# Properties

 Label 231.2.a.d.1.2 Level $231$ Weight $2$ Character 231.1 Self dual yes Analytic conductor $1.845$ Analytic rank $0$ 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] = [231,2,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 231.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: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 1$$ x^3 - 6*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 1.2 Root $$-0.167449$$ of defining polynomial Character $$\chi$$ $$=$$ 231.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} -0.637062 q^{10} +1.00000 q^{11} +1.97196 q^{12} +3.80451 q^{13} +0.167449 q^{14} -3.80451 q^{15} +3.83255 q^{16} +0.334898 q^{17} -0.167449 q^{18} +8.13941 q^{19} -7.50235 q^{20} +1.00000 q^{21} -0.167449 q^{22} -1.66510 q^{23} -0.665102 q^{24} +9.47431 q^{25} -0.637062 q^{26} -1.00000 q^{27} +1.97196 q^{28} +0.195488 q^{29} +0.637062 q^{30} -9.94392 q^{31} -1.97196 q^{32} -1.00000 q^{33} -0.0560785 q^{34} -3.80451 q^{35} -1.97196 q^{36} -4.47431 q^{37} -1.36294 q^{38} -3.80451 q^{39} +2.53039 q^{40} -6.27882 q^{41} -0.167449 q^{42} +2.33490 q^{43} -1.97196 q^{44} +3.80451 q^{45} +0.278820 q^{46} -12.1394 q^{47} -3.83255 q^{48} +1.00000 q^{49} -1.58647 q^{50} -0.334898 q^{51} -7.50235 q^{52} +7.94392 q^{53} +0.167449 q^{54} +3.80451 q^{55} -0.665102 q^{56} -8.13941 q^{57} -0.0327344 q^{58} +3.74843 q^{59} +7.50235 q^{60} +6.00000 q^{61} +1.66510 q^{62} -1.00000 q^{63} -7.33490 q^{64} +14.4743 q^{65} +0.167449 q^{66} -0.139410 q^{67} -0.660406 q^{68} +1.66510 q^{69} +0.637062 q^{70} +4.66980 q^{71} +0.665102 q^{72} +4.19549 q^{73} +0.749219 q^{74} -9.47431 q^{75} -16.0506 q^{76} -1.00000 q^{77} +0.637062 q^{78} +3.33020 q^{79} +14.5810 q^{80} +1.00000 q^{81} +1.05138 q^{82} -13.9439 q^{83} -1.97196 q^{84} +1.27412 q^{85} -0.390977 q^{86} -0.195488 q^{87} +0.665102 q^{88} -9.88784 q^{89} -0.637062 q^{90} -3.80451 q^{91} +3.28352 q^{92} +9.94392 q^{93} +2.03273 q^{94} +30.9665 q^{95} +1.97196 q^{96} +0.0560785 q^{97} -0.167449 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 9 * q^10 + 3 * q^11 - 6 * q^12 + 12 * q^16 + 12 * q^19 - 21 * q^20 + 3 * q^21 - 6 * q^23 - 3 * q^24 + 15 * q^25 + 9 * q^26 - 3 * q^27 - 6 * q^28 + 12 * q^29 - 9 * q^30 - 6 * q^31 + 6 * q^32 - 3 * q^33 - 24 * q^34 + 6 * q^36 - 15 * q^38 + 18 * q^40 + 6 * q^41 + 6 * q^43 + 6 * q^44 - 24 * q^46 - 24 * q^47 - 12 * q^48 + 3 * q^49 - 39 * q^50 - 21 * q^52 - 3 * q^56 - 12 * q^57 - 9 * q^58 - 24 * q^59 + 21 * q^60 + 18 * q^61 + 6 * q^62 - 3 * q^63 - 21 * q^64 + 30 * q^65 + 12 * q^67 - 6 * q^68 + 6 * q^69 - 9 * q^70 + 12 * q^71 + 3 * q^72 + 24 * q^73 + 39 * q^74 - 15 * q^75 - 3 * q^76 - 3 * q^77 - 9 * q^78 + 12 * q^79 + 9 * q^80 + 3 * q^81 + 30 * q^82 - 18 * q^83 + 6 * q^84 - 18 * q^85 - 24 * q^86 - 12 * q^87 + 3 * q^88 + 18 * q^89 + 9 * q^90 - 18 * q^92 + 6 * q^93 + 15 * q^94 + 12 * q^95 - 6 * q^96 + 24 * q^97 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.167449 −0.118404 −0.0592022 0.998246i $$-0.518856\pi$$
−0.0592022 + 0.998246i $$0.518856\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.97196 −0.985980
$$5$$ 3.80451 1.70143 0.850715 0.525628i $$-0.176170\pi$$
0.850715 + 0.525628i $$0.176170\pi$$
$$6$$ 0.167449 0.0683608
$$7$$ −1.00000 −0.377964
$$8$$ 0.665102 0.235149
$$9$$ 1.00000 0.333333
$$10$$ −0.637062 −0.201457
$$11$$ 1.00000 0.301511
$$12$$ 1.97196 0.569256
$$13$$ 3.80451 1.05518 0.527591 0.849499i $$-0.323095\pi$$
0.527591 + 0.849499i $$0.323095\pi$$
$$14$$ 0.167449 0.0447527
$$15$$ −3.80451 −0.982321
$$16$$ 3.83255 0.958138
$$17$$ 0.334898 0.0812248 0.0406124 0.999175i $$-0.487069\pi$$
0.0406124 + 0.999175i $$0.487069\pi$$
$$18$$ −0.167449 −0.0394682
$$19$$ 8.13941 1.86731 0.933654 0.358175i $$-0.116601\pi$$
0.933654 + 0.358175i $$0.116601\pi$$
$$20$$ −7.50235 −1.67758
$$21$$ 1.00000 0.218218
$$22$$ −0.167449 −0.0357003
$$23$$ −1.66510 −0.347198 −0.173599 0.984816i $$-0.555540\pi$$
−0.173599 + 0.984816i $$0.555540\pi$$
$$24$$ −0.665102 −0.135763
$$25$$ 9.47431 1.89486
$$26$$ −0.637062 −0.124938
$$27$$ −1.00000 −0.192450
$$28$$ 1.97196 0.372666
$$29$$ 0.195488 0.0363013 0.0181506 0.999835i $$-0.494222\pi$$
0.0181506 + 0.999835i $$0.494222\pi$$
$$30$$ 0.637062 0.116311
$$31$$ −9.94392 −1.78598 −0.892991 0.450075i $$-0.851397\pi$$
−0.892991 + 0.450075i $$0.851397\pi$$
$$32$$ −1.97196 −0.348597
$$33$$ −1.00000 −0.174078
$$34$$ −0.0560785 −0.00961738
$$35$$ −3.80451 −0.643080
$$36$$ −1.97196 −0.328660
$$37$$ −4.47431 −0.735572 −0.367786 0.929911i $$-0.619884\pi$$
−0.367786 + 0.929911i $$0.619884\pi$$
$$38$$ −1.36294 −0.221098
$$39$$ −3.80451 −0.609209
$$40$$ 2.53039 0.400089
$$41$$ −6.27882 −0.980587 −0.490293 0.871557i $$-0.663110\pi$$
−0.490293 + 0.871557i $$0.663110\pi$$
$$42$$ −0.167449 −0.0258380
$$43$$ 2.33490 0.356069 0.178034 0.984024i $$-0.443026\pi$$
0.178034 + 0.984024i $$0.443026\pi$$
$$44$$ −1.97196 −0.297284
$$45$$ 3.80451 0.567143
$$46$$ 0.278820 0.0411098
$$47$$ −12.1394 −1.77071 −0.885357 0.464911i $$-0.846086\pi$$
−0.885357 + 0.464911i $$0.846086\pi$$
$$48$$ −3.83255 −0.553181
$$49$$ 1.00000 0.142857
$$50$$ −1.58647 −0.224360
$$51$$ −0.334898 −0.0468952
$$52$$ −7.50235 −1.04039
$$53$$ 7.94392 1.09118 0.545591 0.838052i $$-0.316305\pi$$
0.545591 + 0.838052i $$0.316305\pi$$
$$54$$ 0.167449 0.0227869
$$55$$ 3.80451 0.513000
$$56$$ −0.665102 −0.0888779
$$57$$ −8.13941 −1.07809
$$58$$ −0.0327344 −0.00429823
$$59$$ 3.74843 0.488004 0.244002 0.969775i $$-0.421540\pi$$
0.244002 + 0.969775i $$0.421540\pi$$
$$60$$ 7.50235 0.968549
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 1.66510 0.211468
$$63$$ −1.00000 −0.125988
$$64$$ −7.33490 −0.916862
$$65$$ 14.4743 1.79532
$$66$$ 0.167449 0.0206116
$$67$$ −0.139410 −0.0170316 −0.00851582 0.999964i $$-0.502711\pi$$
−0.00851582 + 0.999964i $$0.502711\pi$$
$$68$$ −0.660406 −0.0800860
$$69$$ 1.66510 0.200455
$$70$$ 0.637062 0.0761435
$$71$$ 4.66980 0.554203 0.277101 0.960841i $$-0.410626\pi$$
0.277101 + 0.960841i $$0.410626\pi$$
$$72$$ 0.665102 0.0783830
$$73$$ 4.19549 0.491045 0.245522 0.969391i $$-0.421041\pi$$
0.245522 + 0.969391i $$0.421041\pi$$
$$74$$ 0.749219 0.0870950
$$75$$ −9.47431 −1.09400
$$76$$ −16.0506 −1.84113
$$77$$ −1.00000 −0.113961
$$78$$ 0.637062 0.0721331
$$79$$ 3.33020 0.374677 0.187339 0.982295i $$-0.440014\pi$$
0.187339 + 0.982295i $$0.440014\pi$$
$$80$$ 14.5810 1.63020
$$81$$ 1.00000 0.111111
$$82$$ 1.05138 0.116106
$$83$$ −13.9439 −1.53054 −0.765272 0.643707i $$-0.777396\pi$$
−0.765272 + 0.643707i $$0.777396\pi$$
$$84$$ −1.97196 −0.215159
$$85$$ 1.27412 0.138198
$$86$$ −0.390977 −0.0421601
$$87$$ −0.195488 −0.0209586
$$88$$ 0.665102 0.0709001
$$89$$ −9.88784 −1.04811 −0.524055 0.851685i $$-0.675581\pi$$
−0.524055 + 0.851685i $$0.675581\pi$$
$$90$$ −0.637062 −0.0671523
$$91$$ −3.80451 −0.398821
$$92$$ 3.28352 0.342330
$$93$$ 9.94392 1.03114
$$94$$ 2.03273 0.209661
$$95$$ 30.9665 3.17709
$$96$$ 1.97196 0.201262
$$97$$ 0.0560785 0.00569391 0.00284695 0.999996i $$-0.499094\pi$$
0.00284695 + 0.999996i $$0.499094\pi$$
$$98$$ −0.167449 −0.0169149
$$99$$ 1.00000 0.100504
$$100$$ −18.6830 −1.86830
$$101$$ −18.8831 −1.87894 −0.939472 0.342626i $$-0.888683\pi$$
−0.939472 + 0.342626i $$0.888683\pi$$
$$102$$ 0.0560785 0.00555260
$$103$$ −8.27882 −0.815736 −0.407868 0.913041i $$-0.633728\pi$$
−0.407868 + 0.913041i $$0.633728\pi$$
$$104$$ 2.53039 0.248125
$$105$$ 3.80451 0.371282
$$106$$ −1.33020 −0.129201
$$107$$ −8.13941 −0.786866 −0.393433 0.919353i $$-0.628713\pi$$
−0.393433 + 0.919353i $$0.628713\pi$$
$$108$$ 1.97196 0.189752
$$109$$ 11.5529 1.10657 0.553286 0.832992i $$-0.313374\pi$$
0.553286 + 0.832992i $$0.313374\pi$$
$$110$$ −0.637062 −0.0607415
$$111$$ 4.47431 0.424683
$$112$$ −3.83255 −0.362142
$$113$$ −1.33020 −0.125135 −0.0625675 0.998041i $$-0.519929\pi$$
−0.0625675 + 0.998041i $$0.519929\pi$$
$$114$$ 1.36294 0.127651
$$115$$ −6.33490 −0.590732
$$116$$ −0.385496 −0.0357924
$$117$$ 3.80451 0.351727
$$118$$ −0.627672 −0.0577819
$$119$$ −0.334898 −0.0307001
$$120$$ −2.53039 −0.230992
$$121$$ 1.00000 0.0909091
$$122$$ −1.00470 −0.0909608
$$123$$ 6.27882 0.566142
$$124$$ 19.6090 1.76094
$$125$$ 17.0226 1.52254
$$126$$ 0.167449 0.0149176
$$127$$ 14.6137 1.29676 0.648379 0.761318i $$-0.275447\pi$$
0.648379 + 0.761318i $$0.275447\pi$$
$$128$$ 5.17214 0.457157
$$129$$ −2.33490 −0.205576
$$130$$ −2.42371 −0.212574
$$131$$ 20.5576 1.79613 0.898065 0.439863i $$-0.144973\pi$$
0.898065 + 0.439863i $$0.144973\pi$$
$$132$$ 1.97196 0.171637
$$133$$ −8.13941 −0.705776
$$134$$ 0.0233441 0.00201662
$$135$$ −3.80451 −0.327440
$$136$$ 0.222741 0.0190999
$$137$$ −16.2227 −1.38600 −0.693001 0.720936i $$-0.743712\pi$$
−0.693001 + 0.720936i $$0.743712\pi$$
$$138$$ −0.278820 −0.0237347
$$139$$ 22.5482 1.91252 0.956259 0.292522i $$-0.0944945\pi$$
0.956259 + 0.292522i $$0.0944945\pi$$
$$140$$ 7.50235 0.634064
$$141$$ 12.1394 1.02232
$$142$$ −0.781954 −0.0656201
$$143$$ 3.80451 0.318149
$$144$$ 3.83255 0.319379
$$145$$ 0.743738 0.0617641
$$146$$ −0.702531 −0.0581419
$$147$$ −1.00000 −0.0824786
$$148$$ 8.82316 0.725259
$$149$$ 8.19549 0.671401 0.335700 0.941969i $$-0.391027\pi$$
0.335700 + 0.941969i $$0.391027\pi$$
$$150$$ 1.58647 0.129534
$$151$$ −13.2741 −1.08023 −0.540116 0.841590i $$-0.681620\pi$$
−0.540116 + 0.841590i $$0.681620\pi$$
$$152$$ 5.41353 0.439096
$$153$$ 0.334898 0.0270749
$$154$$ 0.167449 0.0134934
$$155$$ −37.8318 −3.03872
$$156$$ 7.50235 0.600669
$$157$$ −16.9392 −1.35190 −0.675949 0.736949i $$-0.736266\pi$$
−0.675949 + 0.736949i $$0.736266\pi$$
$$158$$ −0.557640 −0.0443634
$$159$$ −7.94392 −0.629994
$$160$$ −7.50235 −0.593113
$$161$$ 1.66510 0.131228
$$162$$ −0.167449 −0.0131561
$$163$$ −6.79982 −0.532603 −0.266301 0.963890i $$-0.585802\pi$$
−0.266301 + 0.963890i $$0.585802\pi$$
$$164$$ 12.3816 0.966839
$$165$$ −3.80451 −0.296181
$$166$$ 2.33490 0.181223
$$167$$ −18.2227 −1.41012 −0.705059 0.709149i $$-0.749080\pi$$
−0.705059 + 0.709149i $$0.749080\pi$$
$$168$$ 0.665102 0.0513137
$$169$$ 1.47431 0.113408
$$170$$ −0.213351 −0.0163633
$$171$$ 8.13941 0.622436
$$172$$ −4.60433 −0.351077
$$173$$ −1.72118 −0.130859 −0.0654294 0.997857i $$-0.520842\pi$$
−0.0654294 + 0.997857i $$0.520842\pi$$
$$174$$ 0.0327344 0.00248159
$$175$$ −9.47431 −0.716190
$$176$$ 3.83255 0.288889
$$177$$ −3.74843 −0.281749
$$178$$ 1.65571 0.124101
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ −7.50235 −0.559192
$$181$$ 0.725875 0.0539539 0.0269769 0.999636i $$-0.491412\pi$$
0.0269769 + 0.999636i $$0.491412\pi$$
$$182$$ 0.637062 0.0472222
$$183$$ −6.00000 −0.443533
$$184$$ −1.10746 −0.0816432
$$185$$ −17.0226 −1.25152
$$186$$ −1.66510 −0.122091
$$187$$ 0.334898 0.0244902
$$188$$ 23.9384 1.74589
$$189$$ 1.00000 0.0727393
$$190$$ −5.18531 −0.376182
$$191$$ 5.27412 0.381622 0.190811 0.981627i $$-0.438888\pi$$
0.190811 + 0.981627i $$0.438888\pi$$
$$192$$ 7.33490 0.529351
$$193$$ −19.8318 −1.42752 −0.713761 0.700390i $$-0.753010\pi$$
−0.713761 + 0.700390i $$0.753010\pi$$
$$194$$ −0.00939029 −0.000674184 0
$$195$$ −14.4743 −1.03653
$$196$$ −1.97196 −0.140854
$$197$$ 2.66980 0.190215 0.0951076 0.995467i $$-0.469680\pi$$
0.0951076 + 0.995467i $$0.469680\pi$$
$$198$$ −0.167449 −0.0119001
$$199$$ 13.5529 0.960743 0.480371 0.877065i $$-0.340502\pi$$
0.480371 + 0.877065i $$0.340502\pi$$
$$200$$ 6.30138 0.445575
$$201$$ 0.139410 0.00983322
$$202$$ 3.16197 0.222475
$$203$$ −0.195488 −0.0137206
$$204$$ 0.660406 0.0462377
$$205$$ −23.8878 −1.66840
$$206$$ 1.38628 0.0965868
$$207$$ −1.66510 −0.115733
$$208$$ 14.5810 1.01101
$$209$$ 8.13941 0.563015
$$210$$ −0.637062 −0.0439615
$$211$$ −4.27882 −0.294566 −0.147283 0.989094i $$-0.547053\pi$$
−0.147283 + 0.989094i $$0.547053\pi$$
$$212$$ −15.6651 −1.07588
$$213$$ −4.66980 −0.319969
$$214$$ 1.36294 0.0931685
$$215$$ 8.88315 0.605826
$$216$$ −0.665102 −0.0452544
$$217$$ 9.94392 0.675037
$$218$$ −1.93453 −0.131023
$$219$$ −4.19549 −0.283505
$$220$$ −7.50235 −0.505808
$$221$$ 1.27412 0.0857069
$$222$$ −0.749219 −0.0502843
$$223$$ −10.2694 −0.687692 −0.343846 0.939026i $$-0.611730\pi$$
−0.343846 + 0.939026i $$0.611730\pi$$
$$224$$ 1.97196 0.131757
$$225$$ 9.47431 0.631621
$$226$$ 0.222741 0.0148165
$$227$$ −0.390977 −0.0259500 −0.0129750 0.999916i $$-0.504130\pi$$
−0.0129750 + 0.999916i $$0.504130\pi$$
$$228$$ 16.0506 1.06298
$$229$$ 7.94392 0.524949 0.262475 0.964939i $$-0.415461\pi$$
0.262475 + 0.964939i $$0.415461\pi$$
$$230$$ 1.06077 0.0699453
$$231$$ 1.00000 0.0657952
$$232$$ 0.130020 0.00853621
$$233$$ 26.5576 1.73985 0.869924 0.493185i $$-0.164167\pi$$
0.869924 + 0.493185i $$0.164167\pi$$
$$234$$ −0.637062 −0.0416461
$$235$$ −46.1845 −3.01275
$$236$$ −7.39176 −0.481163
$$237$$ −3.33020 −0.216320
$$238$$ 0.0560785 0.00363503
$$239$$ 10.7998 0.698582 0.349291 0.937014i $$-0.386422\pi$$
0.349291 + 0.937014i $$0.386422\pi$$
$$240$$ −14.5810 −0.941198
$$241$$ −12.0833 −0.778356 −0.389178 0.921163i $$-0.627241\pi$$
−0.389178 + 0.921163i $$0.627241\pi$$
$$242$$ −0.167449 −0.0107640
$$243$$ −1.00000 −0.0641500
$$244$$ −11.8318 −0.757451
$$245$$ 3.80451 0.243061
$$246$$ −1.05138 −0.0670338
$$247$$ 30.9665 1.97035
$$248$$ −6.61372 −0.419972
$$249$$ 13.9439 0.883660
$$250$$ −2.85041 −0.180276
$$251$$ 4.80921 0.303554 0.151777 0.988415i $$-0.451500\pi$$
0.151777 + 0.988415i $$0.451500\pi$$
$$252$$ 1.97196 0.124222
$$253$$ −1.66510 −0.104684
$$254$$ −2.44706 −0.153542
$$255$$ −1.27412 −0.0797888
$$256$$ 13.8037 0.862733
$$257$$ −16.7531 −1.04503 −0.522516 0.852630i $$-0.675006\pi$$
−0.522516 + 0.852630i $$0.675006\pi$$
$$258$$ 0.390977 0.0243412
$$259$$ 4.47431 0.278020
$$260$$ −28.5428 −1.77015
$$261$$ 0.195488 0.0121004
$$262$$ −3.44236 −0.212670
$$263$$ −12.1394 −0.748548 −0.374274 0.927318i $$-0.622108\pi$$
−0.374274 + 0.927318i $$0.622108\pi$$
$$264$$ −0.665102 −0.0409342
$$265$$ 30.2227 1.85657
$$266$$ 1.36294 0.0835671
$$267$$ 9.88784 0.605126
$$268$$ 0.274911 0.0167929
$$269$$ −25.2180 −1.53757 −0.768786 0.639506i $$-0.779139\pi$$
−0.768786 + 0.639506i $$0.779139\pi$$
$$270$$ 0.637062 0.0387704
$$271$$ 4.13941 0.251451 0.125726 0.992065i $$-0.459874\pi$$
0.125726 + 0.992065i $$0.459874\pi$$
$$272$$ 1.28352 0.0778245
$$273$$ 3.80451 0.230260
$$274$$ 2.71648 0.164109
$$275$$ 9.47431 0.571322
$$276$$ −3.28352 −0.197644
$$277$$ −18.2788 −1.09827 −0.549134 0.835734i $$-0.685042\pi$$
−0.549134 + 0.835734i $$0.685042\pi$$
$$278$$ −3.77569 −0.226451
$$279$$ −9.94392 −0.595327
$$280$$ −2.53039 −0.151220
$$281$$ 6.74374 0.402298 0.201149 0.979561i $$-0.435533\pi$$
0.201149 + 0.979561i $$0.435533\pi$$
$$282$$ −2.03273 −0.121048
$$283$$ 23.3575 1.38846 0.694228 0.719755i $$-0.255746\pi$$
0.694228 + 0.719755i $$0.255746\pi$$
$$284$$ −9.20866 −0.546433
$$285$$ −30.9665 −1.83430
$$286$$ −0.637062 −0.0376703
$$287$$ 6.27882 0.370627
$$288$$ −1.97196 −0.116199
$$289$$ −16.8878 −0.993403
$$290$$ −0.124538 −0.00731314
$$291$$ −0.0560785 −0.00328738
$$292$$ −8.27334 −0.484161
$$293$$ −14.1667 −0.827625 −0.413813 0.910362i $$-0.635803\pi$$
−0.413813 + 0.910362i $$0.635803\pi$$
$$294$$ 0.167449 0.00976584
$$295$$ 14.2610 0.830305
$$296$$ −2.97587 −0.172969
$$297$$ −1.00000 −0.0580259
$$298$$ −1.37233 −0.0794968
$$299$$ −6.33490 −0.366357
$$300$$ 18.6830 1.07866
$$301$$ −2.33490 −0.134581
$$302$$ 2.22274 0.127904
$$303$$ 18.8831 1.08481
$$304$$ 31.1947 1.78914
$$305$$ 22.8271 1.30707
$$306$$ −0.0560785 −0.00320579
$$307$$ −12.5576 −0.716702 −0.358351 0.933587i $$-0.616661\pi$$
−0.358351 + 0.933587i $$0.616661\pi$$
$$308$$ 1.97196 0.112363
$$309$$ 8.27882 0.470966
$$310$$ 6.33490 0.359798
$$311$$ 9.33959 0.529600 0.264800 0.964303i $$-0.414694\pi$$
0.264800 + 0.964303i $$0.414694\pi$$
$$312$$ −2.53039 −0.143255
$$313$$ −2.99530 −0.169305 −0.0846523 0.996411i $$-0.526978\pi$$
−0.0846523 + 0.996411i $$0.526978\pi$$
$$314$$ 2.83646 0.160071
$$315$$ −3.80451 −0.214360
$$316$$ −6.56703 −0.369424
$$317$$ 9.93453 0.557979 0.278989 0.960294i $$-0.410001\pi$$
0.278989 + 0.960294i $$0.410001\pi$$
$$318$$ 1.33020 0.0745941
$$319$$ 0.195488 0.0109453
$$320$$ −27.9057 −1.55998
$$321$$ 8.13941 0.454298
$$322$$ −0.278820 −0.0155380
$$323$$ 2.72588 0.151672
$$324$$ −1.97196 −0.109553
$$325$$ 36.0451 1.99942
$$326$$ 1.13862 0.0630625
$$327$$ −11.5529 −0.638879
$$328$$ −4.17605 −0.230584
$$329$$ 12.1394 0.669267
$$330$$ 0.637062 0.0350691
$$331$$ 22.5482 1.23936 0.619682 0.784853i $$-0.287262\pi$$
0.619682 + 0.784853i $$0.287262\pi$$
$$332$$ 27.4969 1.50909
$$333$$ −4.47431 −0.245191
$$334$$ 3.05138 0.166964
$$335$$ −0.530387 −0.0289781
$$336$$ 3.83255 0.209083
$$337$$ 13.2835 0.723599 0.361800 0.932256i $$-0.382162\pi$$
0.361800 + 0.932256i $$0.382162\pi$$
$$338$$ −0.246872 −0.0134281
$$339$$ 1.33020 0.0722467
$$340$$ −2.51252 −0.136261
$$341$$ −9.94392 −0.538494
$$342$$ −1.36294 −0.0736992
$$343$$ −1.00000 −0.0539949
$$344$$ 1.55294 0.0837292
$$345$$ 6.33490 0.341059
$$346$$ 0.288210 0.0154943
$$347$$ −27.7757 −1.49108 −0.745538 0.666463i $$-0.767808\pi$$
−0.745538 + 0.666463i $$0.767808\pi$$
$$348$$ 0.385496 0.0206647
$$349$$ −2.85589 −0.152873 −0.0764363 0.997074i $$-0.524354\pi$$
−0.0764363 + 0.997074i $$0.524354\pi$$
$$350$$ 1.58647 0.0848001
$$351$$ −3.80451 −0.203070
$$352$$ −1.97196 −0.105106
$$353$$ 24.6410 1.31151 0.655753 0.754975i $$-0.272351\pi$$
0.655753 + 0.754975i $$0.272351\pi$$
$$354$$ 0.627672 0.0333604
$$355$$ 17.7663 0.942937
$$356$$ 19.4984 1.03342
$$357$$ 0.334898 0.0177247
$$358$$ 2.00939 0.106200
$$359$$ −11.8878 −0.627416 −0.313708 0.949519i $$-0.601571\pi$$
−0.313708 + 0.949519i $$0.601571\pi$$
$$360$$ 2.53039 0.133363
$$361$$ 47.2500 2.48684
$$362$$ −0.121547 −0.00638838
$$363$$ −1.00000 −0.0524864
$$364$$ 7.50235 0.393230
$$365$$ 15.9618 0.835478
$$366$$ 1.00470 0.0525163
$$367$$ 3.60902 0.188389 0.0941947 0.995554i $$-0.469972\pi$$
0.0941947 + 0.995554i $$0.469972\pi$$
$$368$$ −6.38159 −0.332663
$$369$$ −6.27882 −0.326862
$$370$$ 2.85041 0.148186
$$371$$ −7.94392 −0.412428
$$372$$ −19.6090 −1.01668
$$373$$ 12.3349 0.638677 0.319338 0.947641i $$-0.396539\pi$$
0.319338 + 0.947641i $$0.396539\pi$$
$$374$$ −0.0560785 −0.00289975
$$375$$ −17.0226 −0.879041
$$376$$ −8.07394 −0.416382
$$377$$ 0.743738 0.0383045
$$378$$ −0.167449 −0.00861266
$$379$$ 23.3575 1.19979 0.599896 0.800078i $$-0.295209\pi$$
0.599896 + 0.800078i $$0.295209\pi$$
$$380$$ −61.0647 −3.13255
$$381$$ −14.6137 −0.748683
$$382$$ −0.883148 −0.0451858
$$383$$ −15.2180 −0.777606 −0.388803 0.921321i $$-0.627111\pi$$
−0.388803 + 0.921321i $$0.627111\pi$$
$$384$$ −5.17214 −0.263940
$$385$$ −3.80451 −0.193896
$$386$$ 3.32081 0.169025
$$387$$ 2.33490 0.118690
$$388$$ −0.110585 −0.00561408
$$389$$ 3.73057 0.189147 0.0945737 0.995518i $$-0.469851\pi$$
0.0945737 + 0.995518i $$0.469851\pi$$
$$390$$ 2.42371 0.122729
$$391$$ −0.557640 −0.0282011
$$392$$ 0.665102 0.0335927
$$393$$ −20.5576 −1.03700
$$394$$ −0.447055 −0.0225223
$$395$$ 12.6698 0.637487
$$396$$ −1.97196 −0.0990948
$$397$$ 20.8925 1.04857 0.524283 0.851544i $$-0.324333\pi$$
0.524283 + 0.851544i $$0.324333\pi$$
$$398$$ −2.26943 −0.113756
$$399$$ 8.13941 0.407480
$$400$$ 36.3108 1.81554
$$401$$ 23.2741 1.16225 0.581127 0.813813i $$-0.302612\pi$$
0.581127 + 0.813813i $$0.302612\pi$$
$$402$$ −0.0233441 −0.00116430
$$403$$ −37.8318 −1.88453
$$404$$ 37.2368 1.85260
$$405$$ 3.80451 0.189048
$$406$$ 0.0327344 0.00162458
$$407$$ −4.47431 −0.221783
$$408$$ −0.222741 −0.0110273
$$409$$ −23.8972 −1.18164 −0.590821 0.806803i $$-0.701196\pi$$
−0.590821 + 0.806803i $$0.701196\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 16.2227 0.800209
$$412$$ 16.3255 0.804300
$$413$$ −3.74843 −0.184448
$$414$$ 0.278820 0.0137033
$$415$$ −53.0498 −2.60411
$$416$$ −7.50235 −0.367833
$$417$$ −22.5482 −1.10419
$$418$$ −1.36294 −0.0666635
$$419$$ −30.2967 −1.48009 −0.740045 0.672557i $$-0.765196\pi$$
−0.740045 + 0.672557i $$0.765196\pi$$
$$420$$ −7.50235 −0.366077
$$421$$ −15.5257 −0.756676 −0.378338 0.925668i $$-0.623504\pi$$
−0.378338 + 0.925668i $$0.623504\pi$$
$$422$$ 0.716485 0.0348779
$$423$$ −12.1394 −0.590238
$$424$$ 5.28352 0.256590
$$425$$ 3.17293 0.153910
$$426$$ 0.781954 0.0378858
$$427$$ −6.00000 −0.290360
$$428$$ 16.0506 0.775835
$$429$$ −3.80451 −0.183684
$$430$$ −1.48748 −0.0717325
$$431$$ 3.07864 0.148293 0.0741463 0.997247i $$-0.476377\pi$$
0.0741463 + 0.997247i $$0.476377\pi$$
$$432$$ −3.83255 −0.184394
$$433$$ 16.9392 0.814047 0.407024 0.913418i $$-0.366567\pi$$
0.407024 + 0.913418i $$0.366567\pi$$
$$434$$ −1.66510 −0.0799274
$$435$$ −0.743738 −0.0356595
$$436$$ −22.7820 −1.09106
$$437$$ −13.5529 −0.648325
$$438$$ 0.702531 0.0335682
$$439$$ −11.0786 −0.528754 −0.264377 0.964419i $$-0.585166\pi$$
−0.264377 + 0.964419i $$0.585166\pi$$
$$440$$ 2.53039 0.120631
$$441$$ 1.00000 0.0476190
$$442$$ −0.213351 −0.0101481
$$443$$ −14.5482 −0.691208 −0.345604 0.938380i $$-0.612326\pi$$
−0.345604 + 0.938380i $$0.612326\pi$$
$$444$$ −8.82316 −0.418729
$$445$$ −37.6184 −1.78328
$$446$$ 1.71961 0.0814258
$$447$$ −8.19549 −0.387633
$$448$$ 7.33490 0.346541
$$449$$ 27.4875 1.29721 0.648607 0.761123i $$-0.275352\pi$$
0.648607 + 0.761123i $$0.275352\pi$$
$$450$$ −1.58647 −0.0747867
$$451$$ −6.27882 −0.295658
$$452$$ 2.62311 0.123381
$$453$$ 13.2741 0.623673
$$454$$ 0.0654688 0.00307260
$$455$$ −14.4743 −0.678566
$$456$$ −5.41353 −0.253512
$$457$$ −7.94392 −0.371601 −0.185800 0.982587i $$-0.559488\pi$$
−0.185800 + 0.982587i $$0.559488\pi$$
$$458$$ −1.33020 −0.0621563
$$459$$ −0.334898 −0.0156317
$$460$$ 12.4922 0.582450
$$461$$ 8.26943 0.385146 0.192573 0.981283i $$-0.438317\pi$$
0.192573 + 0.981283i $$0.438317\pi$$
$$462$$ −0.167449 −0.00779044
$$463$$ 9.73904 0.452612 0.226306 0.974056i $$-0.427335\pi$$
0.226306 + 0.974056i $$0.427335\pi$$
$$464$$ 0.749219 0.0347816
$$465$$ 37.8318 1.75441
$$466$$ −4.44706 −0.206006
$$467$$ −40.6970 −1.88323 −0.941617 0.336685i $$-0.890694\pi$$
−0.941617 + 0.336685i $$0.890694\pi$$
$$468$$ −7.50235 −0.346796
$$469$$ 0.139410 0.00643735
$$470$$ 7.73356 0.356723
$$471$$ 16.9392 0.780518
$$472$$ 2.49309 0.114754
$$473$$ 2.33490 0.107359
$$474$$ 0.557640 0.0256132
$$475$$ 77.1153 3.53829
$$476$$ 0.660406 0.0302697
$$477$$ 7.94392 0.363727
$$478$$ −1.80842 −0.0827152
$$479$$ 37.8318 1.72858 0.864289 0.502996i $$-0.167769\pi$$
0.864289 + 0.502996i $$0.167769\pi$$
$$480$$ 7.50235 0.342434
$$481$$ −17.0226 −0.776162
$$482$$ 2.02334 0.0921608
$$483$$ −1.66510 −0.0757647
$$484$$ −1.97196 −0.0896346
$$485$$ 0.213351 0.00968778
$$486$$ 0.167449 0.00759565
$$487$$ 40.5576 1.83784 0.918921 0.394442i $$-0.129062\pi$$
0.918921 + 0.394442i $$0.129062\pi$$
$$488$$ 3.99061 0.180646
$$489$$ 6.79982 0.307498
$$490$$ −0.637062 −0.0287795
$$491$$ 31.0786 1.40256 0.701280 0.712886i $$-0.252612\pi$$
0.701280 + 0.712886i $$0.252612\pi$$
$$492$$ −12.3816 −0.558205
$$493$$ 0.0654688 0.00294856
$$494$$ −5.18531 −0.233298
$$495$$ 3.80451 0.171000
$$496$$ −38.1106 −1.71122
$$497$$ −4.66980 −0.209469
$$498$$ −2.33490 −0.104629
$$499$$ 15.3575 0.687494 0.343747 0.939062i $$-0.388304\pi$$
0.343747 + 0.939062i $$0.388304\pi$$
$$500$$ −33.5678 −1.50120
$$501$$ 18.2227 0.814132
$$502$$ −0.805298 −0.0359422
$$503$$ 14.8925 0.664025 0.332013 0.943275i $$-0.392272\pi$$
0.332013 + 0.943275i $$0.392272\pi$$
$$504$$ −0.665102 −0.0296260
$$505$$ −71.8412 −3.19689
$$506$$ 0.278820 0.0123951
$$507$$ −1.47431 −0.0654763
$$508$$ −28.8177 −1.27858
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0.213351 0.00944735
$$511$$ −4.19549 −0.185597
$$512$$ −12.6557 −0.559309
$$513$$ −8.13941 −0.359364
$$514$$ 2.80530 0.123736
$$515$$ −31.4969 −1.38792
$$516$$ 4.60433 0.202694
$$517$$ −12.1394 −0.533891
$$518$$ −0.749219 −0.0329188
$$519$$ 1.72118 0.0755514
$$520$$ 9.62689 0.422167
$$521$$ 27.6924 1.21322 0.606612 0.794998i $$-0.292528\pi$$
0.606612 + 0.794998i $$0.292528\pi$$
$$522$$ −0.0327344 −0.00143274
$$523$$ 18.4088 0.804962 0.402481 0.915428i $$-0.368148\pi$$
0.402481 + 0.915428i $$0.368148\pi$$
$$524$$ −40.5389 −1.77095
$$525$$ 9.47431 0.413493
$$526$$ 2.03273 0.0886314
$$527$$ −3.33020 −0.145066
$$528$$ −3.83255 −0.166790
$$529$$ −20.2274 −0.879454
$$530$$ −5.06077 −0.219826
$$531$$ 3.74843 0.162668
$$532$$ 16.0506 0.695882
$$533$$ −23.8878 −1.03470
$$534$$ −1.65571 −0.0716496
$$535$$ −30.9665 −1.33880
$$536$$ −0.0927218 −0.00400497
$$537$$ 12.0000 0.517838
$$538$$ 4.22274 0.182055
$$539$$ 1.00000 0.0430730
$$540$$ 7.50235 0.322850
$$541$$ −4.26943 −0.183557 −0.0917786 0.995779i $$-0.529255\pi$$
−0.0917786 + 0.995779i $$0.529255\pi$$
$$542$$ −0.693141 −0.0297729
$$543$$ −0.725875 −0.0311503
$$544$$ −0.660406 −0.0283147
$$545$$ 43.9533 1.88275
$$546$$ −0.637062 −0.0272638
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 31.9906 1.36657
$$549$$ 6.00000 0.256074
$$550$$ −1.58647 −0.0676471
$$551$$ 1.59116 0.0677857
$$552$$ 1.10746 0.0471367
$$553$$ −3.33020 −0.141615
$$554$$ 3.06077 0.130040
$$555$$ 17.0226 0.722567
$$556$$ −44.4643 −1.88570
$$557$$ −12.4743 −0.528553 −0.264277 0.964447i $$-0.585133\pi$$
−0.264277 + 0.964447i $$0.585133\pi$$
$$558$$ 1.66510 0.0704894
$$559$$ 8.88315 0.375717
$$560$$ −14.5810 −0.616159
$$561$$ −0.334898 −0.0141394
$$562$$ −1.12923 −0.0476338
$$563$$ −32.7710 −1.38113 −0.690566 0.723269i $$-0.742639\pi$$
−0.690566 + 0.723269i $$0.742639\pi$$
$$564$$ −23.9384 −1.00799
$$565$$ −5.06077 −0.212908
$$566$$ −3.91119 −0.164399
$$567$$ −1.00000 −0.0419961
$$568$$ 3.10589 0.130320
$$569$$ −29.2180 −1.22488 −0.612442 0.790515i $$-0.709813\pi$$
−0.612442 + 0.790515i $$0.709813\pi$$
$$570$$ 5.18531 0.217189
$$571$$ 32.4455 1.35780 0.678901 0.734230i $$-0.262457\pi$$
0.678901 + 0.734230i $$0.262457\pi$$
$$572$$ −7.50235 −0.313689
$$573$$ −5.27412 −0.220330
$$574$$ −1.05138 −0.0438839
$$575$$ −15.7757 −0.657892
$$576$$ −7.33490 −0.305621
$$577$$ −14.8831 −0.619594 −0.309797 0.950803i $$-0.600261\pi$$
−0.309797 + 0.950803i $$0.600261\pi$$
$$578$$ 2.82786 0.117623
$$579$$ 19.8318 0.824180
$$580$$ −1.46662 −0.0608982
$$581$$ 13.9439 0.578491
$$582$$ 0.00939029 0.000389240 0
$$583$$ 7.94392 0.329004
$$584$$ 2.79043 0.115469
$$585$$ 14.4743 0.598439
$$586$$ 2.37220 0.0979945
$$587$$ −4.53039 −0.186989 −0.0934945 0.995620i $$-0.529804\pi$$
−0.0934945 + 0.995620i $$0.529804\pi$$
$$588$$ 1.97196 0.0813223
$$589$$ −80.9377 −3.33498
$$590$$ −2.38799 −0.0983118
$$591$$ −2.66980 −0.109821
$$592$$ −17.1480 −0.704779
$$593$$ 8.28821 0.340356 0.170178 0.985413i $$-0.445566\pi$$
0.170178 + 0.985413i $$0.445566\pi$$
$$594$$ 0.167449 0.00687052
$$595$$ −1.27412 −0.0522340
$$596$$ −16.1612 −0.661988
$$597$$ −13.5529 −0.554685
$$598$$ 1.06077 0.0433783
$$599$$ 1.99061 0.0813341 0.0406671 0.999173i $$-0.487052\pi$$
0.0406671 + 0.999173i $$0.487052\pi$$
$$600$$ −6.30138 −0.257253
$$601$$ −8.47431 −0.345674 −0.172837 0.984950i $$-0.555293\pi$$
−0.172837 + 0.984950i $$0.555293\pi$$
$$602$$ 0.390977 0.0159350
$$603$$ −0.139410 −0.00567721
$$604$$ 26.1761 1.06509
$$605$$ 3.80451 0.154675
$$606$$ −3.16197 −0.128446
$$607$$ −22.9665 −0.932181 −0.466090 0.884737i $$-0.654338\pi$$
−0.466090 + 0.884737i $$0.654338\pi$$
$$608$$ −16.0506 −0.650938
$$609$$ 0.195488 0.00792159
$$610$$ −3.82237 −0.154763
$$611$$ −46.1845 −1.86843
$$612$$ −0.660406 −0.0266953
$$613$$ 3.55294 0.143502 0.0717510 0.997423i $$-0.477141\pi$$
0.0717510 + 0.997423i $$0.477141\pi$$
$$614$$ 2.10277 0.0848608
$$615$$ 23.8878 0.963251
$$616$$ −0.665102 −0.0267977
$$617$$ 22.6698 0.912652 0.456326 0.889813i $$-0.349165\pi$$
0.456326 + 0.889813i $$0.349165\pi$$
$$618$$ −1.38628 −0.0557644
$$619$$ 8.71648 0.350345 0.175173 0.984538i $$-0.443952\pi$$
0.175173 + 0.984538i $$0.443952\pi$$
$$620$$ 74.6028 2.99612
$$621$$ 1.66510 0.0668182
$$622$$ −1.56391 −0.0627070
$$623$$ 9.88784 0.396148
$$624$$ −14.5810 −0.583707
$$625$$ 17.3910 0.695639
$$626$$ 0.501561 0.0200464
$$627$$ −8.13941 −0.325057
$$628$$ 33.4035 1.33294
$$629$$ −1.49844 −0.0597467
$$630$$ 0.637062 0.0253812
$$631$$ −11.8878 −0.473248 −0.236624 0.971601i $$-0.576041\pi$$
−0.236624 + 0.971601i $$0.576041\pi$$
$$632$$ 2.21492 0.0881049
$$633$$ 4.27882 0.170068
$$634$$ −1.66353 −0.0660672
$$635$$ 55.5981 2.20634
$$636$$ 15.6651 0.621162
$$637$$ 3.80451 0.150740
$$638$$ −0.0327344 −0.00129597
$$639$$ 4.66980 0.184734
$$640$$ 19.6775 0.777821
$$641$$ −4.89254 −0.193244 −0.0966218 0.995321i $$-0.530804\pi$$
−0.0966218 + 0.995321i $$0.530804\pi$$
$$642$$ −1.36294 −0.0537909
$$643$$ 22.7804 0.898371 0.449185 0.893439i $$-0.351714\pi$$
0.449185 + 0.893439i $$0.351714\pi$$
$$644$$ −3.28352 −0.129389
$$645$$ −8.88315 −0.349774
$$646$$ −0.456446 −0.0179586
$$647$$ 31.2453 1.22838 0.614190 0.789158i $$-0.289483\pi$$
0.614190 + 0.789158i $$0.289483\pi$$
$$648$$ 0.665102 0.0261277
$$649$$ 3.74843 0.147139
$$650$$ −6.03573 −0.236741
$$651$$ −9.94392 −0.389733
$$652$$ 13.4090 0.525136
$$653$$ 28.2882 1.10700 0.553502 0.832848i $$-0.313291\pi$$
0.553502 + 0.832848i $$0.313291\pi$$
$$654$$ 1.93453 0.0756462
$$655$$ 78.2118 3.05599
$$656$$ −24.0639 −0.939537
$$657$$ 4.19549 0.163682
$$658$$ −2.03273 −0.0792442
$$659$$ 30.2967 1.18019 0.590096 0.807333i $$-0.299090\pi$$
0.590096 + 0.807333i $$0.299090\pi$$
$$660$$ 7.50235 0.292028
$$661$$ −35.7196 −1.38933 −0.694666 0.719333i $$-0.744448\pi$$
−0.694666 + 0.719333i $$0.744448\pi$$
$$662$$ −3.77569 −0.146746
$$663$$ −1.27412 −0.0494829
$$664$$ −9.27412 −0.359906
$$665$$ −30.9665 −1.20083
$$666$$ 0.749219 0.0290317
$$667$$ −0.325508 −0.0126037
$$668$$ 35.9345 1.39035
$$669$$ 10.2694 0.397039
$$670$$ 0.0888128 0.00343114
$$671$$ 6.00000 0.231627
$$672$$ −1.97196 −0.0760700
$$673$$ 34.9377 1.34675 0.673374 0.739302i $$-0.264844\pi$$
0.673374 + 0.739302i $$0.264844\pi$$
$$674$$ −2.22431 −0.0856774
$$675$$ −9.47431 −0.364666
$$676$$ −2.90728 −0.111818
$$677$$ 20.0094 0.769023 0.384512 0.923120i $$-0.374370\pi$$
0.384512 + 0.923120i $$0.374370\pi$$
$$678$$ −0.222741 −0.00855433
$$679$$ −0.0560785 −0.00215209
$$680$$ 0.847422 0.0324972
$$681$$ 0.390977 0.0149823
$$682$$ 1.66510 0.0637600
$$683$$ 25.0498 0.958504 0.479252 0.877677i $$-0.340908\pi$$
0.479252 + 0.877677i $$0.340908\pi$$
$$684$$ −16.0506 −0.613710
$$685$$ −61.7196 −2.35818
$$686$$ 0.167449 0.00639324
$$687$$ −7.94392 −0.303080
$$688$$ 8.94862 0.341163
$$689$$ 30.2227 1.15139
$$690$$ −1.06077 −0.0403830
$$691$$ −38.8271 −1.47705 −0.738526 0.674225i $$-0.764478\pi$$
−0.738526 + 0.674225i $$0.764478\pi$$
$$692$$ 3.39410 0.129024
$$693$$ −1.00000 −0.0379869
$$694$$ 4.65102 0.176550
$$695$$ 85.7851 3.25401
$$696$$ −0.130020 −0.00492838
$$697$$ −2.10277 −0.0796480
$$698$$ 0.478217 0.0181008
$$699$$ −26.5576 −1.00450
$$700$$ 18.6830 0.706150
$$701$$ −41.7757 −1.57785 −0.788923 0.614492i $$-0.789361\pi$$
−0.788923 + 0.614492i $$0.789361\pi$$
$$702$$ 0.637062 0.0240444
$$703$$ −36.4182 −1.37354
$$704$$ −7.33490 −0.276444
$$705$$ 46.1845 1.73941
$$706$$ −4.12611 −0.155288
$$707$$ 18.8831 0.710174
$$708$$ 7.39176 0.277799
$$709$$ −13.4041 −0.503403 −0.251702 0.967805i $$-0.580990\pi$$
−0.251702 + 0.967805i $$0.580990\pi$$
$$710$$ −2.97495 −0.111648
$$711$$ 3.33020 0.124892
$$712$$ −6.57642 −0.246462
$$713$$ 16.5576 0.620088
$$714$$ −0.0560785 −0.00209868
$$715$$ 14.4743 0.541308
$$716$$ 23.6635 0.884348
$$717$$ −10.7998 −0.403327
$$718$$ 1.99061 0.0742889
$$719$$ 5.47900 0.204332 0.102166 0.994767i $$-0.467423\pi$$
0.102166 + 0.994767i $$0.467423\pi$$
$$720$$ 14.5810 0.543401
$$721$$ 8.27882 0.308319
$$722$$ −7.91197 −0.294453
$$723$$ 12.0833 0.449384
$$724$$ −1.43140 −0.0531975
$$725$$ 1.85212 0.0687859
$$726$$ 0.167449 0.00621462
$$727$$ −32.8831 −1.21957 −0.609784 0.792567i $$-0.708744\pi$$
−0.609784 + 0.792567i $$0.708744\pi$$
$$728$$ −2.53039 −0.0937824
$$729$$ 1.00000 0.0370370
$$730$$ −2.67279 −0.0989243
$$731$$ 0.781954 0.0289216
$$732$$ 11.8318 0.437315
$$733$$ −35.8972 −1.32589 −0.662947 0.748666i $$-0.730695\pi$$
−0.662947 + 0.748666i $$0.730695\pi$$
$$734$$ −0.604328 −0.0223062
$$735$$ −3.80451 −0.140332
$$736$$ 3.28352 0.121032
$$737$$ −0.139410 −0.00513523
$$738$$ 1.05138 0.0387020
$$739$$ 29.6651 1.09125 0.545624 0.838030i $$-0.316293\pi$$
0.545624 + 0.838030i $$0.316293\pi$$
$$740$$ 33.5678 1.23398
$$741$$ −30.9665 −1.13758
$$742$$ 1.33020 0.0488333
$$743$$ −18.7998 −0.689698 −0.344849 0.938658i $$-0.612070\pi$$
−0.344849 + 0.938658i $$0.612070\pi$$
$$744$$ 6.61372 0.242471
$$745$$ 31.1798 1.14234
$$746$$ −2.06547 −0.0756222
$$747$$ −13.9439 −0.510181
$$748$$ −0.660406 −0.0241469
$$749$$ 8.13941 0.297408
$$750$$ 2.85041 0.104082
$$751$$ −9.59116 −0.349986 −0.174993 0.984570i $$-0.555990\pi$$
−0.174993 + 0.984570i $$0.555990\pi$$
$$752$$ −46.5249 −1.69659
$$753$$ −4.80921 −0.175257
$$754$$ −0.124538 −0.00453542
$$755$$ −50.5016 −1.83794
$$756$$ −1.97196 −0.0717195
$$757$$ 2.74374 0.0997229 0.0498614 0.998756i $$-0.484122\pi$$
0.0498614 + 0.998756i $$0.484122\pi$$
$$758$$ −3.91119 −0.142061
$$759$$ 1.66510 0.0604394
$$760$$ 20.5959 0.747090
$$761$$ 6.39098 0.231673 0.115836 0.993268i $$-0.463045\pi$$
0.115836 + 0.993268i $$0.463045\pi$$
$$762$$ 2.44706 0.0886475
$$763$$ −11.5529 −0.418245
$$764$$ −10.4004 −0.376272
$$765$$ 1.27412 0.0460661
$$766$$ 2.54825 0.0920720
$$767$$ 14.2610 0.514933
$$768$$ −13.8037 −0.498099
$$769$$ 17.7951 0.641708 0.320854 0.947129i $$-0.396030\pi$$
0.320854 + 0.947129i $$0.396030\pi$$
$$770$$ 0.637062 0.0229581
$$771$$ 16.7531 0.603349
$$772$$ 39.1075 1.40751
$$773$$ −31.5257 −1.13390 −0.566950 0.823752i $$-0.691877\pi$$
−0.566950 + 0.823752i $$0.691877\pi$$
$$774$$ −0.390977 −0.0140534
$$775$$ −94.2118 −3.38419
$$776$$ 0.0372979 0.00133892
$$777$$ −4.47431 −0.160515
$$778$$ −0.624681 −0.0223959
$$779$$ −51.1059 −1.83106
$$780$$ 28.5428 1.02200
$$781$$ 4.66980 0.167098
$$782$$ 0.0933763 0.00333913
$$783$$ −0.195488 −0.00698619
$$784$$ 3.83255 0.136877
$$785$$ −64.4455 −2.30016
$$786$$ 3.44236 0.122785
$$787$$ 31.8606 1.13571 0.567854 0.823130i $$-0.307774\pi$$
0.567854 + 0.823130i $$0.307774\pi$$
$$788$$ −5.26473 −0.187548
$$789$$ 12.1394 0.432174
$$790$$ −2.12155 −0.0754813
$$791$$ 1.33020 0.0472966
$$792$$ 0.665102 0.0236334
$$793$$ 22.8271 0.810613
$$794$$ −3.49844 −0.124155
$$795$$ −30.2227 −1.07189
$$796$$ −26.7259 −0.947274
$$797$$ 43.8590 1.55357 0.776783 0.629768i $$-0.216850\pi$$
0.776783 + 0.629768i $$0.216850\pi$$
$$798$$ −1.36294 −0.0482475
$$799$$ −4.06547 −0.143826
$$800$$ −18.6830 −0.660543
$$801$$ −9.88784 −0.349370
$$802$$ −3.89723 −0.137616
$$803$$ 4.19549 0.148056
$$804$$ −0.274911 −0.00969536
$$805$$ 6.33490 0.223276
$$806$$ 6.33490 0.223137
$$807$$ 25.2180 0.887717
$$808$$ −12.5592 −0.441832
$$809$$ 6.74374 0.237097 0.118549 0.992948i $$-0.462176\pi$$
0.118549 + 0.992948i $$0.462176\pi$$
$$810$$ −0.637062 −0.0223841
$$811$$ 3.97275 0.139502 0.0697510 0.997564i $$-0.477780\pi$$
0.0697510 + 0.997564i $$0.477780\pi$$
$$812$$ 0.385496 0.0135282
$$813$$ −4.13941 −0.145175
$$814$$ 0.749219 0.0262601
$$815$$ −25.8700 −0.906186
$$816$$ −1.28352 −0.0449320
$$817$$ 19.0047 0.664890
$$818$$ 4.00157 0.139912
$$819$$ −3.80451 −0.132940
$$820$$ 47.1059 1.64501
$$821$$ 49.5896 1.73069 0.865344 0.501178i $$-0.167100\pi$$
0.865344 + 0.501178i $$0.167100\pi$$
$$822$$ −2.71648 −0.0947483
$$823$$ 23.1908 0.808380 0.404190 0.914675i $$-0.367553\pi$$
0.404190 + 0.914675i $$0.367553\pi$$
$$824$$ −5.50626 −0.191820
$$825$$ −9.47431 −0.329853
$$826$$ 0.627672 0.0218395
$$827$$ 27.7484 0.964908 0.482454 0.875921i $$-0.339746\pi$$
0.482454 + 0.875921i $$0.339746\pi$$
$$828$$ 3.28352 0.114110
$$829$$ 0.269430 0.00935768 0.00467884 0.999989i $$-0.498511\pi$$
0.00467884 + 0.999989i $$0.498511\pi$$
$$830$$ 8.88315 0.308339
$$831$$ 18.2788 0.634085
$$832$$ −27.9057 −0.967456
$$833$$ 0.334898 0.0116035
$$834$$ 3.77569 0.130741
$$835$$ −69.3286 −2.39922
$$836$$ −16.0506 −0.555122
$$837$$ 9.94392 0.343712
$$838$$ 5.07316 0.175249
$$839$$ −27.3575 −0.944484 −0.472242 0.881469i $$-0.656555\pi$$
−0.472242 + 0.881469i $$0.656555\pi$$
$$840$$ 2.53039 0.0873066
$$841$$ −28.9618 −0.998682
$$842$$ 2.59976 0.0895938
$$843$$ −6.74374 −0.232267
$$844$$ 8.43767 0.290436
$$845$$ 5.60902 0.192956
$$846$$ 2.03273 0.0698868
$$847$$ −1.00000 −0.0343604
$$848$$ 30.4455 1.04550
$$849$$ −23.3575 −0.801626
$$850$$ −0.531305 −0.0182236
$$851$$ 7.45018 0.255389
$$852$$ 9.20866 0.315483
$$853$$ −28.5482 −0.977473 −0.488737 0.872431i $$-0.662542\pi$$
−0.488737 + 0.872431i $$0.662542\pi$$
$$854$$ 1.00470 0.0343800
$$855$$ 30.9665 1.05903
$$856$$ −5.41353 −0.185031
$$857$$ −39.8972 −1.36286 −0.681432 0.731882i $$-0.738642\pi$$
−0.681432 + 0.731882i $$0.738642\pi$$
$$858$$ 0.637062 0.0217490
$$859$$ 20.5576 0.701418 0.350709 0.936485i $$-0.385941\pi$$
0.350709 + 0.936485i $$0.385941\pi$$
$$860$$ −17.5172 −0.597332
$$861$$ −6.27882 −0.213982
$$862$$ −0.515515 −0.0175585
$$863$$ −9.66510 −0.329004 −0.164502 0.986377i $$-0.552602\pi$$
−0.164502 + 0.986377i $$0.552602\pi$$
$$864$$ 1.97196 0.0670875
$$865$$ −6.54825 −0.222647
$$866$$ −2.83646 −0.0963868
$$867$$ 16.8878 0.573541
$$868$$ −19.6090 −0.665574
$$869$$ 3.33020 0.112969
$$870$$ 0.124538 0.00422224
$$871$$ −0.530387 −0.0179715
$$872$$ 7.68388 0.260209
$$873$$ 0.0560785 0.00189797
$$874$$ 2.26943 0.0767646
$$875$$ −17.0226 −0.575467
$$876$$ 8.27334 0.279530
$$877$$ −25.7212 −0.868543 −0.434271 0.900782i $$-0.642994\pi$$
−0.434271 + 0.900782i $$0.642994\pi$$
$$878$$ 1.85511 0.0626069
$$879$$ 14.1667 0.477830
$$880$$ 14.5810 0.491525
$$881$$ −14.6316 −0.492950 −0.246475 0.969149i $$-0.579272\pi$$
−0.246475 + 0.969149i $$0.579272\pi$$
$$882$$ −0.167449 −0.00563831
$$883$$ 18.6877 0.628890 0.314445 0.949276i $$-0.398182\pi$$
0.314445 + 0.949276i $$0.398182\pi$$
$$884$$ −2.51252 −0.0845053
$$885$$ −14.2610 −0.479377
$$886$$ 2.43609 0.0818421
$$887$$ 38.6137 1.29652 0.648261 0.761418i $$-0.275497\pi$$
0.648261 + 0.761418i $$0.275497\pi$$
$$888$$ 2.97587 0.0998636
$$889$$ −14.6137 −0.490128
$$890$$ 6.29917 0.211149
$$891$$ 1.00000 0.0335013
$$892$$ 20.2509 0.678051
$$893$$ −98.8076 −3.30647
$$894$$ 1.37233 0.0458975
$$895$$ −45.6541 −1.52605
$$896$$ −5.17214 −0.172789
$$897$$ 6.33490 0.211516
$$898$$ −4.60276 −0.153596
$$899$$ −1.94392 −0.0648334
$$900$$ −18.6830 −0.622765
$$901$$ 2.66041 0.0886310
$$902$$ 1.05138 0.0350072
$$903$$ 2.33490 0.0777006
$$904$$ −0.884720 −0.0294254
$$905$$ 2.76160 0.0917987
$$906$$ −2.22274 −0.0738456
$$907$$ 49.0965 1.63022 0.815111 0.579304i $$-0.196676\pi$$
0.815111 + 0.579304i $$0.196676\pi$$
$$908$$ 0.770991 0.0255862
$$909$$ −18.8831 −0.626314
$$910$$ 2.42371 0.0803452
$$911$$ 29.3753 0.973248 0.486624 0.873612i $$-0.338228\pi$$
0.486624 + 0.873612i $$0.338228\pi$$
$$912$$ −31.1947 −1.03296
$$913$$ −13.9439 −0.461476
$$914$$ 1.33020 0.0439992
$$915$$ −22.8271 −0.754640
$$916$$ −15.6651 −0.517590
$$917$$ −20.5576 −0.678873
$$918$$ 0.0560785 0.00185087
$$919$$ −27.0047 −0.890803 −0.445401 0.895331i $$-0.646939\pi$$
−0.445401 + 0.895331i $$0.646939\pi$$
$$920$$ −4.21335 −0.138910
$$921$$ 12.5576 0.413788
$$922$$ −1.38471 −0.0456030
$$923$$ 17.7663 0.584785
$$924$$ −1.97196 −0.0648727
$$925$$ −42.3910 −1.39381
$$926$$ −1.63079 −0.0535912
$$927$$ −8.27882 −0.271912
$$928$$ −0.385496 −0.0126545
$$929$$ 57.8496 1.89798 0.948992 0.315299i $$-0.102105\pi$$
0.948992 + 0.315299i $$0.102105\pi$$
$$930$$ −6.33490 −0.207730
$$931$$ 8.13941 0.266758
$$932$$ −52.3706 −1.71546
$$933$$ −9.33959 −0.305765
$$934$$ 6.81469 0.222983
$$935$$ 1.27412 0.0416683
$$936$$ 2.53039 0.0827083
$$937$$ 25.2180 0.823838 0.411919 0.911221i $$-0.364859\pi$$
0.411919 + 0.911221i $$0.364859\pi$$
$$938$$ −0.0233441 −0.000762211 0
$$939$$ 2.99530 0.0977481
$$940$$ 91.0741 2.97051
$$941$$ −34.2788 −1.11746 −0.558729 0.829350i $$-0.688711\pi$$
−0.558729 + 0.829350i $$0.688711\pi$$
$$942$$ −2.83646 −0.0924169
$$943$$ 10.4549 0.340458
$$944$$ 14.3661 0.467575
$$945$$ 3.80451 0.123761
$$946$$ −0.390977 −0.0127118
$$947$$ −18.1573 −0.590032 −0.295016 0.955492i $$-0.595325\pi$$
−0.295016 + 0.955492i $$0.595325\pi$$
$$948$$ 6.56703 0.213287
$$949$$ 15.9618 0.518141
$$950$$ −12.9129 −0.418950
$$951$$ −9.93453 −0.322149
$$952$$ −0.222741 −0.00721909
$$953$$ −19.8045 −0.641531 −0.320766 0.947159i $$-0.603940\pi$$
−0.320766 + 0.947159i $$0.603940\pi$$
$$954$$ −1.33020 −0.0430669
$$955$$ 20.0655 0.649303
$$956$$ −21.2968 −0.688788
$$957$$ −0.195488 −0.00631924
$$958$$ −6.33490 −0.204671
$$959$$ 16.2227 0.523860
$$960$$ 27.9057 0.900653
$$961$$ 67.8816 2.18973
$$962$$ 2.85041 0.0919010
$$963$$ −8.13941 −0.262289
$$964$$ 23.8279 0.767444
$$965$$ −75.4502 −2.42883
$$966$$ 0.278820 0.00897088
$$967$$ −0.278820 −0.00896624 −0.00448312 0.999990i $$-0.501427\pi$$
−0.00448312 + 0.999990i $$0.501427\pi$$
$$968$$ 0.665102 0.0213772
$$969$$ −2.72588 −0.0875677
$$970$$ −0.0357255 −0.00114708
$$971$$ 25.3481 0.813458 0.406729 0.913549i $$-0.366669\pi$$
0.406729 + 0.913549i $$0.366669\pi$$
$$972$$ 1.97196 0.0632507
$$973$$ −22.5482 −0.722864
$$974$$ −6.79134 −0.217609
$$975$$ −36.0451 −1.15437
$$976$$ 22.9953 0.736062
$$977$$ 6.71648 0.214879 0.107440 0.994212i $$-0.465735\pi$$
0.107440 + 0.994212i $$0.465735\pi$$
$$978$$ −1.13862 −0.0364092
$$979$$ −9.88784 −0.316017
$$980$$ −7.50235 −0.239654
$$981$$ 11.5529 0.368857
$$982$$ −5.20409 −0.166069
$$983$$ 9.99061 0.318651 0.159325 0.987226i $$-0.449068\pi$$
0.159325 + 0.987226i $$0.449068\pi$$
$$984$$ 4.17605 0.133128
$$985$$ 10.1573 0.323638
$$986$$ −0.0109627 −0.000349123 0
$$987$$ −12.1394 −0.386402
$$988$$ −61.0647 −1.94273
$$989$$ −3.88784 −0.123626
$$990$$ −0.637062 −0.0202472
$$991$$ −26.7064 −0.848358 −0.424179 0.905578i $$-0.639437\pi$$
−0.424179 + 0.905578i $$0.639437\pi$$
$$992$$ 19.6090 0.622587
$$993$$ −22.5482 −0.715547
$$994$$ 0.781954 0.0248021
$$995$$ 51.5623 1.63464
$$996$$ −27.4969 −0.871272
$$997$$ 54.3333 1.72075 0.860377 0.509658i $$-0.170228\pi$$
0.860377 + 0.509658i $$0.170228\pi$$
$$998$$ −2.57159 −0.0814024
$$999$$ 4.47431 0.141561
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 231.2.a.d.1.2 3
3.2 odd 2 693.2.a.m.1.2 3
4.3 odd 2 3696.2.a.bp.1.3 3
5.4 even 2 5775.2.a.bw.1.2 3
7.6 odd 2 1617.2.a.s.1.2 3
11.10 odd 2 2541.2.a.bi.1.2 3
21.20 even 2 4851.2.a.bp.1.2 3
33.32 even 2 7623.2.a.cb.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 1.1 even 1 trivial
693.2.a.m.1.2 3 3.2 odd 2
1617.2.a.s.1.2 3 7.6 odd 2
2541.2.a.bi.1.2 3 11.10 odd 2
3696.2.a.bp.1.3 3 4.3 odd 2
4851.2.a.bp.1.2 3 21.20 even 2
5775.2.a.bw.1.2 3 5.4 even 2
7623.2.a.cb.1.2 3 33.32 even 2