −13.1231 −0.557547
$$555$$ −2.43845 −0.103506
$$556$$ 3.80776 0.161485
$$557$$ −28.9848 −1.22813 −0.614064 0.789257i $$-0.710466\pi$$
−0.614064 + 0.789257i $$0.710466\pi$$
$$558$$ −3.12311 −0.132212
$$559$$ −23.8078 −1.00696
$$560$$ 1.56155 0.0659877
$$561$$ 5.75379 0.242925
$$562$$ −4.93087 −0.207996
$$563$$ 19.5616 0.824421 0.412211 0.911089i $$-0.364757\pi$$
0.412211 + 0.911089i $$0.364757\pi$$
$$564$$ −2.43845 −0.102677
$$565$$ 4.19224 0.176369
$$566$$ 10.4924 0.441029
$$567$$ −1.00000 −0.0419961
$$568$$ 9.36932 0.393128
$$569$$ −44.5464 −1.86748 −0.933741 0.357949i $$-0.883476\pi$$
−0.933741 + 0.357949i $$0.883476\pi$$
$$570$$ 8.00000 0.335083
$$571$$ 43.3693 1.81495 0.907475 0.420107i $$-0.138007\pi$$
0.907475 + 0.420107i $$0.138007\pi$$
$$572$$ −18.2462 −0.762912
$$573$$ 20.0000 0.835512
$$574$$ 3.56155 0.148656
$$575$$ −2.56155 −0.106824
$$576$$ 1.00000 0.0416667
$$577$$ −19.7538 −0.822361 −0.411180 0.911554i $$-0.634883\pi$$
−0.411180 + 0.911554i $$0.634883\pi$$
$$578$$ 15.7386 0.654641
$$579$$ −2.68466 −0.111571
$$580$$ −11.8078 −0.490291
$$581$$ 9.12311 0.378490
$$582$$ −12.4384 −0.515590
$$583$$ 72.9848 3.02272
$$584$$ −9.12311 −0.377517
$$585$$ 5.56155 0.229942
$$586$$ −27.6155 −1.14079
$$587$$ −24.9848 −1.03123 −0.515617 0.856819i $$-0.672437\pi$$
−0.515617 + 0.856819i $$0.672437\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ −16.0000 −0.659269
$$590$$ 7.61553 0.313526
$$591$$ −24.9309 −1.02552
$$592$$ −1.56155 −0.0641794
$$593$$ 30.3002 1.24428 0.622140 0.782906i $$-0.286264\pi$$
0.622140 + 0.782906i $$0.286264\pi$$
$$594$$ 5.12311 0.210204
$$595$$ −1.75379 −0.0718983
$$596$$ −6.24621 −0.255855
$$597$$ 21.1771 0.866720
$$598$$ 3.56155 0.145643
$$599$$ −10.7386 −0.438769 −0.219384 0.975639i $$-0.570405\pi$$
−0.219384 + 0.975639i $$0.570405\pi$$
$$600$$ −2.56155 −0.104575
$$601$$ −3.36932 −0.137437 −0.0687187 0.997636i $$-0.521891\pi$$
−0.0687187 + 0.997636i $$0.521891\pi$$
$$602$$ 6.68466 0.272446
$$603$$ −1.12311 −0.0457364
$$604$$ 7.31534 0.297657
$$605$$ −23.8078 −0.967923
$$606$$ 16.2462 0.659957
$$607$$ −47.6155 −1.93265 −0.966327 0.257316i $$-0.917162\pi$$
−0.966327 + 0.257316i $$0.917162\pi$$
$$608$$ 5.12311 0.207769
$$609$$ 7.56155 0.306410
$$610$$ 1.36932 0.0554420
$$611$$ −8.68466 −0.351344
$$612$$ −1.12311 −0.0453989
$$613$$ 3.80776 0.153794 0.0768971 0.997039i $$-0.475499\pi$$
0.0768971 + 0.997039i $$0.475499\pi$$
$$614$$ −2.93087 −0.118280
$$615$$ 5.56155 0.224263
$$616$$ 5.12311 0.206416
$$617$$ 31.7538 1.27836 0.639180 0.769057i $$-0.279274\pi$$
0.639180 + 0.769057i $$0.279274\pi$$
$$618$$ 18.9309 0.761511
$$619$$ 8.24621 0.331443 0.165722 0.986173i $$-0.447005\pi$$
0.165722 + 0.986173i $$0.447005\pi$$
$$620$$ −4.87689 −0.195861
$$621$$ −1.00000 −0.0401286
$$622$$ −18.7386 −0.751351
$$623$$ −14.0000 −0.560898
$$624$$ 3.56155 0.142576
$$625$$ −5.63068 −0.225227
$$626$$ −4.24621 −0.169713
$$627$$ 26.2462 1.04817
$$628$$ 14.2462 0.568486
$$629$$ 1.75379 0.0699281
$$630$$ −1.56155 −0.0622138
$$631$$ 8.49242 0.338078 0.169039 0.985609i $$-0.445934\pi$$
0.169039 + 0.985609i $$0.445934\pi$$
$$632$$ −14.2462 −0.566684
$$633$$ 16.4924 0.655515
$$634$$ 17.8078 0.707237
$$635$$ 26.0540 1.03392
$$636$$ −14.2462 −0.564899
$$637$$ −3.56155 −0.141114
$$638$$ −38.7386 −1.53368
$$639$$ −9.36932 −0.370644
$$640$$ 1.56155 0.0617258
$$641$$ −13.3153 −0.525924 −0.262962 0.964806i $$-0.584699\pi$$
−0.262962 + 0.964806i $$0.584699\pi$$
$$642$$ −5.12311 −0.202193
$$643$$ −38.0000 −1.49857 −0.749287 0.662246i $$-0.769604\pi$$
−0.749287 + 0.662246i $$0.769604\pi$$
$$644$$ −1.00000 −0.0394055
$$645$$ 10.4384 0.411013
$$646$$ −5.75379 −0.226380
$$647$$ −3.50758 −0.137897 −0.0689486 0.997620i $$-0.521964\pi$$
−0.0689486 + 0.997620i $$0.521964\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 24.9848 0.980741
$$650$$ −9.12311 −0.357838
$$651$$ 3.12311 0.122404
$$652$$ −8.00000 −0.313304
$$653$$ 36.5464 1.43017 0.715086 0.699037i $$-0.246388\pi$$
0.715086 + 0.699037i $$0.246388\pi$$
$$654$$ 10.4384 0.408176
$$655$$ −7.61553 −0.297563
$$656$$ 3.56155 0.139055
$$657$$ 9.12311 0.355926
$$658$$ 2.43845 0.0950606
$$659$$ 5.61553 0.218750 0.109375 0.994001i $$-0.465115\pi$$
0.109375 + 0.994001i $$0.465115\pi$$
$$660$$ 8.00000 0.311400
$$661$$ 32.8769 1.27876 0.639381 0.768890i $$-0.279190\pi$$
0.639381 + 0.768890i $$0.279190\pi$$
$$662$$ 24.4924 0.951925
$$663$$ −4.00000 −0.155347
$$664$$ 9.12311 0.354045
$$665$$ −8.00000 −0.310227
$$666$$ 1.56155 0.0605089
$$667$$ 7.56155 0.292784
$$668$$ −14.2462 −0.551202
$$669$$ −3.12311 −0.120746
$$670$$ −1.75379 −0.0677548
$$671$$ 4.49242 0.173428
$$672$$ −1.00000 −0.0385758
$$673$$ 22.6847 0.874429 0.437215 0.899357i $$-0.355965\pi$$
0.437215 + 0.899357i $$0.355965\pi$$
$$674$$ −27.3693 −1.05423
$$675$$ 2.56155 0.0985942
$$676$$ −0.315342 −0.0121285
$$677$$ 2.63068 0.101105 0.0505527 0.998721i $$-0.483902\pi$$
0.0505527 + 0.998721i $$0.483902\pi$$
$$678$$ −2.68466 −0.103104
$$679$$ 12.4384 0.477344
$$680$$ −1.75379 −0.0672547
$$681$$ 6.19224 0.237287
$$682$$ −16.0000 −0.612672
$$683$$ −7.50758 −0.287269 −0.143635 0.989631i $$-0.545879\pi$$
−0.143635 + 0.989631i $$0.545879\pi$$
$$684$$ −5.12311 −0.195887
$$685$$ 0.684658 0.0261595
$$686$$ 1.00000 0.0381802
$$687$$ −2.24621 −0.0856983
$$688$$ 6.68466 0.254850
$$689$$ −50.7386 −1.93299
$$690$$ −1.56155 −0.0594473
$$691$$ −50.0540 −1.90414 −0.952071 0.305876i $$-0.901051\pi$$
−0.952071 + 0.305876i $$0.901051\pi$$
$$692$$ 18.4924 0.702976
$$693$$ −5.12311 −0.194611
$$694$$ −14.9309 −0.566768
$$695$$ −5.94602 −0.225546
$$696$$ 7.56155 0.286620
$$697$$ −4.00000 −0.151511
$$698$$ 16.2462 0.614928
$$699$$ 18.4924 0.699448
$$700$$ 2.56155 0.0968176
$$701$$ 34.2462 1.29346 0.646731 0.762718i $$-0.276136\pi$$
0.646731 + 0.762718i $$0.276136\pi$$
$$702$$ −3.56155 −0.134422
$$703$$ 8.00000 0.301726
$$704$$ 5.12311 0.193084
$$705$$ 3.80776 0.143409
$$706$$ 33.8078 1.27237
$$707$$ −16.2462 −0.611002
$$708$$ −4.87689 −0.183285
$$709$$ 10.6307 0.399244 0.199622 0.979873i $$-0.436029\pi$$
0.199622 + 0.979873i $$0.436029\pi$$
$$710$$ −14.6307 −0.549080
$$711$$ 14.2462 0.534275
$$712$$ −14.0000 −0.524672
$$713$$ 3.12311 0.116961
$$714$$ 1.12311 0.0420312
$$715$$ 28.4924 1.06556
$$716$$ −3.80776 −0.142303
$$717$$ −4.49242 −0.167773
$$718$$ 6.43845 0.240281
$$719$$ 10.4384 0.389288 0.194644 0.980874i $$-0.437645\pi$$
0.194644 + 0.980874i $$0.437645\pi$$
$$720$$ −1.56155 −0.0581956
$$721$$ −18.9309 −0.705022
$$722$$ −7.24621 −0.269676
$$723$$ 20.0540 0.745815
$$724$$ −15.1231 −0.562046
$$725$$ −19.3693 −0.719358
$$726$$ 15.2462 0.565840
$$727$$ −1.75379 −0.0650444 −0.0325222 0.999471i $$-0.510354\pi$$
−0.0325222 + 0.999471i $$0.510354\pi$$
$$728$$ −3.56155 −0.132000
$$729$$ 1.00000 0.0370370
$$730$$ 14.2462 0.527276
$$731$$ −7.50758 −0.277678
$$732$$ −0.876894 −0.0324109
$$733$$ 25.7538 0.951238 0.475619 0.879651i $$-0.342224\pi$$
0.475619 + 0.879651i $$0.342224\pi$$
$$734$$ 18.4384 0.680576
$$735$$ 1.56155 0.0575987
$$736$$ −1.00000 −0.0368605
$$737$$ −5.75379 −0.211944
$$738$$ −3.56155 −0.131103
$$739$$ 23.6155 0.868711 0.434356 0.900741i $$-0.356976\pi$$
0.434356 + 0.900741i $$0.356976\pi$$
$$740$$ 2.43845 0.0896391
$$741$$ −18.2462 −0.670291
$$742$$ 14.2462 0.522995
$$743$$ −24.9848 −0.916605 −0.458303 0.888796i $$-0.651542\pi$$
−0.458303 + 0.888796i $$0.651542\pi$$
$$744$$ 3.12311 0.114499
$$745$$ 9.75379 0.357351
$$746$$ −15.1231 −0.553696
$$747$$ −9.12311 −0.333797
$$748$$ −5.75379 −0.210379
$$749$$ 5.12311 0.187194
$$750$$ 11.8078 0.431159
$$751$$ 24.4924 0.893741 0.446871 0.894599i $$-0.352538\pi$$
0.446871 + 0.894599i $$0.352538\pi$$
$$752$$ 2.43845 0.0889210
$$753$$ 0.438447 0.0159779
$$754$$ 26.9309 0.980764
$$755$$ −11.4233 −0.415736
$$756$$ 1.00000 0.0363696
$$757$$ −25.3693 −0.922064 −0.461032 0.887384i $$-0.652521\pi$$
−0.461032 + 0.887384i $$0.652521\pi$$
$$758$$ −20.0540 −0.728393
$$759$$ −5.12311 −0.185957
$$760$$ −8.00000 −0.290191
$$761$$ −3.26137 −0.118224 −0.0591122 0.998251i $$-0.518827\pi$$
−0.0591122 + 0.998251i $$0.518827\pi$$
$$762$$ −16.6847 −0.604421
$$763$$ −10.4384 −0.377897
$$764$$ −20.0000 −0.723575
$$765$$ 1.75379 0.0634084
$$766$$ 6.24621 0.225685
$$767$$ −17.3693 −0.627170
$$768$$ −1.00000 −0.0360844
$$769$$ −47.1771 −1.70125 −0.850625 0.525774i $$-0.823776\pi$$
−0.850625 + 0.525774i $$0.823776\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ 2.00000 0.0720282
$$772$$ 2.68466 0.0966230
$$773$$ −32.3002 −1.16176 −0.580878 0.813990i $$-0.697291\pi$$
−0.580878 + 0.813990i $$0.697291\pi$$
$$774$$ −6.68466 −0.240275
$$775$$ −8.00000 −0.287368
$$776$$ 12.4384 0.446514
$$777$$ −1.56155 −0.0560204
$$778$$ 29.3693 1.05294
$$779$$ −18.2462 −0.653738
$$780$$ −5.56155 −0.199136
$$781$$ −48.0000 −1.71758
$$782$$ 1.12311 0.0401622
$$783$$ −7.56155 −0.270228
$$784$$ 1.00000 0.0357143
$$785$$ −22.2462 −0.794001
$$786$$ 4.87689 0.173953
$$787$$ −10.8769 −0.387719 −0.193860 0.981029i $$-0.562101\pi$$
−0.193860 + 0.981029i $$0.562101\pi$$
$$788$$ 24.9309 0.888125
$$789$$ −22.9309 −0.816361
$$790$$ 22.2462 0.791485
$$791$$ 2.68466 0.0954555
$$792$$ −5.12311 −0.182042
$$793$$ −3.12311 −0.110905
$$794$$ 7.75379 0.275172
$$795$$ 22.2462 0.788992
$$796$$ −21.1771 −0.750602
$$797$$ 27.4233 0.971383 0.485691 0.874130i $$-0.338568\pi$$
0.485691 + 0.874130i $$0.338568\pi$$
$$798$$ 5.12311 0.181356
$$799$$ −2.73863 −0.0968859
$$800$$ 2.56155 0.0905646
$$801$$ 14.0000 0.494666
$$802$$ −16.2462 −0.573674
$$803$$ 46.7386 1.64937
$$804$$ 1.12311 0.0396089
$$805$$ 1.56155 0.0550375
$$806$$ 11.1231 0.391795
$$807$$ −12.2462 −0.431087
$$808$$ −16.2462 −0.571540
$$809$$ 23.8617 0.838934 0.419467 0.907771i $$-0.362217\pi$$
0.419467 + 0.907771i $$0.362217\pi$$
$$810$$ 1.56155 0.0548674
$$811$$ −24.6847 −0.866796 −0.433398 0.901203i $$-0.642685\pi$$
−0.433398 + 0.901203i $$0.642685\pi$$
$$812$$ −7.56155 −0.265358
$$813$$ −14.2462 −0.499636
$$814$$ 8.00000 0.280400
$$815$$ 12.4924 0.437590
$$816$$ 1.12311 0.0393166
$$817$$ −34.2462 −1.19812
$$818$$ −33.6155 −1.17534
$$819$$ 3.56155 0.124451
$$820$$ −5.56155 −0.194218
$$821$$ −55.4773 −1.93617 −0.968085 0.250622i $$-0.919365\pi$$
−0.968085 + 0.250622i $$0.919365\pi$$
$$822$$ −0.438447 −0.0152926
$$823$$ 23.3153 0.812722 0.406361 0.913713i $$-0.366798\pi$$
0.406361 + 0.913713i $$0.366798\pi$$
$$824$$ −18.9309 −0.659488
$$825$$ 13.1231 0.456888
$$826$$ 4.87689 0.169689
$$827$$ 48.7386 1.69481 0.847404 0.530948i $$-0.178164\pi$$
0.847404 + 0.530948i $$0.178164\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 24.7386 0.859208 0.429604 0.903017i $$-0.358653\pi$$
0.429604 + 0.903017i $$0.358653\pi$$
$$830$$ −14.2462 −0.494493
$$831$$ −13.1231 −0.455235
$$832$$ −3.56155 −0.123475
$$833$$ −1.12311 −0.0389133
$$834$$ 3.80776 0.131852
$$835$$ 22.2462 0.769862
$$836$$ −26.2462 −0.907744
$$837$$ −3.12311 −0.107950
$$838$$ −23.8617 −0.824290
$$839$$ 7.61553 0.262917 0.131459 0.991322i $$-0.458034\pi$$
0.131459 + 0.991322i $$0.458034\pi$$
$$840$$ 1.56155 0.0538787
$$841$$ 28.1771 0.971623
$$842$$ −33.1771 −1.14336
$$843$$ −4.93087 −0.169828
$$844$$ −16.4924 −0.567693
$$845$$ 0.492423 0.0169398
$$846$$ −2.43845 −0.0838355
$$847$$ −15.2462 −0.523866
$$848$$ 14.2462 0.489217
$$849$$ 10.4924 0.360099
$$850$$ −2.87689 −0.0986767
$$851$$ −1.56155 −0.0535293
$$852$$ 9.36932 0.320988
$$853$$ −0.438447 −0.0150121 −0.00750607 0.999972i $$-0.502389\pi$$
−0.00750607 + 0.999972i $$0.502389\pi$$
$$854$$ 0.876894 0.0300067
$$855$$ 8.00000 0.273594
$$856$$ 5.12311 0.175104
$$857$$ 1.31534 0.0449312 0.0224656 0.999748i $$-0.492848\pi$$
0.0224656 + 0.999748i $$0.492848\pi$$
$$858$$ −18.2462 −0.622915
$$859$$ −33.1771 −1.13199 −0.565994 0.824410i $$-0.691507\pi$$
−0.565994 + 0.824410i $$0.691507\pi$$
$$860$$ −10.4384 −0.355948
$$861$$ 3.56155 0.121377
$$862$$ 3.31534 0.112921
$$863$$ 10.7386 0.365547 0.182774 0.983155i $$-0.441492\pi$$
0.182774 + 0.983155i $$0.441492\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −28.8769 −0.981844
$$866$$ −14.6847 −0.499005
$$867$$ 15.7386 0.534512
$$868$$ −3.12311 −0.106005
$$869$$ 72.9848 2.47584
$$870$$ −11.8078 −0.400321
$$871$$ 4.00000 0.135535
$$872$$ −10.4384 −0.353490
$$873$$ −12.4384 −0.420978
$$874$$ 5.12311 0.173292
$$875$$ −11.8078 −0.399175
$$876$$ −9.12311 −0.308241
$$877$$ −35.8617 −1.21096 −0.605482 0.795859i $$-0.707020\pi$$
−0.605482 + 0.795859i $$0.707020\pi$$
$$878$$ 23.6155 0.796985
$$879$$ −27.6155 −0.931449
$$880$$ −8.00000 −0.269680
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −8.87689 −0.298731 −0.149366 0.988782i $$-0.547723\pi$$
−0.149366 + 0.988782i $$0.547723\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 7.61553 0.255993
$$886$$ −9.06913 −0.304683
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −1.56155 −0.0524023
$$889$$ 16.6847 0.559585
$$890$$ 21.8617 0.732807
$$891$$ 5.12311 0.171630
$$892$$ 3.12311 0.104569
$$893$$ −12.4924 −0.418043
$$894$$ −6.24621 −0.208905
$$895$$ 5.94602 0.198754
$$896$$ 1.00000 0.0334077
$$897$$ 3.56155 0.118917
$$898$$ 0.246211 0.00821618
$$899$$ 23.6155 0.787622
$$900$$ −2.56155 −0.0853851
$$901$$ −16.0000 −0.533037
$$902$$ −18.2462 −0.607532
$$903$$ 6.68466 0.222452
$$904$$ 2.68466 0.0892904
$$905$$ 23.6155 0.785007
$$906$$ 7.31534 0.243036
$$907$$ 52.5464 1.74477 0.872387 0.488815i $$-0.162571\pi$$
0.872387 + 0.488815i $$0.162571\pi$$
$$908$$ −6.19224 −0.205497
$$909$$ 16.2462 0.538853
$$910$$ 5.56155 0.184364
$$911$$ −1.94602 −0.0644747 −0.0322373 0.999480i $$-0.510263\pi$$
−0.0322373 + 0.999480i $$0.510263\pi$$
$$912$$ 5.12311 0.169643
$$913$$ −46.7386 −1.54682
$$914$$ −7.36932 −0.243755
$$915$$ 1.36932 0.0452682
$$916$$ 2.24621 0.0742169
$$917$$ −4.87689 −0.161049
$$918$$ −1.12311 −0.0370680
$$919$$ −50.7386 −1.67371 −0.836857 0.547422i $$-0.815609\pi$$
−0.836857 + 0.547422i $$0.815609\pi$$
$$920$$ 1.56155 0.0514829
$$921$$ −2.93087 −0.0965754
$$922$$ −40.7386 −1.34166
$$923$$ 33.3693 1.09836
$$924$$ 5.12311 0.168538
$$925$$ 4.00000 0.131519
$$926$$ 13.5616 0.445660
$$927$$ 18.9309 0.621771
$$928$$ −7.56155 −0.248220
$$929$$ −36.4384 −1.19551 −0.597753 0.801680i $$-0.703940\pi$$
−0.597753 + 0.801680i $$0.703940\pi$$
$$930$$ −4.87689 −0.159920
$$931$$ −5.12311 −0.167903
$$932$$ −18.4924 −0.605739
$$933$$ −18.7386 −0.613475
$$934$$ −2.68466 −0.0878447
$$935$$ 8.98485 0.293836
$$936$$ 3.56155 0.116413
$$937$$ −27.1771 −0.887837 −0.443918 0.896067i $$-0.646412\pi$$
−0.443918 + 0.896067i $$0.646412\pi$$
$$938$$ −1.12311 −0.0366707
$$939$$ −4.24621 −0.138570
$$940$$ −3.80776 −0.124196
$$941$$ 1.94602 0.0634386 0.0317193 0.999497i $$-0.489902\pi$$
0.0317193 + 0.999497i $$0.489902\pi$$
$$942$$ 14.2462 0.464167
$$943$$ 3.56155 0.115980
$$944$$ 4.87689 0.158729
$$945$$ −1.56155 −0.0507973
$$946$$ −34.2462 −1.11344
$$947$$ −54.9309 −1.78501 −0.892507 0.451034i $$-0.851055\pi$$
−0.892507 + 0.451034i $$0.851055\pi$$
$$948$$ −14.2462 −0.462695
$$949$$ −32.4924 −1.05475
$$950$$ −13.1231 −0.425770
$$951$$ 17.8078 0.577456
$$952$$ −1.12311 −0.0364001
$$953$$ 50.0000 1.61966 0.809829 0.586665i $$-0.199560\pi$$
0.809829 + 0.586665i $$0.199560\pi$$
$$954$$ −14.2462 −0.461238
$$955$$ 31.2311 1.01061
$$956$$ 4.49242 0.145295
$$957$$ −38.7386 −1.25224
$$958$$ −31.6155 −1.02145
$$959$$ 0.438447 0.0141582
$$960$$ 1.56155 0.0503989
$$961$$ −21.2462 −0.685362
$$962$$ −5.56155 −0.179312
$$963$$ −5.12311 −0.165090
$$964$$ −20.0540 −0.645895
$$965$$ −4.19224 −0.134953
$$966$$ −1.00000 −0.0321745
$$967$$ −9.75379 −0.313661 −0.156830 0.987626i $$-0.550128\pi$$
−0.156830 + 0.987626i $$0.550128\pi$$
$$968$$ −15.2462 −0.490032
$$969$$ −5.75379 −0.184838
$$970$$ −19.4233 −0.623644
$$971$$ −1.61553 −0.0518448 −0.0259224 0.999664i $$-0.508252\pi$$
−0.0259224 + 0.999664i $$0.508252\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −3.80776 −0.122071
$$974$$ 5.56155 0.178204
$$975$$ −9.12311 −0.292173
$$976$$ 0.876894 0.0280687
$$977$$ −26.1922 −0.837964 −0.418982 0.907995i $$-0.637613\pi$$
−0.418982 + 0.907995i $$0.637613\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ 71.7235 2.29229
$$980$$ −1.56155 −0.0498820
$$981$$ 10.4384 0.333274
$$982$$ 26.7386 0.853264
$$983$$ 33.8617 1.08002 0.540011 0.841658i $$-0.318420\pi$$
0.540011 + 0.841658i $$0.318420\pi$$
$$984$$ 3.56155 0.113538
$$985$$ −38.9309 −1.24044
$$986$$ 8.49242 0.270454
$$987$$ 2.43845 0.0776166
$$988$$ 18.2462 0.580489
$$989$$ 6.68466 0.212560
$$990$$ 8.00000 0.254257
$$991$$ −6.24621 −0.198417 −0.0992087 0.995067i $$-0.531631\pi$$
−0.0992087 + 0.995067i $$0.531631\pi$$
$$992$$ −3.12311 −0.0991587
$$993$$ 24.4924 0.777244
$$994$$ −9.36932 −0.297177
$$995$$ 33.0691 1.04836
$$996$$ 9.12311 0.289077
$$997$$ 51.4773 1.63030 0.815151 0.579249i $$-0.196654\pi$$
0.815151 + 0.579249i $$0.196654\pi$$
$$998$$ −41.3693 −1.30952
$$999$$ 1.56155 0.0494053
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 966.2.a.l.1.1 2
3.2 odd 2 2898.2.a.ba.1.2 2
4.3 odd 2 7728.2.a.bm.1.1 2
7.6 odd 2 6762.2.a.bw.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.l.1.1 2 1.1 even 1 trivial
2898.2.a.ba.1.2 2 3.2 odd 2
6762.2.a.bw.1.2 2 7.6 odd 2
7728.2.a.bm.1.1 2 4.3 odd 2