# Properties

 Label 1859.2.a.f.1.3 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: 3.3.361.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x + 7$$ x^3 - x^2 - 6*x + 7 Coefficient ring: $$\Z[a_1, a_2]$$ 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 1.3 Root $$-2.50702$$ 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)

 $$f(q)$$ $$=$$ $$q+2.50702 q^{2} +2.28514 q^{3} +4.28514 q^{4} -1.22188 q^{5} +5.72889 q^{6} +0.778124 q^{7} +5.72889 q^{8} +2.22188 q^{9} +O(q^{10})$$ $$q+2.50702 q^{2} +2.28514 q^{3} +4.28514 q^{4} -1.22188 q^{5} +5.72889 q^{6} +0.778124 q^{7} +5.72889 q^{8} +2.22188 q^{9} -3.06327 q^{10} +1.00000 q^{11} +9.79216 q^{12} +1.95077 q^{14} -2.79216 q^{15} +5.79216 q^{16} -4.28514 q^{17} +5.57028 q^{18} +8.29918 q^{19} -5.23591 q^{20} +1.77812 q^{21} +2.50702 q^{22} -2.72889 q^{23} +13.0913 q^{24} -3.50702 q^{25} -1.77812 q^{27} +3.33437 q^{28} +7.36245 q^{29} -7.00000 q^{30} -1.44375 q^{31} +3.06327 q^{32} +2.28514 q^{33} -10.7429 q^{34} -0.950771 q^{35} +9.52106 q^{36} -0.792161 q^{37} +20.8062 q^{38} -7.00000 q^{40} +9.74293 q^{41} +4.45779 q^{42} -9.67967 q^{43} +4.28514 q^{44} -2.71486 q^{45} -6.84139 q^{46} -3.38049 q^{47} +13.2359 q^{48} -6.39452 q^{49} -8.79216 q^{50} -9.79216 q^{51} +1.79216 q^{53} -4.45779 q^{54} -1.22188 q^{55} +4.45779 q^{56} +18.9648 q^{57} +18.4578 q^{58} +5.20784 q^{59} -11.9648 q^{60} -14.5211 q^{61} -3.61951 q^{62} +1.72889 q^{63} -3.90466 q^{64} +5.72889 q^{66} +6.06327 q^{67} -18.3624 q^{68} -6.23591 q^{69} -2.38360 q^{70} -6.50702 q^{71} +12.7289 q^{72} -2.00000 q^{73} -1.98596 q^{74} -8.01404 q^{75} +35.5632 q^{76} +0.778124 q^{77} +13.5843 q^{79} -7.07730 q^{80} -10.7289 q^{81} +24.4257 q^{82} -13.3624 q^{83} +7.61951 q^{84} +5.23591 q^{85} -24.2671 q^{86} +16.8242 q^{87} +5.72889 q^{88} -4.77812 q^{89} -6.80620 q^{90} -11.6937 q^{92} -3.29918 q^{93} -8.47494 q^{94} -10.1406 q^{95} +7.00000 q^{96} -4.14057 q^{97} -16.0312 q^{98} +2.22188 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + q^{3} + 7 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10})$$ 3 * q - q^2 + q^3 + 7 * q^4 - q^5 + 6 * q^6 + 5 * q^7 + 6 * q^8 + 4 * q^9 $$3 q - q^{2} + q^{3} + 7 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} + 6 q^{8} + 4 q^{9} - 6 q^{10} + 3 q^{11} + 15 q^{12} - 8 q^{14} + 6 q^{15} + 3 q^{16} - 7 q^{17} + 5 q^{18} + 2 q^{19} + 4 q^{20} + 8 q^{21} - q^{22} + 3 q^{23} + 2 q^{24} - 2 q^{25} - 8 q^{27} + 18 q^{28} - 4 q^{29} - 21 q^{30} + q^{31} + 6 q^{32} + q^{33} - 4 q^{34} + 11 q^{35} + 3 q^{36} + 12 q^{37} + 31 q^{38} - 21 q^{40} + q^{41} - 9 q^{42} - 4 q^{43} + 7 q^{44} - 14 q^{45} - 20 q^{46} - 8 q^{47} + 20 q^{48} - 12 q^{50} - 15 q^{51} - 9 q^{53} + 9 q^{54} - q^{55} - 9 q^{56} + 26 q^{57} + 33 q^{58} + 30 q^{59} - 5 q^{60} - 18 q^{61} - 13 q^{62} - 6 q^{63} - 8 q^{64} + 6 q^{66} + 15 q^{67} - 29 q^{68} + q^{69} - 29 q^{70} - 11 q^{71} + 27 q^{72} - 6 q^{73} - 23 q^{74} - 7 q^{75} + 30 q^{76} + 5 q^{77} + 12 q^{79} - q^{80} - 21 q^{81} + 44 q^{82} - 14 q^{83} + 25 q^{84} - 4 q^{85} - 43 q^{86} + 43 q^{87} + 6 q^{88} - 17 q^{89} + 11 q^{90} + 7 q^{92} + 13 q^{93} - 10 q^{94} - 7 q^{95} + 21 q^{96} + 11 q^{97} - 38 q^{98} + 4 q^{99}+O(q^{100})$$ 3 * q - q^2 + q^3 + 7 * q^4 - q^5 + 6 * q^6 + 5 * q^7 + 6 * q^8 + 4 * q^9 - 6 * q^10 + 3 * q^11 + 15 * q^12 - 8 * q^14 + 6 * q^15 + 3 * q^16 - 7 * q^17 + 5 * q^18 + 2 * q^19 + 4 * q^20 + 8 * q^21 - q^22 + 3 * q^23 + 2 * q^24 - 2 * q^25 - 8 * q^27 + 18 * q^28 - 4 * q^29 - 21 * q^30 + q^31 + 6 * q^32 + q^33 - 4 * q^34 + 11 * q^35 + 3 * q^36 + 12 * q^37 + 31 * q^38 - 21 * q^40 + q^41 - 9 * q^42 - 4 * q^43 + 7 * q^44 - 14 * q^45 - 20 * q^46 - 8 * q^47 + 20 * q^48 - 12 * q^50 - 15 * q^51 - 9 * q^53 + 9 * q^54 - q^55 - 9 * q^56 + 26 * q^57 + 33 * q^58 + 30 * q^59 - 5 * q^60 - 18 * q^61 - 13 * q^62 - 6 * q^63 - 8 * q^64 + 6 * q^66 + 15 * q^67 - 29 * q^68 + q^69 - 29 * q^70 - 11 * q^71 + 27 * q^72 - 6 * q^73 - 23 * q^74 - 7 * q^75 + 30 * q^76 + 5 * q^77 + 12 * q^79 - q^80 - 21 * q^81 + 44 * q^82 - 14 * q^83 + 25 * q^84 - 4 * q^85 - 43 * q^86 + 43 * q^87 + 6 * q^88 - 17 * q^89 + 11 * q^90 + 7 * q^92 + 13 * q^93 - 10 * q^94 - 7 * q^95 + 21 * q^96 + 11 * q^97 - 38 * q^98 + 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.50702 1.77273 0.886365 0.462987i $$-0.153222\pi$$
0.886365 + 0.462987i $$0.153222\pi$$
$$3$$ 2.28514 1.31933 0.659664 0.751561i $$-0.270699\pi$$
0.659664 + 0.751561i $$0.270699\pi$$
$$4$$ 4.28514 2.14257
$$5$$ −1.22188 −0.546440 −0.273220 0.961952i $$-0.588089\pi$$
−0.273220 + 0.961952i $$0.588089\pi$$
$$6$$ 5.72889 2.33881
$$7$$ 0.778124 0.294103 0.147052 0.989129i $$-0.453022\pi$$
0.147052 + 0.989129i $$0.453022\pi$$
$$8$$ 5.72889 2.02547
$$9$$ 2.22188 0.740625
$$10$$ −3.06327 −0.968690
$$11$$ 1.00000 0.301511
$$12$$ 9.79216 2.82675
$$13$$ 0 0
$$14$$ 1.95077 0.521365
$$15$$ −2.79216 −0.720933
$$16$$ 5.79216 1.44804
$$17$$ −4.28514 −1.03930 −0.519650 0.854379i $$-0.673938\pi$$
−0.519650 + 0.854379i $$0.673938\pi$$
$$18$$ 5.57028 1.31293
$$19$$ 8.29918 1.90396 0.951981 0.306156i $$-0.0990431\pi$$
0.951981 + 0.306156i $$0.0990431\pi$$
$$20$$ −5.23591 −1.17079
$$21$$ 1.77812 0.388018
$$22$$ 2.50702 0.534498
$$23$$ −2.72889 −0.569014 −0.284507 0.958674i $$-0.591830\pi$$
−0.284507 + 0.958674i $$0.591830\pi$$
$$24$$ 13.0913 2.67226
$$25$$ −3.50702 −0.701404
$$26$$ 0 0
$$27$$ −1.77812 −0.342200
$$28$$ 3.33437 0.630137
$$29$$ 7.36245 1.36717 0.683586 0.729870i $$-0.260419\pi$$
0.683586 + 0.729870i $$0.260419\pi$$
$$30$$ −7.00000 −1.27802
$$31$$ −1.44375 −0.259306 −0.129653 0.991559i $$-0.541386\pi$$
−0.129653 + 0.991559i $$0.541386\pi$$
$$32$$ 3.06327 0.541514
$$33$$ 2.28514 0.397792
$$34$$ −10.7429 −1.84240
$$35$$ −0.950771 −0.160710
$$36$$ 9.52106 1.58684
$$37$$ −0.792161 −0.130230 −0.0651152 0.997878i $$-0.520742\pi$$
−0.0651152 + 0.997878i $$0.520742\pi$$
$$38$$ 20.8062 3.37521
$$39$$ 0 0
$$40$$ −7.00000 −1.10680
$$41$$ 9.74293 1.52159 0.760795 0.648992i $$-0.224809\pi$$
0.760795 + 0.648992i $$0.224809\pi$$
$$42$$ 4.45779 0.687852
$$43$$ −9.67967 −1.47614 −0.738068 0.674727i $$-0.764261\pi$$
−0.738068 + 0.674727i $$0.764261\pi$$
$$44$$ 4.28514 0.646010
$$45$$ −2.71486 −0.404707
$$46$$ −6.84139 −1.00871
$$47$$ −3.38049 −0.493095 −0.246547 0.969131i $$-0.579296\pi$$
−0.246547 + 0.969131i $$0.579296\pi$$
$$48$$ 13.2359 1.91044
$$49$$ −6.39452 −0.913503
$$50$$ −8.79216 −1.24340
$$51$$ −9.79216 −1.37118
$$52$$ 0 0
$$53$$ 1.79216 0.246172 0.123086 0.992396i $$-0.460721\pi$$
0.123086 + 0.992396i $$0.460721\pi$$
$$54$$ −4.45779 −0.606628
$$55$$ −1.22188 −0.164758
$$56$$ 4.45779 0.595697
$$57$$ 18.9648 2.51195
$$58$$ 18.4578 2.42363
$$59$$ 5.20784 0.678003 0.339001 0.940786i $$-0.389911\pi$$
0.339001 + 0.940786i $$0.389911\pi$$
$$60$$ −11.9648 −1.54465
$$61$$ −14.5211 −1.85923 −0.929615 0.368531i $$-0.879861\pi$$
−0.929615 + 0.368531i $$0.879861\pi$$
$$62$$ −3.61951 −0.459679
$$63$$ 1.72889 0.217820
$$64$$ −3.90466 −0.488082
$$65$$ 0 0
$$66$$ 5.72889 0.705178
$$67$$ 6.06327 0.740746 0.370373 0.928883i $$-0.379230\pi$$
0.370373 + 0.928883i $$0.379230\pi$$
$$68$$ −18.3624 −2.22677
$$69$$ −6.23591 −0.750716
$$70$$ −2.38360 −0.284895
$$71$$ −6.50702 −0.772241 −0.386121 0.922448i $$-0.626185\pi$$
−0.386121 + 0.922448i $$0.626185\pi$$
$$72$$ 12.7289 1.50011
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −1.98596 −0.230863
$$75$$ −8.01404 −0.925381
$$76$$ 35.5632 4.07938
$$77$$ 0.778124 0.0886754
$$78$$ 0 0
$$79$$ 13.5843 1.52836 0.764178 0.645006i $$-0.223145\pi$$
0.764178 + 0.645006i $$0.223145\pi$$
$$80$$ −7.07730 −0.791267
$$81$$ −10.7289 −1.19210
$$82$$ 24.4257 2.69737
$$83$$ −13.3624 −1.46672 −0.733360 0.679841i $$-0.762049\pi$$
−0.733360 + 0.679841i $$0.762049\pi$$
$$84$$ 7.61951 0.831357
$$85$$ 5.23591 0.567915
$$86$$ −24.2671 −2.61679
$$87$$ 16.8242 1.80375
$$88$$ 5.72889 0.610702
$$89$$ −4.77812 −0.506480 −0.253240 0.967403i $$-0.581496\pi$$
−0.253240 + 0.967403i $$0.581496\pi$$
$$90$$ −6.80620 −0.717436
$$91$$ 0 0
$$92$$ −11.6937 −1.21915
$$93$$ −3.29918 −0.342109
$$94$$ −8.47494 −0.874123
$$95$$ −10.1406 −1.04040
$$96$$ 7.00000 0.714435
$$97$$ −4.14057 −0.420411 −0.210206 0.977657i $$-0.567413\pi$$
−0.210206 + 0.977657i $$0.567413\pi$$
$$98$$ −16.0312 −1.61939
$$99$$ 2.22188 0.223307
$$100$$ −15.0281 −1.50281
$$101$$ 4.82424 0.480030 0.240015 0.970769i $$-0.422848\pi$$
0.240015 + 0.970769i $$0.422848\pi$$
$$102$$ −24.5491 −2.43073
$$103$$ 9.76008 0.961690 0.480845 0.876806i $$-0.340330\pi$$
0.480845 + 0.876806i $$0.340330\pi$$
$$104$$ 0 0
$$105$$ −2.17265 −0.212029
$$106$$ 4.49298 0.436397
$$107$$ 2.36245 0.228386 0.114193 0.993459i $$-0.463572\pi$$
0.114193 + 0.993459i $$0.463572\pi$$
$$108$$ −7.61951 −0.733188
$$109$$ −2.86946 −0.274845 −0.137422 0.990513i $$-0.543882\pi$$
−0.137422 + 0.990513i $$0.543882\pi$$
$$110$$ −3.06327 −0.292071
$$111$$ −1.81020 −0.171817
$$112$$ 4.50702 0.425873
$$113$$ −5.60236 −0.527026 −0.263513 0.964656i $$-0.584881\pi$$
−0.263513 + 0.964656i $$0.584881\pi$$
$$114$$ 47.5451 4.45301
$$115$$ 3.33437 0.310932
$$116$$ 31.5491 2.92926
$$117$$ 0 0
$$118$$ 13.0561 1.20192
$$119$$ −3.33437 −0.305661
$$120$$ −15.9960 −1.46023
$$121$$ 1.00000 0.0909091
$$122$$ −36.4046 −3.29591
$$123$$ 22.2640 2.00748
$$124$$ −6.18668 −0.555581
$$125$$ 10.3945 0.929714
$$126$$ 4.33437 0.386137
$$127$$ 7.23591 0.642083 0.321042 0.947065i $$-0.395967\pi$$
0.321042 + 0.947065i $$0.395967\pi$$
$$128$$ −15.9156 −1.40675
$$129$$ −22.1194 −1.94751
$$130$$ 0 0
$$131$$ −10.3484 −0.904145 −0.452072 0.891981i $$-0.649315\pi$$
−0.452072 + 0.891981i $$0.649315\pi$$
$$132$$ 9.79216 0.852298
$$133$$ 6.45779 0.559961
$$134$$ 15.2007 1.31314
$$135$$ 2.17265 0.186992
$$136$$ −24.5491 −2.10507
$$137$$ −4.93673 −0.421774 −0.210887 0.977510i $$-0.567635\pi$$
−0.210887 + 0.977510i $$0.567635\pi$$
$$138$$ −15.6336 −1.33082
$$139$$ 16.4718 1.39712 0.698561 0.715550i $$-0.253824\pi$$
0.698561 + 0.715550i $$0.253824\pi$$
$$140$$ −4.07419 −0.344332
$$141$$ −7.72489 −0.650553
$$142$$ −16.3132 −1.36897
$$143$$ 0 0
$$144$$ 12.8695 1.07246
$$145$$ −8.99600 −0.747077
$$146$$ −5.01404 −0.414965
$$147$$ −14.6124 −1.20521
$$148$$ −3.39452 −0.279028
$$149$$ −1.90466 −0.156036 −0.0780178 0.996952i $$-0.524859\pi$$
−0.0780178 + 0.996952i $$0.524859\pi$$
$$150$$ −20.0913 −1.64045
$$151$$ 18.4086 1.49807 0.749034 0.662532i $$-0.230518\pi$$
0.749034 + 0.662532i $$0.230518\pi$$
$$152$$ 47.5451 3.85642
$$153$$ −9.52106 −0.769732
$$154$$ 1.95077 0.157198
$$155$$ 1.76409 0.141695
$$156$$ 0 0
$$157$$ −12.1054 −0.966114 −0.483057 0.875589i $$-0.660474\pi$$
−0.483057 + 0.875589i $$0.660474\pi$$
$$158$$ 34.0561 2.70936
$$159$$ 4.09534 0.324782
$$160$$ −3.74293 −0.295905
$$161$$ −2.12342 −0.167349
$$162$$ −26.8975 −2.11327
$$163$$ 1.91869 0.150284 0.0751418 0.997173i $$-0.476059\pi$$
0.0751418 + 0.997173i $$0.476059\pi$$
$$164$$ 41.7499 3.26012
$$165$$ −2.79216 −0.217369
$$166$$ −33.4999 −2.60010
$$167$$ −7.06015 −0.546331 −0.273165 0.961967i $$-0.588071\pi$$
−0.273165 + 0.961967i $$0.588071\pi$$
$$168$$ 10.1867 0.785920
$$169$$ 0 0
$$170$$ 13.1265 1.00676
$$171$$ 18.4397 1.41012
$$172$$ −41.4787 −3.16272
$$173$$ 1.64759 0.125264 0.0626319 0.998037i $$-0.480051\pi$$
0.0626319 + 0.998037i $$0.480051\pi$$
$$174$$ 42.1787 3.19756
$$175$$ −2.72889 −0.206285
$$176$$ 5.79216 0.436601
$$177$$ 11.9007 0.894508
$$178$$ −11.9788 −0.897852
$$179$$ 7.33437 0.548197 0.274098 0.961702i $$-0.411621\pi$$
0.274098 + 0.961702i $$0.411621\pi$$
$$180$$ −11.6336 −0.867114
$$181$$ 11.7750 0.875230 0.437615 0.899163i $$-0.355823\pi$$
0.437615 + 0.899163i $$0.355823\pi$$
$$182$$ 0 0
$$183$$ −33.1827 −2.45293
$$184$$ −15.6336 −1.15252
$$185$$ 0.967923 0.0711631
$$186$$ −8.27111 −0.606467
$$187$$ −4.28514 −0.313361
$$188$$ −14.4859 −1.05649
$$189$$ −1.38360 −0.100642
$$190$$ −25.4226 −1.84435
$$191$$ 14.7429 1.06676 0.533381 0.845875i $$-0.320921\pi$$
0.533381 + 0.845875i $$0.320921\pi$$
$$192$$ −8.92270 −0.643940
$$193$$ −8.90154 −0.640747 −0.320374 0.947291i $$-0.603808\pi$$
−0.320374 + 0.947291i $$0.603808\pi$$
$$194$$ −10.3805 −0.745275
$$195$$ 0 0
$$196$$ −27.4014 −1.95725
$$197$$ −6.11250 −0.435497 −0.217749 0.976005i $$-0.569871\pi$$
−0.217749 + 0.976005i $$0.569871\pi$$
$$198$$ 5.57028 0.395863
$$199$$ 9.34529 0.662470 0.331235 0.943548i $$-0.392535\pi$$
0.331235 + 0.943548i $$0.392535\pi$$
$$200$$ −20.0913 −1.42067
$$201$$ 13.8554 0.977287
$$202$$ 12.0945 0.850963
$$203$$ 5.72889 0.402090
$$204$$ −41.9608 −2.93784
$$205$$ −11.9047 −0.831457
$$206$$ 24.4687 1.70482
$$207$$ −6.06327 −0.421426
$$208$$ 0 0
$$209$$ 8.29918 0.574066
$$210$$ −5.44687 −0.375870
$$211$$ −5.18668 −0.357066 −0.178533 0.983934i $$-0.557135\pi$$
−0.178533 + 0.983934i $$0.557135\pi$$
$$212$$ 7.67967 0.527442
$$213$$ −14.8695 −1.01884
$$214$$ 5.92270 0.404867
$$215$$ 11.8274 0.806619
$$216$$ −10.1867 −0.693116
$$217$$ −1.12342 −0.0762626
$$218$$ −7.19380 −0.487226
$$219$$ −4.57028 −0.308831
$$220$$ −5.23591 −0.353005
$$221$$ 0 0
$$222$$ −4.53821 −0.304585
$$223$$ 16.8343 1.12731 0.563653 0.826012i $$-0.309395\pi$$
0.563653 + 0.826012i $$0.309395\pi$$
$$224$$ 2.38360 0.159261
$$225$$ −7.79216 −0.519477
$$226$$ −14.0452 −0.934275
$$227$$ −17.6164 −1.16924 −0.584621 0.811307i $$-0.698757\pi$$
−0.584621 + 0.811307i $$0.698757\pi$$
$$228$$ 81.2669 5.38203
$$229$$ 26.8554 1.77466 0.887328 0.461138i $$-0.152559\pi$$
0.887328 + 0.461138i $$0.152559\pi$$
$$230$$ 8.35933 0.551198
$$231$$ 1.77812 0.116992
$$232$$ 42.1787 2.76917
$$233$$ −27.1406 −1.77804 −0.889019 0.457870i $$-0.848612\pi$$
−0.889019 + 0.457870i $$0.848612\pi$$
$$234$$ 0 0
$$235$$ 4.13054 0.269446
$$236$$ 22.3163 1.45267
$$237$$ 31.0421 2.01640
$$238$$ −8.35933 −0.541855
$$239$$ 0.746047 0.0482577 0.0241289 0.999709i $$-0.492319\pi$$
0.0241289 + 0.999709i $$0.492319\pi$$
$$240$$ −16.1726 −1.04394
$$241$$ 16.1726 1.04177 0.520886 0.853626i $$-0.325602\pi$$
0.520886 + 0.853626i $$0.325602\pi$$
$$242$$ 2.50702 0.161157
$$243$$ −19.1827 −1.23057
$$244$$ −62.2248 −3.98353
$$245$$ 7.81332 0.499174
$$246$$ 55.8162 3.55871
$$247$$ 0 0
$$248$$ −8.27111 −0.525216
$$249$$ −30.5351 −1.93508
$$250$$ 26.0593 1.64813
$$251$$ −11.0281 −0.696086 −0.348043 0.937479i $$-0.613154\pi$$
−0.348043 + 0.937479i $$0.613154\pi$$
$$252$$ 7.40856 0.466695
$$253$$ −2.72889 −0.171564
$$254$$ 18.1406 1.13824
$$255$$ 11.9648 0.749265
$$256$$ −32.0913 −2.00571
$$257$$ 18.8523 1.17597 0.587987 0.808870i $$-0.299920\pi$$
0.587987 + 0.808870i $$0.299920\pi$$
$$258$$ −55.4538 −3.45240
$$259$$ −0.616399 −0.0383012
$$260$$ 0 0
$$261$$ 16.3584 1.01256
$$262$$ −25.9437 −1.60280
$$263$$ −15.5522 −0.958993 −0.479496 0.877544i $$-0.659181\pi$$
−0.479496 + 0.877544i $$0.659181\pi$$
$$264$$ 13.0913 0.805716
$$265$$ −2.18980 −0.134518
$$266$$ 16.1898 0.992660
$$267$$ −10.9187 −0.668213
$$268$$ 25.9820 1.58710
$$269$$ −5.22188 −0.318383 −0.159192 0.987248i $$-0.550889\pi$$
−0.159192 + 0.987248i $$0.550889\pi$$
$$270$$ 5.44687 0.331486
$$271$$ 25.9367 1.57554 0.787772 0.615967i $$-0.211234\pi$$
0.787772 + 0.615967i $$0.211234\pi$$
$$272$$ −24.8202 −1.50495
$$273$$ 0 0
$$274$$ −12.3765 −0.747691
$$275$$ −3.50702 −0.211481
$$276$$ −26.7218 −1.60846
$$277$$ −17.7741 −1.06794 −0.533972 0.845502i $$-0.679301\pi$$
−0.533972 + 0.845502i $$0.679301\pi$$
$$278$$ 41.2952 2.47672
$$279$$ −3.20784 −0.192048
$$280$$ −5.44687 −0.325513
$$281$$ 7.41168 0.442143 0.221072 0.975258i $$-0.429045\pi$$
0.221072 + 0.975258i $$0.429045\pi$$
$$282$$ −19.3664 −1.15326
$$283$$ 9.31722 0.553851 0.276926 0.960891i $$-0.410684\pi$$
0.276926 + 0.960891i $$0.410684\pi$$
$$284$$ −27.8835 −1.65458
$$285$$ −23.1726 −1.37263
$$286$$ 0 0
$$287$$ 7.58121 0.447505
$$288$$ 6.80620 0.401059
$$289$$ 1.36245 0.0801439
$$290$$ −22.5531 −1.32437
$$291$$ −9.46179 −0.554660
$$292$$ −8.57028 −0.501538
$$293$$ −28.7530 −1.67977 −0.839883 0.542767i $$-0.817377\pi$$
−0.839883 + 0.542767i $$0.817377\pi$$
$$294$$ −36.6336 −2.13651
$$295$$ −6.36333 −0.370488
$$296$$ −4.53821 −0.263778
$$297$$ −1.77812 −0.103177
$$298$$ −4.77501 −0.276609
$$299$$ 0 0
$$300$$ −34.3413 −1.98270
$$301$$ −7.53198 −0.434136
$$302$$ 46.1506 2.65567
$$303$$ 11.0241 0.633316
$$304$$ 48.0702 2.75701
$$305$$ 17.7429 1.01596
$$306$$ −23.8695 −1.36453
$$307$$ −0.792161 −0.0452110 −0.0226055 0.999744i $$-0.507196\pi$$
−0.0226055 + 0.999744i $$0.507196\pi$$
$$308$$ 3.33437 0.189993
$$309$$ 22.3032 1.26878
$$310$$ 4.42260 0.251187
$$311$$ 25.8022 1.46311 0.731554 0.681783i $$-0.238795\pi$$
0.731554 + 0.681783i $$0.238795\pi$$
$$312$$ 0 0
$$313$$ −26.2780 −1.48532 −0.742661 0.669668i $$-0.766437\pi$$
−0.742661 + 0.669668i $$0.766437\pi$$
$$314$$ −30.3484 −1.71266
$$315$$ −2.11250 −0.119026
$$316$$ 58.2108 3.27461
$$317$$ 10.9788 0.616633 0.308317 0.951284i $$-0.400234\pi$$
0.308317 + 0.951284i $$0.400234\pi$$
$$318$$ 10.2671 0.575751
$$319$$ 7.36245 0.412218
$$320$$ 4.77101 0.266707
$$321$$ 5.39853 0.301316
$$322$$ −5.32345 −0.296664
$$323$$ −35.5632 −1.97879
$$324$$ −45.9748 −2.55416
$$325$$ 0 0
$$326$$ 4.81020 0.266412
$$327$$ −6.55714 −0.362610
$$328$$ 55.8162 3.08194
$$329$$ −2.63044 −0.145021
$$330$$ −7.00000 −0.385337
$$331$$ 10.0561 0.552736 0.276368 0.961052i $$-0.410869\pi$$
0.276368 + 0.961052i $$0.410869\pi$$
$$332$$ −57.2600 −3.14255
$$333$$ −1.76008 −0.0964520
$$334$$ −17.6999 −0.968497
$$335$$ −7.40856 −0.404773
$$336$$ 10.2992 0.561866
$$337$$ 19.7398 1.07530 0.537648 0.843169i $$-0.319313\pi$$
0.537648 + 0.843169i $$0.319313\pi$$
$$338$$ 0 0
$$339$$ −12.8022 −0.695320
$$340$$ 22.4366 1.21680
$$341$$ −1.44375 −0.0781836
$$342$$ 46.2288 2.49977
$$343$$ −10.4226 −0.562767
$$344$$ −55.4538 −2.98987
$$345$$ 7.61951 0.410221
$$346$$ 4.13054 0.222059
$$347$$ 18.3132 0.983105 0.491553 0.870848i $$-0.336430\pi$$
0.491553 + 0.870848i $$0.336430\pi$$
$$348$$ 72.0943 3.86466
$$349$$ 17.3765 0.930142 0.465071 0.885273i $$-0.346029\pi$$
0.465071 + 0.885273i $$0.346029\pi$$
$$350$$ −6.84139 −0.365688
$$351$$ 0 0
$$352$$ 3.06327 0.163273
$$353$$ 33.7670 1.79724 0.898618 0.438732i $$-0.144572\pi$$
0.898618 + 0.438732i $$0.144572\pi$$
$$354$$ 29.8352 1.58572
$$355$$ 7.95077 0.421983
$$356$$ −20.4749 −1.08517
$$357$$ −7.61951 −0.403267
$$358$$ 18.3874 0.971805
$$359$$ −4.72889 −0.249582 −0.124791 0.992183i $$-0.539826\pi$$
−0.124791 + 0.992183i $$0.539826\pi$$
$$360$$ −15.5531 −0.819722
$$361$$ 49.8764 2.62507
$$362$$ 29.5202 1.55155
$$363$$ 2.28514 0.119939
$$364$$ 0 0
$$365$$ 2.44375 0.127912
$$366$$ −83.1896 −4.34839
$$367$$ 6.42660 0.335466 0.167733 0.985832i $$-0.446355\pi$$
0.167733 + 0.985832i $$0.446355\pi$$
$$368$$ −15.8062 −0.823955
$$369$$ 21.6476 1.12693
$$370$$ 2.42660 0.126153
$$371$$ 1.39452 0.0724000
$$372$$ −14.1375 −0.732993
$$373$$ −4.41168 −0.228428 −0.114214 0.993456i $$-0.536435\pi$$
−0.114214 + 0.993456i $$0.536435\pi$$
$$374$$ −10.7429 −0.555504
$$375$$ 23.7530 1.22660
$$376$$ −19.3664 −0.998748
$$377$$ 0 0
$$378$$ −3.46871 −0.178411
$$379$$ 19.5070 1.00201 0.501004 0.865445i $$-0.332964\pi$$
0.501004 + 0.865445i $$0.332964\pi$$
$$380$$ −43.4538 −2.22913
$$381$$ 16.5351 0.847118
$$382$$ 36.9608 1.89108
$$383$$ 27.6304 1.41185 0.705925 0.708287i $$-0.250532\pi$$
0.705925 + 0.708287i $$0.250532\pi$$
$$384$$ −36.3694 −1.85597
$$385$$ −0.950771 −0.0484558
$$386$$ −22.3163 −1.13587
$$387$$ −21.5070 −1.09326
$$388$$ −17.7429 −0.900761
$$389$$ −28.9960 −1.47016 −0.735078 0.677983i $$-0.762854\pi$$
−0.735078 + 0.677983i $$0.762854\pi$$
$$390$$ 0 0
$$391$$ 11.6937 0.591376
$$392$$ −36.6336 −1.85027
$$393$$ −23.6476 −1.19286
$$394$$ −15.3241 −0.772019
$$395$$ −16.5984 −0.835154
$$396$$ 9.52106 0.478451
$$397$$ 34.6585 1.73946 0.869730 0.493527i $$-0.164293\pi$$
0.869730 + 0.493527i $$0.164293\pi$$
$$398$$ 23.4288 1.17438
$$399$$ 14.7570 0.738773
$$400$$ −20.3132 −1.01566
$$401$$ −14.5874 −0.728462 −0.364231 0.931309i $$-0.618668\pi$$
−0.364231 + 0.931309i $$0.618668\pi$$
$$402$$ 34.7358 1.73246
$$403$$ 0 0
$$404$$ 20.6725 1.02850
$$405$$ 13.1094 0.651410
$$406$$ 14.3624 0.712796
$$407$$ −0.792161 −0.0392660
$$408$$ −56.0983 −2.77728
$$409$$ 16.6015 0.820890 0.410445 0.911885i $$-0.365373\pi$$
0.410445 + 0.911885i $$0.365373\pi$$
$$410$$ −29.8452 −1.47395
$$411$$ −11.2811 −0.556458
$$412$$ 41.8234 2.06049
$$413$$ 4.05234 0.199403
$$414$$ −15.2007 −0.747075
$$415$$ 16.3273 0.801473
$$416$$ 0 0
$$417$$ 37.6405 1.84326
$$418$$ 20.8062 1.01766
$$419$$ −22.2952 −1.08919 −0.544595 0.838699i $$-0.683317\pi$$
−0.544595 + 0.838699i $$0.683317\pi$$
$$420$$ −9.31010 −0.454286
$$421$$ −3.45779 −0.168522 −0.0842612 0.996444i $$-0.526853\pi$$
−0.0842612 + 0.996444i $$0.526853\pi$$
$$422$$ −13.0031 −0.632982
$$423$$ −7.51102 −0.365198
$$424$$ 10.2671 0.498615
$$425$$ 15.0281 0.728969
$$426$$ −37.2780 −1.80613
$$427$$ −11.2992 −0.546806
$$428$$ 10.1234 0.489334
$$429$$ 0 0
$$430$$ 29.6514 1.42992
$$431$$ 25.8804 1.24661 0.623307 0.781977i $$-0.285789\pi$$
0.623307 + 0.781977i $$0.285789\pi$$
$$432$$ −10.2992 −0.495520
$$433$$ 14.3805 0.691082 0.345541 0.938404i $$-0.387695\pi$$
0.345541 + 0.938404i $$0.387695\pi$$
$$434$$ −2.81643 −0.135193
$$435$$ −20.5571 −0.985639
$$436$$ −12.2961 −0.588875
$$437$$ −22.6476 −1.08338
$$438$$ −11.4578 −0.547474
$$439$$ 4.56628 0.217937 0.108968 0.994045i $$-0.465245\pi$$
0.108968 + 0.994045i $$0.465245\pi$$
$$440$$ −7.00000 −0.333712
$$441$$ −14.2078 −0.676564
$$442$$ 0 0
$$443$$ 18.1718 0.863366 0.431683 0.902025i $$-0.357920\pi$$
0.431683 + 0.902025i $$0.357920\pi$$
$$444$$ −7.75697 −0.368129
$$445$$ 5.83828 0.276761
$$446$$ 42.2038 1.99841
$$447$$ −4.35241 −0.205862
$$448$$ −3.03831 −0.143546
$$449$$ −12.6295 −0.596025 −0.298013 0.954562i $$-0.596324\pi$$
−0.298013 + 0.954562i $$0.596324\pi$$
$$450$$ −19.5351 −0.920893
$$451$$ 9.74293 0.458777
$$452$$ −24.0069 −1.12919
$$453$$ 42.0662 1.97644
$$454$$ −44.1646 −2.07275
$$455$$ 0 0
$$456$$ 108.647 5.08788
$$457$$ −25.7882 −1.20632 −0.603160 0.797621i $$-0.706092\pi$$
−0.603160 + 0.797621i $$0.706092\pi$$
$$458$$ 67.3271 3.14599
$$459$$ 7.61951 0.355648
$$460$$ 14.2883 0.666193
$$461$$ −12.6336 −0.588403 −0.294202 0.955743i $$-0.595054\pi$$
−0.294202 + 0.955743i $$0.595054\pi$$
$$462$$ 4.45779 0.207395
$$463$$ −3.47494 −0.161494 −0.0807471 0.996735i $$-0.525731\pi$$
−0.0807471 + 0.996735i $$0.525731\pi$$
$$464$$ 42.6445 1.97972
$$465$$ 4.03119 0.186942
$$466$$ −68.0419 −3.15198
$$467$$ −18.7258 −0.866526 −0.433263 0.901268i $$-0.642638\pi$$
−0.433263 + 0.901268i $$0.642638\pi$$
$$468$$ 0 0
$$469$$ 4.71797 0.217856
$$470$$ 10.3553 0.477656
$$471$$ −27.6625 −1.27462
$$472$$ 29.8352 1.37327
$$473$$ −9.67967 −0.445072
$$474$$ 77.8232 3.57454
$$475$$ −29.1054 −1.33545
$$476$$ −14.2883 −0.654901
$$477$$ 3.98196 0.182321
$$478$$ 1.87035 0.0855479
$$479$$ 9.28514 0.424249 0.212124 0.977243i $$-0.431962\pi$$
0.212124 + 0.977243i $$0.431962\pi$$
$$480$$ −8.55313 −0.390395
$$481$$ 0 0
$$482$$ 40.5451 1.84678
$$483$$ −4.85231 −0.220788
$$484$$ 4.28514 0.194779
$$485$$ 5.05926 0.229729
$$486$$ −48.0913 −2.18147
$$487$$ −39.9296 −1.80938 −0.904692 0.426067i $$-0.859899\pi$$
−0.904692 + 0.426067i $$0.859899\pi$$
$$488$$ −83.1896 −3.76582
$$489$$ 4.38449 0.198273
$$490$$ 19.5881 0.884901
$$491$$ −41.1827 −1.85855 −0.929274 0.369391i $$-0.879566\pi$$
−0.929274 + 0.369391i $$0.879566\pi$$
$$492$$ 95.4044 4.30116
$$493$$ −31.5491 −1.42090
$$494$$ 0 0
$$495$$ −2.71486 −0.122024
$$496$$ −8.36245 −0.375485
$$497$$ −5.06327 −0.227119
$$498$$ −76.5520 −3.43038
$$499$$ 19.3132 0.864578 0.432289 0.901735i $$-0.357706\pi$$
0.432289 + 0.901735i $$0.357706\pi$$
$$500$$ 44.5420 1.99198
$$501$$ −16.1335 −0.720790
$$502$$ −27.6476 −1.23397
$$503$$ 19.6897 0.877920 0.438960 0.898507i $$-0.355347\pi$$
0.438960 + 0.898507i $$0.355347\pi$$
$$504$$ 9.90466 0.441188
$$505$$ −5.89462 −0.262307
$$506$$ −6.84139 −0.304137
$$507$$ 0 0
$$508$$ 31.0069 1.37571
$$509$$ 13.4749 0.597266 0.298633 0.954368i $$-0.403469\pi$$
0.298633 + 0.954368i $$0.403469\pi$$
$$510$$ 29.9960 1.32825
$$511$$ −1.55625 −0.0688443
$$512$$ −48.6224 −2.14883
$$513$$ −14.7570 −0.651536
$$514$$ 47.2631 2.08469
$$515$$ −11.9256 −0.525505
$$516$$ −94.7848 −4.17267
$$517$$ −3.38049 −0.148674
$$518$$ −1.54532 −0.0678977
$$519$$ 3.76497 0.165264
$$520$$ 0 0
$$521$$ 19.4266 0.851095 0.425547 0.904936i $$-0.360082\pi$$
0.425547 + 0.904936i $$0.360082\pi$$
$$522$$ 41.0109 1.79500
$$523$$ 10.9227 0.477616 0.238808 0.971067i $$-0.423243\pi$$
0.238808 + 0.971067i $$0.423243\pi$$
$$524$$ −44.3444 −1.93719
$$525$$ −6.23591 −0.272158
$$526$$ −38.9898 −1.70003
$$527$$ 6.18668 0.269496
$$528$$ 13.2359 0.576019
$$529$$ −15.5531 −0.676223
$$530$$ −5.48987 −0.238465
$$531$$ 11.5712 0.502146
$$532$$ 27.6725 1.19976
$$533$$ 0 0
$$534$$ −27.3734 −1.18456
$$535$$ −2.88662 −0.124799
$$536$$ 34.7358 1.50036
$$537$$ 16.7601 0.723251
$$538$$ −13.0913 −0.564408
$$539$$ −6.39452 −0.275432
$$540$$ 9.31010 0.400643
$$541$$ −16.9820 −0.730111 −0.365056 0.930986i $$-0.618950\pi$$
−0.365056 + 0.930986i $$0.618950\pi$$
$$542$$ 65.0239 2.79301
$$543$$ 26.9076 1.15471
$$544$$ −13.1265 −0.562795
$$545$$ 3.50613 0.150186
$$546$$ 0 0
$$547$$ 1.13345 0.0484629 0.0242315 0.999706i $$-0.492286\pi$$
0.0242315 + 0.999706i $$0.492286\pi$$
$$548$$ −21.1546 −0.903680
$$549$$ −32.2640 −1.37699
$$550$$ −8.79216 −0.374899
$$551$$ 61.1023 2.60304
$$552$$ −35.7249 −1.52055
$$553$$ 10.5703 0.449494
$$554$$ −44.5601 −1.89318
$$555$$ 2.21184 0.0938874
$$556$$ 70.5841 2.99343
$$557$$ −20.2671 −0.858745 −0.429372 0.903128i $$-0.641265\pi$$
−0.429372 + 0.903128i $$0.641265\pi$$
$$558$$ −8.04211 −0.340450
$$559$$ 0 0
$$560$$ −5.50702 −0.232714
$$561$$ −9.79216 −0.413425
$$562$$ 18.5812 0.783801
$$563$$ 0.313217 0.0132005 0.00660026 0.999978i $$-0.497899\pi$$
0.00660026 + 0.999978i $$0.497899\pi$$
$$564$$ −33.1023 −1.39386
$$565$$ 6.84539 0.287988
$$566$$ 23.3584 0.981829
$$567$$ −8.34841 −0.350600
$$568$$ −37.2780 −1.56415
$$569$$ −11.0842 −0.464675 −0.232337 0.972635i $$-0.574637\pi$$
−0.232337 + 0.972635i $$0.574637\pi$$
$$570$$ −58.0943 −2.43330
$$571$$ 41.7037 1.74525 0.872624 0.488393i $$-0.162417\pi$$
0.872624 + 0.488393i $$0.162417\pi$$
$$572$$ 0 0
$$573$$ 33.6897 1.40741
$$574$$ 19.0062 0.793305
$$575$$ 9.57028 0.399108
$$576$$ −8.67566 −0.361486
$$577$$ 19.1867 0.798752 0.399376 0.916787i $$-0.369227\pi$$
0.399376 + 0.916787i $$0.369227\pi$$
$$578$$ 3.41568 0.142073
$$579$$ −20.3413 −0.845355
$$580$$ −38.5491 −1.60067
$$581$$ −10.3976 −0.431367
$$582$$ −23.7209 −0.983263
$$583$$ 1.79216 0.0742237
$$584$$ −11.4578 −0.474127
$$585$$ 0 0
$$586$$ −72.0842 −2.97777
$$587$$ −12.5843 −0.519411 −0.259705 0.965688i $$-0.583625\pi$$
−0.259705 + 0.965688i $$0.583625\pi$$
$$588$$ −62.6162 −2.58225
$$589$$ −11.9820 −0.493708
$$590$$ −15.9530 −0.656775
$$591$$ −13.9679 −0.574564
$$592$$ −4.58832 −0.188579
$$593$$ −13.7117 −0.563074 −0.281537 0.959550i $$-0.590844\pi$$
−0.281537 + 0.959550i $$0.590844\pi$$
$$594$$ −4.45779 −0.182905
$$595$$ 4.07419 0.167025
$$596$$ −8.16172 −0.334317
$$597$$ 21.3553 0.874015
$$598$$ 0 0
$$599$$ −15.3905 −0.628840 −0.314420 0.949284i $$-0.601810\pi$$
−0.314420 + 0.949284i $$0.601810\pi$$
$$600$$ −45.9116 −1.87433
$$601$$ −16.2148 −0.661414 −0.330707 0.943734i $$-0.607287\pi$$
−0.330707 + 0.943734i $$0.607287\pi$$
$$602$$ −18.8828 −0.769606
$$603$$ 13.4718 0.548615
$$604$$ 78.8833 3.20972
$$605$$ −1.22188 −0.0496763
$$606$$ 27.6376 1.12270
$$607$$ 2.68278 0.108891 0.0544453 0.998517i $$-0.482661\pi$$
0.0544453 + 0.998517i $$0.482661\pi$$
$$608$$ 25.4226 1.03102
$$609$$ 13.0913 0.530488
$$610$$ 44.4819 1.80102
$$611$$ 0 0
$$612$$ −40.7991 −1.64921
$$613$$ −36.0913 −1.45772 −0.728858 0.684665i $$-0.759948\pi$$
−0.728858 + 0.684665i $$0.759948\pi$$
$$614$$ −1.98596 −0.0801469
$$615$$ −27.2038 −1.09696
$$616$$ 4.45779 0.179609
$$617$$ −19.1718 −0.771826 −0.385913 0.922535i $$-0.626114\pi$$
−0.385913 + 0.922535i $$0.626114\pi$$
$$618$$ 55.9145 2.24921
$$619$$ 6.38049 0.256453 0.128227 0.991745i $$-0.459071\pi$$
0.128227 + 0.991745i $$0.459071\pi$$
$$620$$ 7.55936 0.303591
$$621$$ 4.85231 0.194717
$$622$$ 64.6866 2.59370
$$623$$ −3.71797 −0.148957
$$624$$ 0 0
$$625$$ 4.83427 0.193371
$$626$$ −65.8795 −2.63307
$$627$$ 18.9648 0.757381
$$628$$ −51.8733 −2.06997
$$629$$ 3.39452 0.135349
$$630$$ −5.29607 −0.211000
$$631$$ −20.9007 −0.832042 −0.416021 0.909355i $$-0.636576\pi$$
−0.416021 + 0.909355i $$0.636576\pi$$
$$632$$ 77.8232 3.09564
$$633$$ −11.8523 −0.471087
$$634$$ 27.5242 1.09312
$$635$$ −8.84139 −0.350860
$$636$$ 17.5491 0.695868
$$637$$ 0 0
$$638$$ 18.4578 0.730751
$$639$$ −14.4578 −0.571941
$$640$$ 19.4469 0.768705
$$641$$ 19.1085 0.754740 0.377370 0.926063i $$-0.376828\pi$$
0.377370 + 0.926063i $$0.376828\pi$$
$$642$$ 13.5342 0.534152
$$643$$ −3.50702 −0.138303 −0.0691517 0.997606i $$-0.522029\pi$$
−0.0691517 + 0.997606i $$0.522029\pi$$
$$644$$ −9.09915 −0.358557
$$645$$ 27.0272 1.06419
$$646$$ −89.1575 −3.50786
$$647$$ 20.8343 0.819080 0.409540 0.912292i $$-0.365689\pi$$
0.409540 + 0.912292i $$0.365689\pi$$
$$648$$ −61.4647 −2.41456
$$649$$ 5.20784 0.204426
$$650$$ 0 0
$$651$$ −2.56717 −0.100615
$$652$$ 8.22188 0.321994
$$653$$ −12.6757 −0.496037 −0.248019 0.968755i $$-0.579779\pi$$
−0.248019 + 0.968755i $$0.579779\pi$$
$$654$$ −16.4389 −0.642810
$$655$$ 12.6445 0.494060
$$656$$ 56.4326 2.20332
$$657$$ −4.44375 −0.173367
$$658$$ −6.59455 −0.257082
$$659$$ 4.37960 0.170605 0.0853025 0.996355i $$-0.472814\pi$$
0.0853025 + 0.996355i $$0.472814\pi$$
$$660$$ −11.9648 −0.465730
$$661$$ −1.12253 −0.0436614 −0.0218307 0.999762i $$-0.506949\pi$$
−0.0218307 + 0.999762i $$0.506949\pi$$
$$662$$ 25.2110 0.979852
$$663$$ 0 0
$$664$$ −76.5520 −2.97080
$$665$$ −7.89062 −0.305985
$$666$$ −4.41256 −0.170983
$$667$$ −20.0913 −0.777940
$$668$$ −30.2538 −1.17055
$$669$$ 38.4687 1.48729
$$670$$ −18.5734 −0.717553
$$671$$ −14.5211 −0.560579
$$672$$ 5.44687 0.210117
$$673$$ 18.0492 0.695747 0.347873 0.937542i $$-0.386904\pi$$
0.347873 + 0.937542i $$0.386904\pi$$
$$674$$ 49.4881 1.90621
$$675$$ 6.23591 0.240020
$$676$$ 0 0
$$677$$ −26.7037 −1.02631 −0.513154 0.858297i $$-0.671523\pi$$
−0.513154 + 0.858297i $$0.671523\pi$$
$$678$$ −32.0953 −1.23261
$$679$$ −3.22188 −0.123644
$$680$$ 29.9960 1.15029
$$681$$ −40.2560 −1.54261
$$682$$ −3.61951 −0.138598
$$683$$ 30.9788 1.18537 0.592686 0.805433i $$-0.298067\pi$$
0.592686 + 0.805433i $$0.298067\pi$$
$$684$$ 79.0170 3.02129
$$685$$ 6.03208 0.230474
$$686$$ −26.1296 −0.997635
$$687$$ 61.3685 2.34135
$$688$$ −56.0662 −2.13750
$$689$$ 0 0
$$690$$ 19.1023 0.727211
$$691$$ −2.33126 −0.0886852 −0.0443426 0.999016i $$-0.514119\pi$$
−0.0443426 + 0.999016i $$0.514119\pi$$
$$692$$ 7.06015 0.268387
$$693$$ 1.72889 0.0656753
$$694$$ 45.9116 1.74278
$$695$$ −20.1265 −0.763443
$$696$$ 96.3843 3.65344
$$697$$ −41.7499 −1.58139
$$698$$ 43.5632 1.64889
$$699$$ −62.0201 −2.34581
$$700$$ −11.6937 −0.441980
$$701$$ −22.5834 −0.852965 −0.426482 0.904496i $$-0.640247\pi$$
−0.426482 + 0.904496i $$0.640247\pi$$
$$702$$ 0 0
$$703$$ −6.57429 −0.247954
$$704$$ −3.90466 −0.147162
$$705$$ 9.43886 0.355488
$$706$$ 84.6545 3.18601
$$707$$ 3.75385 0.141178
$$708$$ 50.9960 1.91655
$$709$$ −15.7218 −0.590444 −0.295222 0.955429i $$-0.595394\pi$$
−0.295222 + 0.955429i $$0.595394\pi$$
$$710$$ 19.9327 0.748062
$$711$$ 30.1827 1.13194
$$712$$ −27.3734 −1.02586
$$713$$ 3.93985 0.147548
$$714$$ −19.1023 −0.714884
$$715$$ 0 0
$$716$$ 31.4288 1.17455
$$717$$ 1.70482 0.0636678
$$718$$ −11.8554 −0.442441
$$719$$ −29.8303 −1.11248 −0.556241 0.831021i $$-0.687757\pi$$
−0.556241 + 0.831021i $$0.687757\pi$$
$$720$$ −15.7249 −0.586032
$$721$$ 7.59455 0.282836
$$722$$ 125.041 4.65355
$$723$$ 36.9568 1.37444
$$724$$ 50.4576 1.87524
$$725$$ −25.8202 −0.958939
$$726$$ 5.72889 0.212619
$$727$$ −36.6053 −1.35761 −0.678807 0.734316i $$-0.737503\pi$$
−0.678807 + 0.734316i $$0.737503\pi$$
$$728$$ 0 0
$$729$$ −11.6485 −0.431425
$$730$$ 6.12653 0.226753
$$731$$ 41.4787 1.53415
$$732$$ −142.193 −5.25559
$$733$$ 29.8695 1.10325 0.551627 0.834091i $$-0.314007\pi$$
0.551627 + 0.834091i $$0.314007\pi$$
$$734$$ 16.1116 0.594690
$$735$$ 17.8545 0.658575
$$736$$ −8.35933 −0.308129
$$737$$ 6.06327 0.223343
$$738$$ 54.2709 1.99774
$$739$$ 30.2569 1.11302 0.556508 0.830842i $$-0.312141\pi$$
0.556508 + 0.830842i $$0.312141\pi$$
$$740$$ 4.14769 0.152472
$$741$$ 0 0
$$742$$ 3.49610 0.128346
$$743$$ 39.2419 1.43965 0.719824 0.694157i $$-0.244223\pi$$
0.719824 + 0.694157i $$0.244223\pi$$
$$744$$ −18.9007 −0.692932
$$745$$ 2.32725 0.0852640
$$746$$ −11.0602 −0.404941
$$747$$ −29.6897 −1.08629
$$748$$ −18.3624 −0.671398
$$749$$ 1.83828 0.0671691
$$750$$ 59.5491 2.17443
$$751$$ 6.82335 0.248988 0.124494 0.992220i $$-0.460269\pi$$
0.124494 + 0.992220i $$0.460269\pi$$
$$752$$ −19.5803 −0.714021
$$753$$ −25.2007 −0.918365
$$754$$ 0 0
$$755$$ −22.4930 −0.818603
$$756$$ −5.92893 −0.215633
$$757$$ −0.752967 −0.0273670 −0.0136835 0.999906i $$-0.504356\pi$$
−0.0136835 + 0.999906i $$0.504356\pi$$
$$758$$ 48.9045 1.77629
$$759$$ −6.23591 −0.226349
$$760$$ −58.0943 −2.10730
$$761$$ 34.9748 1.26784 0.633919 0.773400i $$-0.281445\pi$$
0.633919 + 0.773400i $$0.281445\pi$$
$$762$$ 41.4538 1.50171
$$763$$ −2.23280 −0.0808327
$$764$$ 63.1756 2.28561
$$765$$ 11.6336 0.420612
$$766$$ 69.2700 2.50283
$$767$$ 0 0
$$768$$ −73.3333 −2.64619
$$769$$ 15.4649 0.557679 0.278839 0.960338i $$-0.410050\pi$$
0.278839 + 0.960338i $$0.410050\pi$$
$$770$$ −2.38360 −0.0858990
$$771$$ 43.0802 1.55150
$$772$$ −38.1444 −1.37285
$$773$$ −36.1366 −1.29974 −0.649871 0.760045i $$-0.725177\pi$$
−0.649871 + 0.760045i $$0.725177\pi$$
$$774$$ −53.9185 −1.93806
$$775$$ 5.06327 0.181878
$$776$$ −23.7209 −0.851530
$$777$$ −1.40856 −0.0505318
$$778$$ −72.6935 −2.60619
$$779$$ 80.8583 2.89705
$$780$$ 0 0
$$781$$ −6.50702 −0.232839
$$782$$ 29.3163 1.04835
$$783$$ −13.0913 −0.467846
$$784$$ −37.0381 −1.32279
$$785$$ 14.7913 0.527923
$$786$$ −59.2849 −2.11462
$$787$$ −22.9499 −0.818075 −0.409037 0.912518i $$-0.634135\pi$$
−0.409037 + 0.912518i $$0.634135\pi$$
$$788$$ −26.1929 −0.933084
$$789$$ −35.5391 −1.26523
$$790$$ −41.6124 −1.48050
$$791$$ −4.35933 −0.155000
$$792$$ 12.7289 0.452302
$$793$$ 0 0
$$794$$ 86.8895 3.08359
$$795$$ −5.00400 −0.177474
$$796$$ 40.0459 1.41939
$$797$$ 43.4498 1.53907 0.769535 0.638604i $$-0.220488\pi$$
0.769535 + 0.638604i $$0.220488\pi$$
$$798$$ 36.9960 1.30964
$$799$$ 14.4859 0.512473
$$800$$ −10.7429 −0.379820
$$801$$ −10.6164 −0.375112
$$802$$ −36.5710 −1.29137
$$803$$ −2.00000 −0.0705785
$$804$$ 59.3725 2.09391
$$805$$ 2.59455 0.0914460
$$806$$ 0 0
$$807$$ −11.9327 −0.420052
$$808$$ 27.6376 0.972286
$$809$$ −43.1094 −1.51565 −0.757823 0.652461i $$-0.773737\pi$$
−0.757823 + 0.652461i $$0.773737\pi$$
$$810$$ 32.8655 1.15477
$$811$$ −17.9428 −0.630056 −0.315028 0.949082i $$-0.602014\pi$$
−0.315028 + 0.949082i $$0.602014\pi$$
$$812$$ 24.5491 0.861506
$$813$$ 59.2691 2.07866
$$814$$ −1.98596 −0.0696080
$$815$$ −2.34441 −0.0821210
$$816$$ −56.7178 −1.98552
$$817$$ −80.3333 −2.81051
$$818$$ 41.6202 1.45522
$$819$$ 0 0
$$820$$ −51.0131 −1.78146
$$821$$ 15.6647 0.546703 0.273352 0.961914i $$-0.411868\pi$$
0.273352 + 0.961914i $$0.411868\pi$$
$$822$$ −28.2820 −0.986449
$$823$$ 3.69771 0.128894 0.0644470 0.997921i $$-0.479472\pi$$
0.0644470 + 0.997921i $$0.479472\pi$$
$$824$$ 55.9145 1.94787
$$825$$ −8.01404 −0.279013
$$826$$ 10.1593 0.353487
$$827$$ −42.3201 −1.47162 −0.735808 0.677191i $$-0.763197\pi$$
−0.735808 + 0.677191i $$0.763197\pi$$
$$828$$ −25.9820 −0.902936
$$829$$ 7.15461 0.248490 0.124245 0.992252i $$-0.460349\pi$$
0.124245 + 0.992252i $$0.460349\pi$$
$$830$$ 40.9327 1.42080
$$831$$ −40.6164 −1.40897
$$832$$ 0 0
$$833$$ 27.4014 0.949404
$$834$$ 94.3654 3.26761
$$835$$ 8.62663 0.298537
$$836$$ 35.5632 1.22998
$$837$$ 2.56717 0.0887344
$$838$$ −55.8944 −1.93084
$$839$$ 2.76008 0.0952887 0.0476443 0.998864i $$-0.484829\pi$$
0.0476443 + 0.998864i $$0.484829\pi$$
$$840$$ −12.4469 −0.429458
$$841$$ 25.2056 0.869159
$$842$$ −8.66874 −0.298745
$$843$$ 16.9367 0.583332
$$844$$ −22.2257 −0.765040
$$845$$ 0 0
$$846$$ −18.8303 −0.647398
$$847$$ 0.778124 0.0267367
$$848$$ 10.3805 0.356467
$$849$$ 21.2912 0.730711
$$850$$ 37.6757 1.29226
$$851$$ 2.16172 0.0741030
$$852$$ −63.7178 −2.18294
$$853$$ 46.1615 1.58054 0.790270 0.612758i $$-0.209940\pi$$
0.790270 + 0.612758i $$0.209940\pi$$
$$854$$ −28.3273 −0.969339
$$855$$ −22.5311 −0.770547
$$856$$ 13.5342 0.462590
$$857$$ 26.2680 0.897297 0.448649 0.893708i $$-0.351905\pi$$
0.448649 + 0.893708i $$0.351905\pi$$
$$858$$ 0 0
$$859$$ −32.0381 −1.09313 −0.546563 0.837418i $$-0.684064\pi$$
−0.546563 + 0.837418i $$0.684064\pi$$
$$860$$ 50.6819 1.72824
$$861$$ 17.3241 0.590405
$$862$$ 64.8826 2.20991
$$863$$ −24.2078 −0.824044 −0.412022 0.911174i $$-0.635177\pi$$
−0.412022 + 0.911174i $$0.635177\pi$$
$$864$$ −5.44687 −0.185306
$$865$$ −2.01315 −0.0684491
$$866$$ 36.0521 1.22510
$$867$$ 3.11338 0.105736
$$868$$ −4.81401 −0.163398
$$869$$ 13.5843 0.460817
$$870$$ −51.5371 −1.74727
$$871$$ 0 0
$$872$$ −16.4389 −0.556690
$$873$$ −9.19983 −0.311367
$$874$$ −56.7779 −1.92054
$$875$$ 8.08823 0.273432
$$876$$ −19.5843 −0.661693
$$877$$ 1.05635 0.0356703 0.0178351 0.999841i $$-0.494323\pi$$
0.0178351 + 0.999841i $$0.494323\pi$$
$$878$$ 11.4478 0.386343
$$879$$ −65.7046 −2.21616
$$880$$ −7.07730 −0.238576
$$881$$ −24.1655 −0.814157 −0.407079 0.913393i $$-0.633453\pi$$
−0.407079 + 0.913393i $$0.633453\pi$$
$$882$$ −35.6193 −1.19936
$$883$$ 5.84831 0.196811 0.0984057 0.995146i $$-0.468626\pi$$
0.0984057 + 0.995146i $$0.468626\pi$$
$$884$$ 0 0
$$885$$ −14.5411 −0.488795
$$886$$ 45.5569 1.53052
$$887$$ −8.67566 −0.291300 −0.145650 0.989336i $$-0.546527\pi$$
−0.145650 + 0.989336i $$0.546527\pi$$
$$888$$ −10.3705 −0.348010
$$889$$ 5.63044 0.188839
$$890$$ 14.6367 0.490622
$$891$$ −10.7289 −0.359431
$$892$$ 72.1373 2.41533
$$893$$ −28.0553 −0.938834
$$894$$ −10.9116 −0.364938
$$895$$ −8.96169 −0.299556
$$896$$ −12.3843 −0.413730
$$897$$ 0 0
$$898$$ −31.6625 −1.05659
$$899$$ −10.6295 −0.354515
$$900$$ −33.3905 −1.11302
$$901$$ −7.67967 −0.255847
$$902$$ 24.4257 0.813287
$$903$$ −17.2116 −0.572768
$$904$$ −32.0953 −1.06748
$$905$$ −14.3876 −0.478260
$$906$$ 105.461 3.50370
$$907$$ −15.5952 −0.517832 −0.258916 0.965900i $$-0.583365\pi$$
−0.258916 + 0.965900i $$0.583365\pi$$
$$908$$ −75.4888 −2.50518
$$909$$ 10.7189 0.355522
$$910$$ 0 0
$$911$$ 24.5672 0.813947 0.406973 0.913440i $$-0.366584\pi$$
0.406973 + 0.913440i $$0.366584\pi$$
$$912$$ 109.847 3.63741
$$913$$ −13.3624 −0.442232
$$914$$ −64.6514 −2.13848
$$915$$ 40.5451 1.34038
$$916$$ 115.079 3.80233
$$917$$ −8.05234 −0.265912
$$918$$ 19.1023 0.630469
$$919$$ 17.2811 0.570052 0.285026 0.958520i $$-0.407998\pi$$
0.285026 + 0.958520i $$0.407998\pi$$
$$920$$ 19.1023 0.629783
$$921$$ −1.81020 −0.0596482
$$922$$ −31.6725 −1.04308
$$923$$ 0 0
$$924$$ 7.61951 0.250664
$$925$$ 2.77812 0.0913441
$$926$$ −8.71174 −0.286286
$$927$$ 21.6857 0.712252
$$928$$ 22.5531 0.740343
$$929$$ −48.2569 −1.58326 −0.791628 0.611003i $$-0.790766\pi$$
−0.791628 + 0.611003i $$0.790766\pi$$
$$930$$ 10.1063 0.331398
$$931$$ −53.0693 −1.73928
$$932$$ −116.301 −3.80957
$$933$$ 58.9617 1.93032
$$934$$ −46.9459 −1.53612
$$935$$ 5.23591 0.171233
$$936$$ 0 0
$$937$$ −17.7209 −0.578916 −0.289458 0.957191i $$-0.593475\pi$$
−0.289458 + 0.957191i $$0.593475\pi$$
$$938$$ 11.8280 0.386199
$$939$$ −60.0490 −1.95963
$$940$$ 17.6999 0.577308
$$941$$ 7.96392 0.259616 0.129808 0.991539i $$-0.458564\pi$$
0.129808 + 0.991539i $$0.458564\pi$$
$$942$$ −69.3504 −2.25956
$$943$$ −26.5874 −0.865806
$$944$$ 30.1646 0.981775
$$945$$ 1.69059 0.0549948
$$946$$ −24.2671 −0.788992
$$947$$ 15.5632 0.505735 0.252867 0.967501i $$-0.418626\pi$$
0.252867 + 0.967501i $$0.418626\pi$$
$$948$$ 133.020 4.32028
$$949$$ 0 0
$$950$$ −72.9677 −2.36739
$$951$$ 25.0882 0.813541
$$952$$ −19.1023 −0.619108
$$953$$ −31.8483 −1.03167 −0.515834 0.856689i $$-0.672518\pi$$
−0.515834 + 0.856689i $$0.672518\pi$$
$$954$$ 9.98285 0.323207
$$955$$ −18.0140 −0.582921
$$956$$ 3.19692 0.103396
$$957$$ 16.8242 0.543850
$$958$$ 23.2780 0.752079
$$959$$ −3.84139 −0.124045
$$960$$ 10.9024 0.351874
$$961$$ −28.9156 −0.932761
$$962$$ 0 0
$$963$$ 5.24906 0.169149
$$964$$ 69.3021 2.23207
$$965$$ 10.8766 0.350130
$$966$$ −12.1648 −0.391397
$$967$$ −46.1023 −1.48255 −0.741274 0.671202i $$-0.765778\pi$$
−0.741274 + 0.671202i $$0.765778\pi$$
$$968$$ 5.72889 0.184134
$$969$$ −81.2669 −2.61067
$$970$$ 12.6837 0.407248
$$971$$ −14.7882 −0.474575 −0.237287 0.971440i $$-0.576258\pi$$
−0.237287 + 0.971440i $$0.576258\pi$$
$$972$$ −82.2005 −2.63658
$$973$$ 12.8171 0.410898
$$974$$ −100.104 −3.20755
$$975$$ 0 0
$$976$$ −84.1083 −2.69224
$$977$$ 1.35933 0.0434889 0.0217444 0.999764i $$-0.493078\pi$$
0.0217444 + 0.999764i $$0.493078\pi$$
$$978$$ 10.9920 0.351485
$$979$$ −4.77812 −0.152710
$$980$$ 33.4812 1.06952
$$981$$ −6.37560 −0.203557
$$982$$ −103.246 −3.29470
$$983$$ 27.4709 0.876187 0.438093 0.898929i $$-0.355654\pi$$
0.438093 + 0.898929i $$0.355654\pi$$
$$984$$ 127.548 4.06608
$$985$$ 7.46871 0.237973
$$986$$ −79.0943 −2.51887
$$987$$ −6.01092 −0.191330
$$988$$ 0 0
$$989$$ 26.4148 0.839941
$$990$$ −6.80620 −0.216315
$$991$$ 0.173535 0.00551253 0.00275626 0.999996i $$-0.499123\pi$$
0.00275626 + 0.999996i $$0.499123\pi$$
$$992$$ −4.42260 −0.140418
$$993$$ 22.9797 0.729240
$$994$$ −12.6937 −0.402620
$$995$$ −11.4188 −0.362000
$$996$$ −130.847 −4.14605
$$997$$ 44.1896 1.39950 0.699749 0.714388i $$-0.253295\pi$$
0.699749 + 0.714388i $$0.253295\pi$$
$$998$$ 48.4186 1.53266
$$999$$ 1.40856 0.0445649
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.f.1.3 3
13.3 even 3 143.2.e.b.100.1 6
13.9 even 3 143.2.e.b.133.1 yes 6
13.12 even 2 1859.2.a.g.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
143.2.e.b.100.1 6 13.3 even 3
143.2.e.b.133.1 yes 6 13.9 even 3
1859.2.a.f.1.3 3 1.1 even 1 trivial
1859.2.a.g.1.1 3 13.12 even 2