# Properties

 Label 1859.2.a.h.1.1 Level $1859$ Weight $2$ Character 1859.1 Self dual yes Analytic conductor $14.844$ 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] = [1859,2,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1859.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1859.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: $$14.8441897358$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.756.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 2$$ x^3 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.26180$$ of defining polynomial Character $$\chi$$ $$=$$ 1859.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.26180 q^{3} -1.00000 q^{4} +2.11575 q^{5} -2.26180 q^{6} +3.37755 q^{7} -3.00000 q^{8} +2.11575 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.26180 q^{3} -1.00000 q^{4} +2.11575 q^{5} -2.26180 q^{6} +3.37755 q^{7} -3.00000 q^{8} +2.11575 q^{9} +2.11575 q^{10} +1.00000 q^{11} +2.26180 q^{12} +3.37755 q^{14} -4.78541 q^{15} -1.00000 q^{16} -2.14605 q^{17} +2.11575 q^{18} +3.14605 q^{19} -2.11575 q^{20} -7.63935 q^{21} +1.00000 q^{22} +4.52360 q^{23} +6.78541 q^{24} -0.523604 q^{25} +2.00000 q^{27} -3.37755 q^{28} +3.49330 q^{29} -4.78541 q^{30} -9.27871 q^{31} +5.00000 q^{32} -2.26180 q^{33} -2.14605 q^{34} +7.14605 q^{35} -2.11575 q^{36} -9.16296 q^{37} +3.14605 q^{38} -6.34725 q^{40} +10.0169 q^{41} -7.63935 q^{42} +2.52360 q^{43} -1.00000 q^{44} +4.47640 q^{45} +4.52360 q^{46} -1.96970 q^{47} +2.26180 q^{48} +4.40786 q^{49} -0.523604 q^{50} +4.85395 q^{51} +7.75510 q^{53} +2.00000 q^{54} +2.11575 q^{55} -10.1327 q^{56} -7.11575 q^{57} +3.49330 q^{58} -6.03030 q^{59} +4.78541 q^{60} +9.78541 q^{61} -9.27871 q^{62} +7.14605 q^{63} +7.00000 q^{64} -2.26180 q^{66} +4.03030 q^{67} +2.14605 q^{68} -10.2315 q^{69} +7.14605 q^{70} +6.00000 q^{71} -6.34725 q^{72} +10.8405 q^{73} -9.16296 q^{74} +1.18429 q^{75} -3.14605 q^{76} +3.37755 q^{77} +1.66966 q^{79} -2.11575 q^{80} -10.8709 q^{81} +10.0169 q^{82} +5.66966 q^{83} +7.63935 q^{84} -4.54051 q^{85} +2.52360 q^{86} -7.90116 q^{87} -3.00000 q^{88} +9.34725 q^{89} +4.47640 q^{90} -4.52360 q^{92} +20.9866 q^{93} -1.96970 q^{94} +6.65626 q^{95} -11.3090 q^{96} +11.6394 q^{97} +4.40786 q^{98} +2.11575 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 9 * q^8 + 3 * q^9 $$3 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 9 q^{8} + 3 q^{9} + 3 q^{10} + 3 q^{11} + 6 q^{15} - 3 q^{16} - 3 q^{17} + 3 q^{18} + 6 q^{19} - 3 q^{20} - 6 q^{21} + 3 q^{22} + 12 q^{25} + 6 q^{27} - 3 q^{29} + 6 q^{30} + 6 q^{31} + 15 q^{32} - 3 q^{34} + 18 q^{35} - 3 q^{36} + 3 q^{37} + 6 q^{38} - 9 q^{40} + 3 q^{41} - 6 q^{42} - 6 q^{43} - 3 q^{44} + 27 q^{45} - 6 q^{47} + 3 q^{49} + 12 q^{50} + 18 q^{51} + 3 q^{53} + 6 q^{54} + 3 q^{55} - 18 q^{57} - 3 q^{58} - 18 q^{59} - 6 q^{60} + 9 q^{61} + 6 q^{62} + 18 q^{63} + 21 q^{64} + 12 q^{67} + 3 q^{68} - 24 q^{69} + 18 q^{70} + 18 q^{71} - 9 q^{72} + 9 q^{73} + 3 q^{74} + 24 q^{75} - 6 q^{76} - 12 q^{79} - 3 q^{80} - 9 q^{81} + 3 q^{82} + 6 q^{84} + 27 q^{85} - 6 q^{86} - 9 q^{88} + 18 q^{89} + 27 q^{90} + 36 q^{93} - 6 q^{94} - 24 q^{95} + 18 q^{97} + 3 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 9 * q^8 + 3 * q^9 + 3 * q^10 + 3 * q^11 + 6 * q^15 - 3 * q^16 - 3 * q^17 + 3 * q^18 + 6 * q^19 - 3 * q^20 - 6 * q^21 + 3 * q^22 + 12 * q^25 + 6 * q^27 - 3 * q^29 + 6 * q^30 + 6 * q^31 + 15 * q^32 - 3 * q^34 + 18 * q^35 - 3 * q^36 + 3 * q^37 + 6 * q^38 - 9 * q^40 + 3 * q^41 - 6 * q^42 - 6 * q^43 - 3 * q^44 + 27 * q^45 - 6 * q^47 + 3 * q^49 + 12 * q^50 + 18 * q^51 + 3 * q^53 + 6 * q^54 + 3 * q^55 - 18 * q^57 - 3 * q^58 - 18 * q^59 - 6 * q^60 + 9 * q^61 + 6 * q^62 + 18 * q^63 + 21 * q^64 + 12 * q^67 + 3 * q^68 - 24 * q^69 + 18 * q^70 + 18 * q^71 - 9 * q^72 + 9 * q^73 + 3 * q^74 + 24 * q^75 - 6 * q^76 - 12 * q^79 - 3 * q^80 - 9 * q^81 + 3 * q^82 + 6 * q^84 + 27 * q^85 - 6 * q^86 - 9 * q^88 + 18 * q^89 + 27 * q^90 + 36 * q^93 - 6 * q^94 - 24 * q^95 + 18 * q^97 + 3 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ −2.26180 −1.30585 −0.652926 0.757422i $$-0.726459\pi$$
−0.652926 + 0.757422i $$0.726459\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 2.11575 0.946192 0.473096 0.881011i $$-0.343136\pi$$
0.473096 + 0.881011i $$0.343136\pi$$
$$6$$ −2.26180 −0.923377
$$7$$ 3.37755 1.27659 0.638297 0.769790i $$-0.279639\pi$$
0.638297 + 0.769790i $$0.279639\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 2.11575 0.705250
$$10$$ 2.11575 0.669059
$$11$$ 1.00000 0.301511
$$12$$ 2.26180 0.652926
$$13$$ 0 0
$$14$$ 3.37755 0.902689
$$15$$ −4.78541 −1.23559
$$16$$ −1.00000 −0.250000
$$17$$ −2.14605 −0.520494 −0.260247 0.965542i $$-0.583804\pi$$
−0.260247 + 0.965542i $$0.583804\pi$$
$$18$$ 2.11575 0.498687
$$19$$ 3.14605 0.721754 0.360877 0.932613i $$-0.382477\pi$$
0.360877 + 0.932613i $$0.382477\pi$$
$$20$$ −2.11575 −0.473096
$$21$$ −7.63935 −1.66704
$$22$$ 1.00000 0.213201
$$23$$ 4.52360 0.943237 0.471618 0.881803i $$-0.343670\pi$$
0.471618 + 0.881803i $$0.343670\pi$$
$$24$$ 6.78541 1.38507
$$25$$ −0.523604 −0.104721
$$26$$ 0 0
$$27$$ 2.00000 0.384900
$$28$$ −3.37755 −0.638297
$$29$$ 3.49330 0.648690 0.324345 0.945939i $$-0.394856\pi$$
0.324345 + 0.945939i $$0.394856\pi$$
$$30$$ −4.78541 −0.873692
$$31$$ −9.27871 −1.66651 −0.833253 0.552893i $$-0.813524\pi$$
−0.833253 + 0.552893i $$0.813524\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −2.26180 −0.393729
$$34$$ −2.14605 −0.368045
$$35$$ 7.14605 1.20790
$$36$$ −2.11575 −0.352625
$$37$$ −9.16296 −1.50638 −0.753191 0.657802i $$-0.771486\pi$$
−0.753191 + 0.657802i $$0.771486\pi$$
$$38$$ 3.14605 0.510357
$$39$$ 0 0
$$40$$ −6.34725 −1.00359
$$41$$ 10.0169 1.56438 0.782189 0.623041i $$-0.214103\pi$$
0.782189 + 0.623041i $$0.214103\pi$$
$$42$$ −7.63935 −1.17878
$$43$$ 2.52360 0.384846 0.192423 0.981312i $$-0.438365\pi$$
0.192423 + 0.981312i $$0.438365\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 4.47640 0.667302
$$46$$ 4.52360 0.666969
$$47$$ −1.96970 −0.287310 −0.143655 0.989628i $$-0.545886\pi$$
−0.143655 + 0.989628i $$0.545886\pi$$
$$48$$ 2.26180 0.326463
$$49$$ 4.40786 0.629694
$$50$$ −0.523604 −0.0740489
$$51$$ 4.85395 0.679689
$$52$$ 0 0
$$53$$ 7.75510 1.06525 0.532623 0.846353i $$-0.321206\pi$$
0.532623 + 0.846353i $$0.321206\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 2.11575 0.285288
$$56$$ −10.1327 −1.35403
$$57$$ −7.11575 −0.942504
$$58$$ 3.49330 0.458693
$$59$$ −6.03030 −0.785079 −0.392539 0.919735i $$-0.628403\pi$$
−0.392539 + 0.919735i $$0.628403\pi$$
$$60$$ 4.78541 0.617793
$$61$$ 9.78541 1.25289 0.626446 0.779464i $$-0.284509\pi$$
0.626446 + 0.779464i $$0.284509\pi$$
$$62$$ −9.27871 −1.17840
$$63$$ 7.14605 0.900318
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ −2.26180 −0.278409
$$67$$ 4.03030 0.492380 0.246190 0.969222i $$-0.420821\pi$$
0.246190 + 0.969222i $$0.420821\pi$$
$$68$$ 2.14605 0.260247
$$69$$ −10.2315 −1.23173
$$70$$ 7.14605 0.854117
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −6.34725 −0.748030
$$73$$ 10.8405 1.26879 0.634395 0.773009i $$-0.281249\pi$$
0.634395 + 0.773009i $$0.281249\pi$$
$$74$$ −9.16296 −1.06517
$$75$$ 1.18429 0.136750
$$76$$ −3.14605 −0.360877
$$77$$ 3.37755 0.384908
$$78$$ 0 0
$$79$$ 1.66966 0.187851 0.0939256 0.995579i $$-0.470058\pi$$
0.0939256 + 0.995579i $$0.470058\pi$$
$$80$$ −2.11575 −0.236548
$$81$$ −10.8709 −1.20787
$$82$$ 10.0169 1.10618
$$83$$ 5.66966 0.622326 0.311163 0.950357i $$-0.399282\pi$$
0.311163 + 0.950357i $$0.399282\pi$$
$$84$$ 7.63935 0.833522
$$85$$ −4.54051 −0.492487
$$86$$ 2.52360 0.272127
$$87$$ −7.90116 −0.847093
$$88$$ −3.00000 −0.319801
$$89$$ 9.34725 0.990806 0.495403 0.868663i $$-0.335020\pi$$
0.495403 + 0.868663i $$0.335020\pi$$
$$90$$ 4.47640 0.471854
$$91$$ 0 0
$$92$$ −4.52360 −0.471618
$$93$$ 20.9866 2.17621
$$94$$ −1.96970 −0.203159
$$95$$ 6.65626 0.682918
$$96$$ −11.3090 −1.15422
$$97$$ 11.6394 1.18180 0.590899 0.806746i $$-0.298773\pi$$
0.590899 + 0.806746i $$0.298773\pi$$
$$98$$ 4.40786 0.445261
$$99$$ 2.11575 0.212641
$$100$$ 0.523604 0.0523604
$$101$$ 0.901156 0.0896684 0.0448342 0.998994i $$-0.485724\pi$$
0.0448342 + 0.998994i $$0.485724\pi$$
$$102$$ 4.85395 0.480612
$$103$$ 13.0472 1.28558 0.642790 0.766043i $$-0.277777\pi$$
0.642790 + 0.766043i $$0.277777\pi$$
$$104$$ 0 0
$$105$$ −16.1630 −1.57734
$$106$$ 7.75510 0.753242
$$107$$ −13.6091 −1.31564 −0.657818 0.753177i $$-0.728521\pi$$
−0.657818 + 0.753177i $$0.728521\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ 18.6866 1.78985 0.894924 0.446218i $$-0.147230\pi$$
0.894924 + 0.446218i $$0.147230\pi$$
$$110$$ 2.11575 0.201729
$$111$$ 20.7248 1.96711
$$112$$ −3.37755 −0.319149
$$113$$ 12.6394 1.18901 0.594505 0.804092i $$-0.297348\pi$$
0.594505 + 0.804092i $$0.297348\pi$$
$$114$$ −7.11575 −0.666451
$$115$$ 9.57081 0.892483
$$116$$ −3.49330 −0.324345
$$117$$ 0 0
$$118$$ −6.03030 −0.555134
$$119$$ −7.24840 −0.664460
$$120$$ 14.3562 1.31054
$$121$$ 1.00000 0.0909091
$$122$$ 9.78541 0.885929
$$123$$ −22.6563 −2.04285
$$124$$ 9.27871 0.833253
$$125$$ −11.6866 −1.04528
$$126$$ 7.14605 0.636621
$$127$$ 11.5102 1.02137 0.510683 0.859769i $$-0.329393\pi$$
0.510683 + 0.859769i $$0.329393\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ −5.70789 −0.502552
$$130$$ 0 0
$$131$$ 13.6091 1.18903 0.594514 0.804085i $$-0.297344\pi$$
0.594514 + 0.804085i $$0.297344\pi$$
$$132$$ 2.26180 0.196865
$$133$$ 10.6260 0.921387
$$134$$ 4.03030 0.348165
$$135$$ 4.23150 0.364189
$$136$$ 6.43816 0.552068
$$137$$ −15.1024 −1.29028 −0.645140 0.764064i $$-0.723201\pi$$
−0.645140 + 0.764064i $$0.723201\pi$$
$$138$$ −10.2315 −0.870963
$$139$$ −15.2787 −1.29592 −0.647962 0.761673i $$-0.724378\pi$$
−0.647962 + 0.761673i $$0.724378\pi$$
$$140$$ −7.14605 −0.603952
$$141$$ 4.45506 0.375184
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ −2.11575 −0.176312
$$145$$ 7.39095 0.613785
$$146$$ 10.8405 0.897170
$$147$$ −9.96970 −0.822287
$$148$$ 9.16296 0.753191
$$149$$ 4.37755 0.358623 0.179312 0.983792i $$-0.442613\pi$$
0.179312 + 0.983792i $$0.442613\pi$$
$$150$$ 1.18429 0.0966969
$$151$$ 2.29211 0.186529 0.0932645 0.995641i $$-0.470270\pi$$
0.0932645 + 0.995641i $$0.470270\pi$$
$$152$$ −9.43816 −0.765536
$$153$$ −4.54051 −0.367078
$$154$$ 3.37755 0.272171
$$155$$ −19.6314 −1.57683
$$156$$ 0 0
$$157$$ −24.0472 −1.91918 −0.959588 0.281408i $$-0.909198\pi$$
−0.959588 + 0.281408i $$0.909198\pi$$
$$158$$ 1.66966 0.132831
$$159$$ −17.5405 −1.39105
$$160$$ 10.5787 0.836323
$$161$$ 15.2787 1.20413
$$162$$ −10.8709 −0.854095
$$163$$ −6.55391 −0.513342 −0.256671 0.966499i $$-0.582626\pi$$
−0.256671 + 0.966499i $$0.582626\pi$$
$$164$$ −10.0169 −0.782189
$$165$$ −4.78541 −0.372543
$$166$$ 5.66966 0.440051
$$167$$ −18.3259 −1.41810 −0.709051 0.705157i $$-0.750876\pi$$
−0.709051 + 0.705157i $$0.750876\pi$$
$$168$$ 22.9181 1.76817
$$169$$ 0 0
$$170$$ −4.54051 −0.348241
$$171$$ 6.65626 0.509017
$$172$$ −2.52360 −0.192423
$$173$$ 14.6866 1.11660 0.558299 0.829640i $$-0.311454\pi$$
0.558299 + 0.829640i $$0.311454\pi$$
$$174$$ −7.90116 −0.598985
$$175$$ −1.76850 −0.133686
$$176$$ −1.00000 −0.0753778
$$177$$ 13.6394 1.02520
$$178$$ 9.34725 0.700606
$$179$$ −6.49330 −0.485332 −0.242666 0.970110i $$-0.578022\pi$$
−0.242666 + 0.970110i $$0.578022\pi$$
$$180$$ −4.47640 −0.333651
$$181$$ −17.1630 −1.27571 −0.637856 0.770155i $$-0.720179\pi$$
−0.637856 + 0.770155i $$0.720179\pi$$
$$182$$ 0 0
$$183$$ −22.1327 −1.63609
$$184$$ −13.5708 −1.00045
$$185$$ −19.3865 −1.42533
$$186$$ 20.9866 1.53881
$$187$$ −2.14605 −0.156935
$$188$$ 1.96970 0.143655
$$189$$ 6.75510 0.491361
$$190$$ 6.65626 0.482896
$$191$$ 17.9697 1.30024 0.650121 0.759831i $$-0.274718\pi$$
0.650121 + 0.759831i $$0.274718\pi$$
$$192$$ −15.8326 −1.14262
$$193$$ −2.96970 −0.213763 −0.106882 0.994272i $$-0.534087\pi$$
−0.106882 + 0.994272i $$0.534087\pi$$
$$194$$ 11.6394 0.835657
$$195$$ 0 0
$$196$$ −4.40786 −0.314847
$$197$$ 12.2315 0.871458 0.435729 0.900078i $$-0.356491\pi$$
0.435729 + 0.900078i $$0.356491\pi$$
$$198$$ 2.11575 0.150360
$$199$$ −0.201195 −0.0142624 −0.00713118 0.999975i $$-0.502270\pi$$
−0.00713118 + 0.999975i $$0.502270\pi$$
$$200$$ 1.57081 0.111073
$$201$$ −9.11575 −0.642975
$$202$$ 0.901156 0.0634051
$$203$$ 11.7988 0.828114
$$204$$ −4.85395 −0.339844
$$205$$ 21.1933 1.48020
$$206$$ 13.0472 0.909042
$$207$$ 9.57081 0.665218
$$208$$ 0 0
$$209$$ 3.14605 0.217617
$$210$$ −16.1630 −1.11535
$$211$$ 11.6697 0.803372 0.401686 0.915777i $$-0.368424\pi$$
0.401686 + 0.915777i $$0.368424\pi$$
$$212$$ −7.75510 −0.532623
$$213$$ −13.5708 −0.929857
$$214$$ −13.6091 −0.930296
$$215$$ 5.33931 0.364138
$$216$$ −6.00000 −0.408248
$$217$$ −31.3393 −2.12745
$$218$$ 18.6866 1.26561
$$219$$ −24.5192 −1.65685
$$220$$ −2.11575 −0.142644
$$221$$ 0 0
$$222$$ 20.7248 1.39096
$$223$$ −7.50670 −0.502686 −0.251343 0.967898i $$-0.580872\pi$$
−0.251343 + 0.967898i $$0.580872\pi$$
$$224$$ 16.8878 1.12836
$$225$$ −1.10782 −0.0738544
$$226$$ 12.6394 0.840757
$$227$$ −3.47640 −0.230736 −0.115368 0.993323i $$-0.536805\pi$$
−0.115368 + 0.993323i $$0.536805\pi$$
$$228$$ 7.11575 0.471252
$$229$$ 13.4079 0.886016 0.443008 0.896518i $$-0.353911\pi$$
0.443008 + 0.896518i $$0.353911\pi$$
$$230$$ 9.57081 0.631081
$$231$$ −7.63935 −0.502633
$$232$$ −10.4799 −0.688039
$$233$$ 0.292106 0.0191365 0.00956824 0.999954i $$-0.496954\pi$$
0.00956824 + 0.999954i $$0.496954\pi$$
$$234$$ 0 0
$$235$$ −4.16738 −0.271850
$$236$$ 6.03030 0.392539
$$237$$ −3.77643 −0.245306
$$238$$ −7.24840 −0.469844
$$239$$ −21.8023 −1.41027 −0.705137 0.709071i $$-0.749115\pi$$
−0.705137 + 0.709071i $$0.749115\pi$$
$$240$$ 4.78541 0.308897
$$241$$ −2.54844 −0.164160 −0.0820798 0.996626i $$-0.526156\pi$$
−0.0820798 + 0.996626i $$0.526156\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 18.5877 1.19240
$$244$$ −9.78541 −0.626446
$$245$$ 9.32592 0.595811
$$246$$ −22.6563 −1.44451
$$247$$ 0 0
$$248$$ 27.8361 1.76760
$$249$$ −12.8236 −0.812665
$$250$$ −11.6866 −0.739123
$$251$$ −26.8192 −1.69281 −0.846407 0.532537i $$-0.821239\pi$$
−0.846407 + 0.532537i $$0.821239\pi$$
$$252$$ −7.14605 −0.450159
$$253$$ 4.52360 0.284397
$$254$$ 11.5102 0.722215
$$255$$ 10.2697 0.643116
$$256$$ −17.0000 −1.06250
$$257$$ 3.06854 0.191410 0.0957051 0.995410i $$-0.469489\pi$$
0.0957051 + 0.995410i $$0.469489\pi$$
$$258$$ −5.70789 −0.355358
$$259$$ −30.9484 −1.92304
$$260$$ 0 0
$$261$$ 7.39095 0.457488
$$262$$ 13.6091 0.840770
$$263$$ 2.29211 0.141337 0.0706686 0.997500i $$-0.477487\pi$$
0.0706686 + 0.997500i $$0.477487\pi$$
$$264$$ 6.78541 0.417613
$$265$$ 16.4079 1.00793
$$266$$ 10.6260 0.651519
$$267$$ −21.1416 −1.29385
$$268$$ −4.03030 −0.246190
$$269$$ 11.2449 0.685613 0.342807 0.939406i $$-0.388622\pi$$
0.342807 + 0.939406i $$0.388622\pi$$
$$270$$ 4.23150 0.257521
$$271$$ −4.29211 −0.260727 −0.130363 0.991466i $$-0.541614\pi$$
−0.130363 + 0.991466i $$0.541614\pi$$
$$272$$ 2.14605 0.130124
$$273$$ 0 0
$$274$$ −15.1024 −0.912366
$$275$$ −0.523604 −0.0315745
$$276$$ 10.2315 0.615864
$$277$$ −4.83262 −0.290364 −0.145182 0.989405i $$-0.546377\pi$$
−0.145182 + 0.989405i $$0.546377\pi$$
$$278$$ −15.2787 −0.916356
$$279$$ −19.6314 −1.17530
$$280$$ −21.4382 −1.28118
$$281$$ 3.78541 0.225818 0.112909 0.993605i $$-0.463983\pi$$
0.112909 + 0.993605i $$0.463983\pi$$
$$282$$ 4.45506 0.265295
$$283$$ −12.4248 −0.738575 −0.369288 0.929315i $$-0.620398\pi$$
−0.369288 + 0.929315i $$0.620398\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ −15.0551 −0.891790
$$286$$ 0 0
$$287$$ 33.8326 1.99708
$$288$$ 10.5787 0.623359
$$289$$ −12.3945 −0.729086
$$290$$ 7.39095 0.434012
$$291$$ −26.3259 −1.54325
$$292$$ −10.8405 −0.634395
$$293$$ 16.9012 0.987376 0.493688 0.869639i $$-0.335648\pi$$
0.493688 + 0.869639i $$0.335648\pi$$
$$294$$ −9.96970 −0.581445
$$295$$ −12.7586 −0.742835
$$296$$ 27.4889 1.59776
$$297$$ 2.00000 0.116052
$$298$$ 4.37755 0.253585
$$299$$ 0 0
$$300$$ −1.18429 −0.0683750
$$301$$ 8.52360 0.491292
$$302$$ 2.29211 0.131896
$$303$$ −2.03824 −0.117094
$$304$$ −3.14605 −0.180439
$$305$$ 20.7035 1.18548
$$306$$ −4.54051 −0.259564
$$307$$ −8.42476 −0.480826 −0.240413 0.970671i $$-0.577283\pi$$
−0.240413 + 0.970671i $$0.577283\pi$$
$$308$$ −3.37755 −0.192454
$$309$$ −29.5102 −1.67878
$$310$$ −19.6314 −1.11499
$$311$$ −28.5877 −1.62106 −0.810530 0.585697i $$-0.800821\pi$$
−0.810530 + 0.585697i $$0.800821\pi$$
$$312$$ 0 0
$$313$$ 22.3945 1.26581 0.632905 0.774230i $$-0.281862\pi$$
0.632905 + 0.774230i $$0.281862\pi$$
$$314$$ −24.0472 −1.35706
$$315$$ 15.1193 0.851874
$$316$$ −1.66966 −0.0939256
$$317$$ −26.2708 −1.47551 −0.737757 0.675067i $$-0.764115\pi$$
−0.737757 + 0.675067i $$0.764115\pi$$
$$318$$ −17.5405 −0.983623
$$319$$ 3.49330 0.195587
$$320$$ 14.8102 0.827918
$$321$$ 30.7810 1.71803
$$322$$ 15.2787 0.851449
$$323$$ −6.75160 −0.375669
$$324$$ 10.8709 0.603936
$$325$$ 0 0
$$326$$ −6.55391 −0.362987
$$327$$ −42.2653 −2.33728
$$328$$ −30.0507 −1.65927
$$329$$ −6.65275 −0.366778
$$330$$ −4.78541 −0.263428
$$331$$ −8.29211 −0.455775 −0.227888 0.973687i $$-0.573182\pi$$
−0.227888 + 0.973687i $$0.573182\pi$$
$$332$$ −5.66966 −0.311163
$$333$$ −19.3865 −1.06237
$$334$$ −18.3259 −1.00275
$$335$$ 8.52711 0.465886
$$336$$ 7.63935 0.416761
$$337$$ −0.0516349 −0.00281273 −0.00140637 0.999999i $$-0.500448\pi$$
−0.00140637 + 0.999999i $$0.500448\pi$$
$$338$$ 0 0
$$339$$ −28.5877 −1.55267
$$340$$ 4.54051 0.246244
$$341$$ −9.27871 −0.502470
$$342$$ 6.65626 0.359929
$$343$$ −8.75510 −0.472731
$$344$$ −7.57081 −0.408191
$$345$$ −21.6473 −1.16545
$$346$$ 14.6866 0.789555
$$347$$ −8.62245 −0.462877 −0.231439 0.972850i $$-0.574343\pi$$
−0.231439 + 0.972850i $$0.574343\pi$$
$$348$$ 7.90116 0.423546
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ −1.76850 −0.0945304
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ −0.476396 −0.0253560 −0.0126780 0.999920i $$-0.504036\pi$$
−0.0126780 + 0.999920i $$0.504036\pi$$
$$354$$ 13.6394 0.724923
$$355$$ 12.6945 0.673754
$$356$$ −9.34725 −0.495403
$$357$$ 16.3945 0.867687
$$358$$ −6.49330 −0.343182
$$359$$ −10.2315 −0.539998 −0.269999 0.962861i $$-0.587023\pi$$
−0.269999 + 0.962861i $$0.587023\pi$$
$$360$$ −13.4292 −0.707780
$$361$$ −9.10235 −0.479071
$$362$$ −17.1630 −0.902065
$$363$$ −2.26180 −0.118714
$$364$$ 0 0
$$365$$ 22.9359 1.20052
$$366$$ −22.1327 −1.15689
$$367$$ 21.3393 1.11390 0.556952 0.830545i $$-0.311971\pi$$
0.556952 + 0.830545i $$0.311971\pi$$
$$368$$ −4.52360 −0.235809
$$369$$ 21.1933 1.10328
$$370$$ −19.3865 −1.00786
$$371$$ 26.1933 1.35989
$$372$$ −20.9866 −1.08810
$$373$$ 21.2271 1.09910 0.549548 0.835462i $$-0.314800\pi$$
0.549548 + 0.835462i $$0.314800\pi$$
$$374$$ −2.14605 −0.110970
$$375$$ 26.4327 1.36498
$$376$$ 5.90909 0.304738
$$377$$ 0 0
$$378$$ 6.75510 0.347445
$$379$$ −1.04721 −0.0537915 −0.0268958 0.999638i $$-0.508562\pi$$
−0.0268958 + 0.999638i $$0.508562\pi$$
$$380$$ −6.65626 −0.341459
$$381$$ −26.0338 −1.33375
$$382$$ 17.9697 0.919410
$$383$$ −28.9866 −1.48115 −0.740573 0.671976i $$-0.765446\pi$$
−0.740573 + 0.671976i $$0.765446\pi$$
$$384$$ 6.78541 0.346266
$$385$$ 7.14605 0.364197
$$386$$ −2.96970 −0.151154
$$387$$ 5.33931 0.271413
$$388$$ −11.6394 −0.590899
$$389$$ 15.5653 0.789195 0.394597 0.918854i $$-0.370884\pi$$
0.394597 + 0.918854i $$0.370884\pi$$
$$390$$ 0 0
$$391$$ −9.70789 −0.490949
$$392$$ −13.2236 −0.667891
$$393$$ −30.7810 −1.55270
$$394$$ 12.2315 0.616214
$$395$$ 3.53258 0.177743
$$396$$ −2.11575 −0.106320
$$397$$ 18.9787 0.952512 0.476256 0.879307i $$-0.341994\pi$$
0.476256 + 0.879307i $$0.341994\pi$$
$$398$$ −0.201195 −0.0100850
$$399$$ −24.0338 −1.20320
$$400$$ 0.523604 0.0261802
$$401$$ −9.52360 −0.475586 −0.237793 0.971316i $$-0.576424\pi$$
−0.237793 + 0.971316i $$0.576424\pi$$
$$402$$ −9.11575 −0.454652
$$403$$ 0 0
$$404$$ −0.901156 −0.0448342
$$405$$ −23.0000 −1.14288
$$406$$ 11.7988 0.585565
$$407$$ −9.16296 −0.454191
$$408$$ −14.5618 −0.720919
$$409$$ 9.91455 0.490243 0.245122 0.969492i $$-0.421172\pi$$
0.245122 + 0.969492i $$0.421172\pi$$
$$410$$ 21.1933 1.04666
$$411$$ 34.1585 1.68492
$$412$$ −13.0472 −0.642790
$$413$$ −20.3677 −1.00223
$$414$$ 9.57081 0.470380
$$415$$ 11.9956 0.588840
$$416$$ 0 0
$$417$$ 34.5574 1.69228
$$418$$ 3.14605 0.153878
$$419$$ 7.01340 0.342627 0.171313 0.985217i $$-0.445199\pi$$
0.171313 + 0.985217i $$0.445199\pi$$
$$420$$ 16.1630 0.788672
$$421$$ 33.9787 1.65602 0.828009 0.560714i $$-0.189473\pi$$
0.828009 + 0.560714i $$0.189473\pi$$
$$422$$ 11.6697 0.568070
$$423$$ −4.16738 −0.202625
$$424$$ −23.2653 −1.12986
$$425$$ 1.12368 0.0545066
$$426$$ −13.5708 −0.657508
$$427$$ 33.0507 1.59944
$$428$$ 13.6091 0.657818
$$429$$ 0 0
$$430$$ 5.33931 0.257485
$$431$$ 7.86735 0.378957 0.189478 0.981885i $$-0.439320\pi$$
0.189478 + 0.981885i $$0.439320\pi$$
$$432$$ −2.00000 −0.0962250
$$433$$ −13.6528 −0.656109 −0.328055 0.944659i $$-0.606393\pi$$
−0.328055 + 0.944659i $$0.606393\pi$$
$$434$$ −31.3393 −1.50434
$$435$$ −16.7169 −0.801512
$$436$$ −18.6866 −0.894924
$$437$$ 14.2315 0.680785
$$438$$ −24.5192 −1.17157
$$439$$ −8.33034 −0.397586 −0.198793 0.980042i $$-0.563702\pi$$
−0.198793 + 0.980042i $$0.563702\pi$$
$$440$$ −6.34725 −0.302593
$$441$$ 9.32592 0.444091
$$442$$ 0 0
$$443$$ −25.2484 −1.19959 −0.599794 0.800154i $$-0.704751\pi$$
−0.599794 + 0.800154i $$0.704751\pi$$
$$444$$ −20.7248 −0.983555
$$445$$ 19.7764 0.937493
$$446$$ −7.50670 −0.355452
$$447$$ −9.90116 −0.468309
$$448$$ 23.6429 1.11702
$$449$$ −37.2708 −1.75892 −0.879458 0.475976i $$-0.842095\pi$$
−0.879458 + 0.475976i $$0.842095\pi$$
$$450$$ −1.10782 −0.0522229
$$451$$ 10.0169 0.471678
$$452$$ −12.6394 −0.594505
$$453$$ −5.18429 −0.243579
$$454$$ −3.47640 −0.163155
$$455$$ 0 0
$$456$$ 21.3472 0.999676
$$457$$ −3.09091 −0.144587 −0.0722933 0.997383i $$-0.523032\pi$$
−0.0722933 + 0.997383i $$0.523032\pi$$
$$458$$ 13.4079 0.626508
$$459$$ −4.29211 −0.200338
$$460$$ −9.57081 −0.446241
$$461$$ −22.4799 −1.04699 −0.523497 0.852028i $$-0.675373\pi$$
−0.523497 + 0.852028i $$0.675373\pi$$
$$462$$ −7.63935 −0.355415
$$463$$ −29.0507 −1.35010 −0.675051 0.737771i $$-0.735878\pi$$
−0.675051 + 0.737771i $$0.735878\pi$$
$$464$$ −3.49330 −0.162172
$$465$$ 44.4024 2.05911
$$466$$ 0.292106 0.0135315
$$467$$ −7.47990 −0.346129 −0.173064 0.984911i $$-0.555367\pi$$
−0.173064 + 0.984911i $$0.555367\pi$$
$$468$$ 0 0
$$469$$ 13.6126 0.628570
$$470$$ −4.16738 −0.192227
$$471$$ 54.3900 2.50616
$$472$$ 18.0909 0.832702
$$473$$ 2.52360 0.116035
$$474$$ −3.77643 −0.173457
$$475$$ −1.64729 −0.0755827
$$476$$ 7.24840 0.332230
$$477$$ 16.4079 0.751264
$$478$$ −21.8023 −0.997215
$$479$$ 8.62245 0.393970 0.196985 0.980407i $$-0.436885\pi$$
0.196985 + 0.980407i $$0.436885\pi$$
$$480$$ −23.9270 −1.09211
$$481$$ 0 0
$$482$$ −2.54844 −0.116078
$$483$$ −34.5574 −1.57242
$$484$$ −1.00000 −0.0454545
$$485$$ 24.6260 1.11821
$$486$$ 18.5877 0.843156
$$487$$ 8.09091 0.366634 0.183317 0.983054i $$-0.441317\pi$$
0.183317 + 0.983054i $$0.441317\pi$$
$$488$$ −29.3562 −1.32889
$$489$$ 14.8236 0.670348
$$490$$ 9.32592 0.421302
$$491$$ 4.62245 0.208608 0.104304 0.994545i $$-0.466738\pi$$
0.104304 + 0.994545i $$0.466738\pi$$
$$492$$ 22.6563 1.02142
$$493$$ −7.49681 −0.337639
$$494$$ 0 0
$$495$$ 4.47640 0.201199
$$496$$ 9.27871 0.416626
$$497$$ 20.2653 0.909023
$$498$$ −12.8236 −0.574641
$$499$$ 39.5137 1.76888 0.884438 0.466657i $$-0.154542\pi$$
0.884438 + 0.466657i $$0.154542\pi$$
$$500$$ 11.6866 0.522639
$$501$$ 41.4496 1.85183
$$502$$ −26.8192 −1.19700
$$503$$ −37.7035 −1.68111 −0.840557 0.541723i $$-0.817772\pi$$
−0.840557 + 0.541723i $$0.817772\pi$$
$$504$$ −21.4382 −0.954931
$$505$$ 1.90662 0.0848435
$$506$$ 4.52360 0.201099
$$507$$ 0 0
$$508$$ −11.5102 −0.510683
$$509$$ −7.72129 −0.342240 −0.171120 0.985250i $$-0.554739\pi$$
−0.171120 + 0.985250i $$0.554739\pi$$
$$510$$ 10.2697 0.454752
$$511$$ 36.6145 1.61973
$$512$$ −11.0000 −0.486136
$$513$$ 6.29211 0.277803
$$514$$ 3.06854 0.135348
$$515$$ 27.6046 1.21641
$$516$$ 5.70789 0.251276
$$517$$ −1.96970 −0.0866272
$$518$$ −30.9484 −1.35979
$$519$$ −33.2181 −1.45811
$$520$$ 0 0
$$521$$ 5.48979 0.240512 0.120256 0.992743i $$-0.461628\pi$$
0.120256 + 0.992743i $$0.461628\pi$$
$$522$$ 7.39095 0.323493
$$523$$ 41.2137 1.80215 0.901074 0.433665i $$-0.142780\pi$$
0.901074 + 0.433665i $$0.142780\pi$$
$$524$$ −13.6091 −0.594514
$$525$$ 4.00000 0.174574
$$526$$ 2.29211 0.0999406
$$527$$ 19.9126 0.867406
$$528$$ 2.26180 0.0984323
$$529$$ −2.53700 −0.110304
$$530$$ 16.4079 0.712712
$$531$$ −12.7586 −0.553677
$$532$$ −10.6260 −0.460694
$$533$$ 0 0
$$534$$ −21.1416 −0.914888
$$535$$ −28.7933 −1.24484
$$536$$ −12.0909 −0.522248
$$537$$ 14.6866 0.633772
$$538$$ 11.2449 0.484802
$$539$$ 4.40786 0.189860
$$540$$ −4.23150 −0.182095
$$541$$ 31.2539 1.34371 0.671854 0.740683i $$-0.265498\pi$$
0.671854 + 0.740683i $$0.265498\pi$$
$$542$$ −4.29211 −0.184362
$$543$$ 38.8192 1.66589
$$544$$ −10.7303 −0.460056
$$545$$ 39.5361 1.69354
$$546$$ 0 0
$$547$$ 2.40239 0.102719 0.0513594 0.998680i $$-0.483645\pi$$
0.0513594 + 0.998680i $$0.483645\pi$$
$$548$$ 15.1024 0.645140
$$549$$ 20.7035 0.883602
$$550$$ −0.523604 −0.0223266
$$551$$ 10.9901 0.468194
$$552$$ 30.6945 1.30644
$$553$$ 5.63935 0.239810
$$554$$ −4.83262 −0.205318
$$555$$ 43.8485 1.86126
$$556$$ 15.2787 0.647962
$$557$$ 2.44609 0.103644 0.0518221 0.998656i $$-0.483497\pi$$
0.0518221 + 0.998656i $$0.483497\pi$$
$$558$$ −19.6314 −0.831064
$$559$$ 0 0
$$560$$ −7.14605 −0.301976
$$561$$ 4.85395 0.204934
$$562$$ 3.78541 0.159678
$$563$$ 38.4586 1.62084 0.810418 0.585852i $$-0.199240\pi$$
0.810418 + 0.585852i $$0.199240\pi$$
$$564$$ −4.45506 −0.187592
$$565$$ 26.7417 1.12503
$$566$$ −12.4248 −0.522252
$$567$$ −36.7169 −1.54196
$$568$$ −18.0000 −0.755263
$$569$$ 39.2519 1.64553 0.822763 0.568385i $$-0.192431\pi$$
0.822763 + 0.568385i $$0.192431\pi$$
$$570$$ −15.0551 −0.630591
$$571$$ −39.3349 −1.64611 −0.823057 0.567959i $$-0.807733\pi$$
−0.823057 + 0.567959i $$0.807733\pi$$
$$572$$ 0 0
$$573$$ −40.6439 −1.69792
$$574$$ 33.8326 1.41215
$$575$$ −2.36858 −0.0987766
$$576$$ 14.8102 0.617094
$$577$$ −24.0890 −1.00284 −0.501418 0.865205i $$-0.667188\pi$$
−0.501418 + 0.865205i $$0.667188\pi$$
$$578$$ −12.3945 −0.515541
$$579$$ 6.71687 0.279143
$$580$$ −7.39095 −0.306892
$$581$$ 19.1496 0.794458
$$582$$ −26.3259 −1.09124
$$583$$ 7.75510 0.321184
$$584$$ −32.5216 −1.34576
$$585$$ 0 0
$$586$$ 16.9012 0.698180
$$587$$ 24.4665 1.00984 0.504920 0.863166i $$-0.331522\pi$$
0.504920 + 0.863166i $$0.331522\pi$$
$$588$$ 9.96970 0.411143
$$589$$ −29.1913 −1.20281
$$590$$ −12.7586 −0.525264
$$591$$ −27.6652 −1.13800
$$592$$ 9.16296 0.376595
$$593$$ 33.6215 1.38067 0.690335 0.723490i $$-0.257463\pi$$
0.690335 + 0.723490i $$0.257463\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ −15.3358 −0.628707
$$596$$ −4.37755 −0.179312
$$597$$ 0.455064 0.0186245
$$598$$ 0 0
$$599$$ 3.01340 0.123124 0.0615621 0.998103i $$-0.480392\pi$$
0.0615621 + 0.998103i $$0.480392\pi$$
$$600$$ −3.55287 −0.145045
$$601$$ −18.5743 −0.757662 −0.378831 0.925466i $$-0.623674\pi$$
−0.378831 + 0.925466i $$0.623674\pi$$
$$602$$ 8.52360 0.347396
$$603$$ 8.52711 0.347251
$$604$$ −2.29211 −0.0932645
$$605$$ 2.11575 0.0860174
$$606$$ −2.03824 −0.0827977
$$607$$ −19.2137 −0.779859 −0.389930 0.920845i $$-0.627501\pi$$
−0.389930 + 0.920845i $$0.627501\pi$$
$$608$$ 15.7303 0.637946
$$609$$ −26.6866 −1.08139
$$610$$ 20.7035 0.838259
$$611$$ 0 0
$$612$$ 4.54051 0.183539
$$613$$ 18.9350 0.764776 0.382388 0.924002i $$-0.375102\pi$$
0.382388 + 0.924002i $$0.375102\pi$$
$$614$$ −8.42476 −0.339996
$$615$$ −47.9350 −1.93292
$$616$$ −10.1327 −0.408256
$$617$$ −1.72129 −0.0692966 −0.0346483 0.999400i $$-0.511031\pi$$
−0.0346483 + 0.999400i $$0.511031\pi$$
$$618$$ −29.5102 −1.18707
$$619$$ 16.3562 0.657412 0.328706 0.944432i $$-0.393387\pi$$
0.328706 + 0.944432i $$0.393387\pi$$
$$620$$ 19.6314 0.788417
$$621$$ 9.04721 0.363052
$$622$$ −28.5877 −1.14626
$$623$$ 31.5708 1.26486
$$624$$ 0 0
$$625$$ −22.1078 −0.884313
$$626$$ 22.3945 0.895063
$$627$$ −7.11575 −0.284176
$$628$$ 24.0472 0.959588
$$629$$ 19.6642 0.784063
$$630$$ 15.1193 0.602366
$$631$$ −21.0775 −0.839083 −0.419541 0.907736i $$-0.637809\pi$$
−0.419541 + 0.907736i $$0.637809\pi$$
$$632$$ −5.00897 −0.199246
$$633$$ −26.3945 −1.04909
$$634$$ −26.2708 −1.04335
$$635$$ 24.3527 0.966408
$$636$$ 17.5405 0.695526
$$637$$ 0 0
$$638$$ 3.49330 0.138301
$$639$$ 12.6945 0.502187
$$640$$ −6.34725 −0.250897
$$641$$ −4.53700 −0.179201 −0.0896004 0.995978i $$-0.528559\pi$$
−0.0896004 + 0.995978i $$0.528559\pi$$
$$642$$ 30.7810 1.21483
$$643$$ −0.983094 −0.0387695 −0.0193847 0.999812i $$-0.506171\pi$$
−0.0193847 + 0.999812i $$0.506171\pi$$
$$644$$ −15.2787 −0.602065
$$645$$ −12.0765 −0.475511
$$646$$ −6.75160 −0.265638
$$647$$ −46.6856 −1.83540 −0.917701 0.397272i $$-0.869957\pi$$
−0.917701 + 0.397272i $$0.869957\pi$$
$$648$$ 32.6126 1.28114
$$649$$ −6.03030 −0.236710
$$650$$ 0 0
$$651$$ 70.8833 2.77814
$$652$$ 6.55391 0.256671
$$653$$ −25.8709 −1.01240 −0.506202 0.862415i $$-0.668951\pi$$
−0.506202 + 0.862415i $$0.668951\pi$$
$$654$$ −42.2653 −1.65270
$$655$$ 28.7933 1.12505
$$656$$ −10.0169 −0.391094
$$657$$ 22.9359 0.894814
$$658$$ −6.65275 −0.259351
$$659$$ 17.4988 0.681655 0.340828 0.940126i $$-0.389293\pi$$
0.340828 + 0.940126i $$0.389293\pi$$
$$660$$ 4.78541 0.186272
$$661$$ 21.6528 0.842194 0.421097 0.907016i $$-0.361645\pi$$
0.421097 + 0.907016i $$0.361645\pi$$
$$662$$ −8.29211 −0.322282
$$663$$ 0 0
$$664$$ −17.0090 −0.660076
$$665$$ 22.4819 0.871809
$$666$$ −19.3865 −0.751213
$$667$$ 15.8023 0.611868
$$668$$ 18.3259 0.709051
$$669$$ 16.9787 0.656433
$$670$$ 8.52711 0.329431
$$671$$ 9.78541 0.377761
$$672$$ −38.1968 −1.47347
$$673$$ −23.8798 −0.920500 −0.460250 0.887789i $$-0.652240\pi$$
−0.460250 + 0.887789i $$0.652240\pi$$
$$674$$ −0.0516349 −0.00198890
$$675$$ −1.04721 −0.0403071
$$676$$ 0 0
$$677$$ −12.6945 −0.487889 −0.243945 0.969789i $$-0.578441\pi$$
−0.243945 + 0.969789i $$0.578441\pi$$
$$678$$ −28.5877 −1.09790
$$679$$ 39.3125 1.50868
$$680$$ 13.6215 0.522362
$$681$$ 7.86292 0.301308
$$682$$ −9.27871 −0.355300
$$683$$ −37.8699 −1.44905 −0.724526 0.689247i $$-0.757941\pi$$
−0.724526 + 0.689247i $$0.757941\pi$$
$$684$$ −6.65626 −0.254508
$$685$$ −31.9528 −1.22085
$$686$$ −8.75510 −0.334271
$$687$$ −30.3259 −1.15701
$$688$$ −2.52360 −0.0962115
$$689$$ 0 0
$$690$$ −21.6473 −0.824098
$$691$$ 25.7417 0.979261 0.489630 0.871930i $$-0.337132\pi$$
0.489630 + 0.871930i $$0.337132\pi$$
$$692$$ −14.6866 −0.558299
$$693$$ 7.14605 0.271456
$$694$$ −8.62245 −0.327304
$$695$$ −32.3259 −1.22619
$$696$$ 23.7035 0.898478
$$697$$ −21.4968 −0.814250
$$698$$ 30.0000 1.13552
$$699$$ −0.660685 −0.0249894
$$700$$ 1.76850 0.0668431
$$701$$ −33.5023 −1.26536 −0.632682 0.774412i $$-0.718046\pi$$
−0.632682 + 0.774412i $$0.718046\pi$$
$$702$$ 0 0
$$703$$ −28.8272 −1.08724
$$704$$ 7.00000 0.263822
$$705$$ 9.42580 0.354996
$$706$$ −0.476396 −0.0179294
$$707$$ 3.04370 0.114470
$$708$$ −13.6394 −0.512598
$$709$$ −30.9235 −1.16136 −0.580679 0.814133i $$-0.697213\pi$$
−0.580679 + 0.814133i $$0.697213\pi$$
$$710$$ 12.6945 0.476416
$$711$$ 3.53258 0.132482
$$712$$ −28.0417 −1.05091
$$713$$ −41.9732 −1.57191
$$714$$ 16.3945 0.613547
$$715$$ 0 0
$$716$$ 6.49330 0.242666
$$717$$ 49.3125 1.84161
$$718$$ −10.2315 −0.381836
$$719$$ 9.21810 0.343777 0.171889 0.985116i $$-0.445013\pi$$
0.171889 + 0.985116i $$0.445013\pi$$
$$720$$ −4.47640 −0.166825
$$721$$ 44.0676 1.64116
$$722$$ −9.10235 −0.338754
$$723$$ 5.76408 0.214368
$$724$$ 17.1630 0.637856
$$725$$ −1.82911 −0.0679314
$$726$$ −2.26180 −0.0839434
$$727$$ 33.5102 1.24282 0.621412 0.783484i $$-0.286559\pi$$
0.621412 + 0.783484i $$0.286559\pi$$
$$728$$ 0 0
$$729$$ −9.42919 −0.349229
$$730$$ 22.9359 0.848895
$$731$$ −5.41579 −0.200310
$$732$$ 22.1327 0.818046
$$733$$ −1.29561 −0.0478546 −0.0239273 0.999714i $$-0.507617\pi$$
−0.0239273 + 0.999714i $$0.507617\pi$$
$$734$$ 21.3393 0.787648
$$735$$ −21.0934 −0.778041
$$736$$ 22.6180 0.833711
$$737$$ 4.03030 0.148458
$$738$$ 21.1933 0.780135
$$739$$ 28.8495 1.06125 0.530623 0.847608i $$-0.321958\pi$$
0.530623 + 0.847608i $$0.321958\pi$$
$$740$$ 19.3865 0.712663
$$741$$ 0 0
$$742$$ 26.1933 0.961585
$$743$$ −31.7373 −1.16433 −0.582164 0.813071i $$-0.697794\pi$$
−0.582164 + 0.813071i $$0.697794\pi$$
$$744$$ −62.9598 −2.30822
$$745$$ 9.26180 0.339326
$$746$$ 21.2271 0.777178
$$747$$ 11.9956 0.438895
$$748$$ 2.14605 0.0784675
$$749$$ −45.9653 −1.67953
$$750$$ 26.4327 0.965186
$$751$$ −36.1282 −1.31834 −0.659169 0.751995i $$-0.729092\pi$$
−0.659169 + 0.751995i $$0.729092\pi$$
$$752$$ 1.96970 0.0718274
$$753$$ 60.6598 2.21056
$$754$$ 0 0
$$755$$ 4.84952 0.176492
$$756$$ −6.75510 −0.245681
$$757$$ −22.0527 −0.801518 −0.400759 0.916183i $$-0.631254\pi$$
−0.400759 + 0.916183i $$0.631254\pi$$
$$758$$ −1.04721 −0.0380363
$$759$$ −10.2315 −0.371380
$$760$$ −19.9688 −0.724344
$$761$$ 40.1620 1.45587 0.727936 0.685645i $$-0.240480\pi$$
0.727936 + 0.685645i $$0.240480\pi$$
$$762$$ −26.0338 −0.943105
$$763$$ 63.1148 2.28491
$$764$$ −17.9697 −0.650121
$$765$$ −9.60658 −0.347327
$$766$$ −28.9866 −1.04733
$$767$$ 0 0
$$768$$ 38.4506 1.38747
$$769$$ −24.9598 −0.900074 −0.450037 0.893010i $$-0.648589\pi$$
−0.450037 + 0.893010i $$0.648589\pi$$
$$770$$ 7.14605 0.257526
$$771$$ −6.94043 −0.249954
$$772$$ 2.96970 0.106882
$$773$$ 5.46846 0.196687 0.0983435 0.995153i $$-0.468646\pi$$
0.0983435 + 0.995153i $$0.468646\pi$$
$$774$$ 5.33931 0.191918
$$775$$ 4.85837 0.174518
$$776$$ −34.9181 −1.25349
$$777$$ 69.9991 2.51120
$$778$$ 15.5653 0.558045
$$779$$ 31.5137 1.12910
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ −9.70789 −0.347154
$$783$$ 6.98660 0.249681
$$784$$ −4.40786 −0.157423
$$785$$ −50.8779 −1.81591
$$786$$ −30.7810 −1.09792
$$787$$ 37.5484 1.33846 0.669229 0.743056i $$-0.266625\pi$$
0.669229 + 0.743056i $$0.266625\pi$$
$$788$$ −12.2315 −0.435729
$$789$$ −5.18429 −0.184566
$$790$$ 3.53258 0.125683
$$791$$ 42.6901 1.51788
$$792$$ −6.34725 −0.225540
$$793$$ 0 0
$$794$$ 18.9787 0.673528
$$795$$ −37.1113 −1.31620
$$796$$ 0.201195 0.00713118
$$797$$ 7.76057 0.274893 0.137447 0.990509i $$-0.456110\pi$$
0.137447 + 0.990509i $$0.456110\pi$$
$$798$$ −24.0338 −0.850788
$$799$$ 4.22707 0.149543
$$800$$ −2.61802 −0.0925611
$$801$$ 19.7764 0.698766
$$802$$ −9.52360 −0.336290
$$803$$ 10.8405 0.382555
$$804$$ 9.11575 0.321488
$$805$$ 32.3259 1.13934
$$806$$ 0 0
$$807$$ −25.4337 −0.895310
$$808$$ −2.70347 −0.0951077
$$809$$ −6.80674 −0.239312 −0.119656 0.992815i $$-0.538179\pi$$
−0.119656 + 0.992815i $$0.538179\pi$$
$$810$$ −23.0000 −0.808138
$$811$$ −37.0810 −1.30209 −0.651045 0.759039i $$-0.725669\pi$$
−0.651045 + 0.759039i $$0.725669\pi$$
$$812$$ −11.7988 −0.414057
$$813$$ 9.70789 0.340471
$$814$$ −9.16296 −0.321162
$$815$$ −13.8664 −0.485720
$$816$$ −4.85395 −0.169922
$$817$$ 7.93939 0.277764
$$818$$ 9.91455 0.346654
$$819$$ 0 0
$$820$$ −21.1933 −0.740101
$$821$$ −0.170892 −0.00596417 −0.00298208 0.999996i $$-0.500949\pi$$
−0.00298208 + 0.999996i $$0.500949\pi$$
$$822$$ 34.1585 1.19142
$$823$$ 14.9260 0.520287 0.260144 0.965570i $$-0.416230\pi$$
0.260144 + 0.965570i $$0.416230\pi$$
$$824$$ −39.1416 −1.36356
$$825$$ 1.18429 0.0412317
$$826$$ −20.3677 −0.708682
$$827$$ 10.0114 0.348132 0.174066 0.984734i $$-0.444309\pi$$
0.174066 + 0.984734i $$0.444309\pi$$
$$828$$ −9.57081 −0.332609
$$829$$ −3.59214 −0.124760 −0.0623802 0.998052i $$-0.519869\pi$$
−0.0623802 + 0.998052i $$0.519869\pi$$
$$830$$ 11.9956 0.416372
$$831$$ 10.9304 0.379172
$$832$$ 0 0
$$833$$ −9.45949 −0.327752
$$834$$ 34.5574 1.19663
$$835$$ −38.7730 −1.34180
$$836$$ −3.14605 −0.108809
$$837$$ −18.5574 −0.641438
$$838$$ 7.01340 0.242274
$$839$$ −27.2216 −0.939794 −0.469897 0.882721i $$-0.655709\pi$$
−0.469897 + 0.882721i $$0.655709\pi$$
$$840$$ 48.4889 1.67303
$$841$$ −16.7968 −0.579202
$$842$$ 33.9787 1.17098
$$843$$ −8.56184 −0.294885
$$844$$ −11.6697 −0.401686
$$845$$ 0 0
$$846$$ −4.16738 −0.143278
$$847$$ 3.37755 0.116054
$$848$$ −7.75510 −0.266311
$$849$$ 28.1024 0.964470
$$850$$ 1.12368 0.0385420
$$851$$ −41.4496 −1.42087
$$852$$ 13.5708 0.464928
$$853$$ 26.7114 0.914581 0.457290 0.889317i $$-0.348820\pi$$
0.457290 + 0.889317i $$0.348820\pi$$
$$854$$ 33.0507 1.13097
$$855$$ 14.0830 0.481628
$$856$$ 40.8272 1.39544
$$857$$ −26.6955 −0.911902 −0.455951 0.890005i $$-0.650701\pi$$
−0.455951 + 0.890005i $$0.650701\pi$$
$$858$$ 0 0
$$859$$ −21.3855 −0.729663 −0.364832 0.931073i $$-0.618873\pi$$
−0.364832 + 0.931073i $$0.618873\pi$$
$$860$$ −5.33931 −0.182069
$$861$$ −76.5227 −2.60789
$$862$$ 7.86735 0.267963
$$863$$ 0.694496 0.0236409 0.0118205 0.999930i $$-0.496237\pi$$
0.0118205 + 0.999930i $$0.496237\pi$$
$$864$$ 10.0000 0.340207
$$865$$ 31.0731 1.05652
$$866$$ −13.6528 −0.463939
$$867$$ 28.0338 0.952078
$$868$$ 31.3393 1.06373
$$869$$ 1.66966 0.0566392
$$870$$ −16.7169 −0.566755
$$871$$ 0 0
$$872$$ −56.0597 −1.89842
$$873$$ 24.6260 0.833462
$$874$$ 14.2315 0.481388
$$875$$ −39.4720 −1.33440
$$876$$ 24.5192 0.828426
$$877$$ −17.2609 −0.582859 −0.291429 0.956592i $$-0.594131\pi$$
−0.291429 + 0.956592i $$0.594131\pi$$
$$878$$ −8.33034 −0.281135
$$879$$ −38.2271 −1.28937
$$880$$ −2.11575 −0.0713219
$$881$$ 27.7889 0.936232 0.468116 0.883667i $$-0.344933\pi$$
0.468116 + 0.883667i $$0.344933\pi$$
$$882$$ 9.32592 0.314020
$$883$$ 13.1505 0.442549 0.221274 0.975212i $$-0.428978\pi$$
0.221274 + 0.975212i $$0.428978\pi$$
$$884$$ 0 0
$$885$$ 28.8575 0.970033
$$886$$ −25.2484 −0.848237
$$887$$ −53.0204 −1.78025 −0.890126 0.455715i $$-0.849384\pi$$
−0.890126 + 0.455715i $$0.849384\pi$$
$$888$$ −62.1744 −2.08644
$$889$$ 38.8763 1.30387
$$890$$ 19.7764 0.662908
$$891$$ −10.8709 −0.364187
$$892$$ 7.50670 0.251343
$$893$$ −6.19677 −0.207367
$$894$$ −9.90116 −0.331144
$$895$$ −13.7382 −0.459217
$$896$$ −10.1327 −0.338508
$$897$$ 0 0
$$898$$ −37.2708 −1.24374
$$899$$ −32.4133 −1.08104
$$900$$ 1.10782 0.0369272
$$901$$ −16.6429 −0.554454
$$902$$ 10.0169 0.333526
$$903$$ −19.2787 −0.641555
$$904$$ −37.9181 −1.26114
$$905$$ −36.3125 −1.20707
$$906$$ −5.18429 −0.172236
$$907$$ −54.8192 −1.82024 −0.910121 0.414342i $$-0.864012\pi$$
−0.910121 + 0.414342i $$0.864012\pi$$
$$908$$ 3.47640 0.115368
$$909$$ 1.90662 0.0632386
$$910$$ 0 0
$$911$$ −25.4799 −0.844187 −0.422093 0.906552i $$-0.638705\pi$$
−0.422093 + 0.906552i $$0.638705\pi$$
$$912$$ 7.11575 0.235626
$$913$$ 5.66966 0.187638
$$914$$ −3.09091 −0.102238
$$915$$ −46.8272 −1.54806
$$916$$ −13.4079 −0.443008
$$917$$ 45.9653 1.51791
$$918$$ −4.29211 −0.141661
$$919$$ −14.2156 −0.468930 −0.234465 0.972125i $$-0.575334\pi$$
−0.234465 + 0.972125i $$0.575334\pi$$
$$920$$ −28.7124 −0.946621
$$921$$ 19.0551 0.627888
$$922$$ −22.4799 −0.740336
$$923$$ 0 0
$$924$$ 7.63935 0.251316
$$925$$ 4.79777 0.157750
$$926$$ −29.0507 −0.954666
$$927$$ 27.6046 0.906655
$$928$$ 17.4665 0.573366
$$929$$ −4.60462 −0.151073 −0.0755364 0.997143i $$-0.524067\pi$$
−0.0755364 + 0.997143i $$0.524067\pi$$
$$930$$ 44.4024 1.45601
$$931$$ 13.8673 0.454484
$$932$$ −0.292106 −0.00956824
$$933$$ 64.6598 2.11687
$$934$$ −7.47990 −0.244750
$$935$$ −4.54051 −0.148491
$$936$$ 0 0
$$937$$ −54.9002 −1.79351 −0.896756 0.442525i $$-0.854083\pi$$
−0.896756 + 0.442525i $$0.854083\pi$$
$$938$$ 13.6126 0.444466
$$939$$ −50.6518 −1.65296
$$940$$ 4.16738 0.135925
$$941$$ −0.292106 −0.00952237 −0.00476119 0.999989i $$-0.501516\pi$$
−0.00476119 + 0.999989i $$0.501516\pi$$
$$942$$ 54.3900 1.77212
$$943$$ 45.3125 1.47558
$$944$$ 6.03030 0.196270
$$945$$ 14.2921 0.464922
$$946$$ 2.52360 0.0820495
$$947$$ −4.46300 −0.145028 −0.0725140 0.997367i $$-0.523102\pi$$
−0.0725140 + 0.997367i $$0.523102\pi$$
$$948$$ 3.77643 0.122653
$$949$$ 0 0
$$950$$ −1.64729 −0.0534451
$$951$$ 59.4193 1.92680
$$952$$ 21.7452 0.704766
$$953$$ −40.4809 −1.31131 −0.655653 0.755062i $$-0.727607\pi$$
−0.655653 + 0.755062i $$0.727607\pi$$
$$954$$ 16.4079 0.531224
$$955$$ 38.0194 1.23028
$$956$$ 21.8023 0.705137
$$957$$ −7.90116 −0.255408
$$958$$ 8.62245 0.278579
$$959$$ −51.0090 −1.64717
$$960$$ −33.4978 −1.08114
$$961$$ 55.0944 1.77724
$$962$$ 0 0
$$963$$ −28.7933 −0.927852
$$964$$ 2.54844 0.0820798
$$965$$ −6.28313 −0.202261
$$966$$ −34.5574 −1.11187
$$967$$ −1.00897 −0.0324464 −0.0162232 0.999868i $$-0.505164\pi$$
−0.0162232 + 0.999868i $$0.505164\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 15.2708 0.490568
$$970$$ 24.6260 0.790692
$$971$$ 25.8664 0.830093 0.415047 0.909800i $$-0.363765\pi$$
0.415047 + 0.909800i $$0.363765\pi$$
$$972$$ −18.5877 −0.596201
$$973$$ −51.6046 −1.65437
$$974$$ 8.09091 0.259249
$$975$$ 0 0
$$976$$ −9.78541 −0.313223
$$977$$ −43.3677 −1.38745 −0.693727 0.720238i $$-0.744033\pi$$
−0.693727 + 0.720238i $$0.744033\pi$$
$$978$$ 14.8236 0.474008
$$979$$ 9.34725 0.298739
$$980$$ −9.32592 −0.297905
$$981$$ 39.5361 1.26229
$$982$$ 4.62245 0.147508
$$983$$ −17.0472 −0.543722 −0.271861 0.962337i $$-0.587639\pi$$
−0.271861 + 0.962337i $$0.587639\pi$$
$$984$$ 67.9688 2.16677
$$985$$ 25.8788 0.824567
$$986$$ −7.49681 −0.238747
$$987$$ 15.0472 0.478958
$$988$$ 0 0
$$989$$ 11.4158 0.363001
$$990$$ 4.47640 0.142269
$$991$$ −40.0979 −1.27375 −0.636876 0.770966i $$-0.719774\pi$$
−0.636876 + 0.770966i $$0.719774\pi$$
$$992$$ −46.3935 −1.47300
$$993$$ 18.7551 0.595175
$$994$$ 20.2653 0.642777
$$995$$ −0.425679 −0.0134949
$$996$$ 12.8236 0.406333
$$997$$ −54.0428 −1.71155 −0.855776 0.517346i $$-0.826920\pi$$
−0.855776 + 0.517346i $$0.826920\pi$$
$$998$$ 39.5137 1.25078
$$999$$ −18.3259 −0.579806
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 1859.2.a.h.1.1 3
13.3 even 3 143.2.e.a.100.3 6
13.9 even 3 143.2.e.a.133.3 yes 6
13.12 even 2 1859.2.a.e.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
143.2.e.a.100.3 6 13.3 even 3
143.2.e.a.133.3 yes 6 13.9 even 3
1859.2.a.e.1.1 3 13.12 even 2
1859.2.a.h.1.1 3 1.1 even 1 trivial