# Properties

 Label 285.2.a.d.1.1 Level $285$ Weight $2$ Character 285.1 Self dual yes Analytic conductor $2.276$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, 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("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.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: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 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 1.1 Root $$-2.64575$$ of defining polynomial Character $$\chi$$ $$=$$ 285.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.64575 q^{2} -1.00000 q^{3} +5.00000 q^{4} +1.00000 q^{5} +2.64575 q^{6} +1.64575 q^{7} -7.93725 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.64575 q^{2} -1.00000 q^{3} +5.00000 q^{4} +1.00000 q^{5} +2.64575 q^{6} +1.64575 q^{7} -7.93725 q^{8} +1.00000 q^{9} -2.64575 q^{10} +0.354249 q^{11} -5.00000 q^{12} -0.354249 q^{13} -4.35425 q^{14} -1.00000 q^{15} +11.0000 q^{16} -4.00000 q^{17} -2.64575 q^{18} -1.00000 q^{19} +5.00000 q^{20} -1.64575 q^{21} -0.937254 q^{22} +9.29150 q^{23} +7.93725 q^{24} +1.00000 q^{25} +0.937254 q^{26} -1.00000 q^{27} +8.22876 q^{28} +8.93725 q^{29} +2.64575 q^{30} +6.00000 q^{31} -13.2288 q^{32} -0.354249 q^{33} +10.5830 q^{34} +1.64575 q^{35} +5.00000 q^{36} +3.64575 q^{37} +2.64575 q^{38} +0.354249 q^{39} -7.93725 q^{40} -9.64575 q^{41} +4.35425 q^{42} +5.64575 q^{43} +1.77124 q^{44} +1.00000 q^{45} -24.5830 q^{46} -1.29150 q^{47} -11.0000 q^{48} -4.29150 q^{49} -2.64575 q^{50} +4.00000 q^{51} -1.77124 q^{52} +11.2915 q^{53} +2.64575 q^{54} +0.354249 q^{55} -13.0627 q^{56} +1.00000 q^{57} -23.6458 q^{58} +11.2915 q^{59} -5.00000 q^{60} -11.2915 q^{61} -15.8745 q^{62} +1.64575 q^{63} +13.0000 q^{64} -0.354249 q^{65} +0.937254 q^{66} -6.58301 q^{67} -20.0000 q^{68} -9.29150 q^{69} -4.35425 q^{70} +7.29150 q^{71} -7.93725 q^{72} +10.0000 q^{73} -9.64575 q^{74} -1.00000 q^{75} -5.00000 q^{76} +0.583005 q^{77} -0.937254 q^{78} +6.58301 q^{79} +11.0000 q^{80} +1.00000 q^{81} +25.5203 q^{82} +6.00000 q^{83} -8.22876 q^{84} -4.00000 q^{85} -14.9373 q^{86} -8.93725 q^{87} -2.81176 q^{88} -16.9373 q^{89} -2.64575 q^{90} -0.583005 q^{91} +46.4575 q^{92} -6.00000 q^{93} +3.41699 q^{94} -1.00000 q^{95} +13.2288 q^{96} -2.93725 q^{97} +11.3542 q^{98} +0.354249 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 10 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 10 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 10 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} - 10 q^{12} - 6 q^{13} - 14 q^{14} - 2 q^{15} + 22 q^{16} - 8 q^{17} - 2 q^{19} + 10 q^{20} + 2 q^{21} + 14 q^{22} + 8 q^{23} + 2 q^{25} - 14 q^{26} - 2 q^{27} - 10 q^{28} + 2 q^{29} + 12 q^{31} - 6 q^{33} - 2 q^{35} + 10 q^{36} + 2 q^{37} + 6 q^{39} - 14 q^{41} + 14 q^{42} + 6 q^{43} + 30 q^{44} + 2 q^{45} - 28 q^{46} + 8 q^{47} - 22 q^{48} + 2 q^{49} + 8 q^{51} - 30 q^{52} + 12 q^{53} + 6 q^{55} - 42 q^{56} + 2 q^{57} - 42 q^{58} + 12 q^{59} - 10 q^{60} - 12 q^{61} - 2 q^{63} + 26 q^{64} - 6 q^{65} - 14 q^{66} + 8 q^{67} - 40 q^{68} - 8 q^{69} - 14 q^{70} + 4 q^{71} + 20 q^{73} - 14 q^{74} - 2 q^{75} - 10 q^{76} - 20 q^{77} + 14 q^{78} - 8 q^{79} + 22 q^{80} + 2 q^{81} + 14 q^{82} + 12 q^{83} + 10 q^{84} - 8 q^{85} - 14 q^{86} - 2 q^{87} + 42 q^{88} - 18 q^{89} + 20 q^{91} + 40 q^{92} - 12 q^{93} + 28 q^{94} - 2 q^{95} + 10 q^{97} + 28 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 10 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 - 10 * q^12 - 6 * q^13 - 14 * q^14 - 2 * q^15 + 22 * q^16 - 8 * q^17 - 2 * q^19 + 10 * q^20 + 2 * q^21 + 14 * q^22 + 8 * q^23 + 2 * q^25 - 14 * q^26 - 2 * q^27 - 10 * q^28 + 2 * q^29 + 12 * q^31 - 6 * q^33 - 2 * q^35 + 10 * q^36 + 2 * q^37 + 6 * q^39 - 14 * q^41 + 14 * q^42 + 6 * q^43 + 30 * q^44 + 2 * q^45 - 28 * q^46 + 8 * q^47 - 22 * q^48 + 2 * q^49 + 8 * q^51 - 30 * q^52 + 12 * q^53 + 6 * q^55 - 42 * q^56 + 2 * q^57 - 42 * q^58 + 12 * q^59 - 10 * q^60 - 12 * q^61 - 2 * q^63 + 26 * q^64 - 6 * q^65 - 14 * q^66 + 8 * q^67 - 40 * q^68 - 8 * q^69 - 14 * q^70 + 4 * q^71 + 20 * q^73 - 14 * q^74 - 2 * q^75 - 10 * q^76 - 20 * q^77 + 14 * q^78 - 8 * q^79 + 22 * q^80 + 2 * q^81 + 14 * q^82 + 12 * q^83 + 10 * q^84 - 8 * q^85 - 14 * q^86 - 2 * q^87 + 42 * q^88 - 18 * q^89 + 20 * q^91 + 40 * q^92 - 12 * q^93 + 28 * q^94 - 2 * q^95 + 10 * q^97 + 28 * q^98 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.64575 −1.87083 −0.935414 0.353553i $$-0.884973\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.00000 2.50000
$$5$$ 1.00000 0.447214
$$6$$ 2.64575 1.08012
$$7$$ 1.64575 0.622036 0.311018 0.950404i $$-0.399330\pi$$
0.311018 + 0.950404i $$0.399330\pi$$
$$8$$ −7.93725 −2.80624
$$9$$ 1.00000 0.333333
$$10$$ −2.64575 −0.836660
$$11$$ 0.354249 0.106810 0.0534050 0.998573i $$-0.482993\pi$$
0.0534050 + 0.998573i $$0.482993\pi$$
$$12$$ −5.00000 −1.44338
$$13$$ −0.354249 −0.0982509 −0.0491255 0.998793i $$-0.515643\pi$$
−0.0491255 + 0.998793i $$0.515643\pi$$
$$14$$ −4.35425 −1.16372
$$15$$ −1.00000 −0.258199
$$16$$ 11.0000 2.75000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −2.64575 −0.623610
$$19$$ −1.00000 −0.229416
$$20$$ 5.00000 1.11803
$$21$$ −1.64575 −0.359132
$$22$$ −0.937254 −0.199823
$$23$$ 9.29150 1.93741 0.968706 0.248211i $$-0.0798425\pi$$
0.968706 + 0.248211i $$0.0798425\pi$$
$$24$$ 7.93725 1.62019
$$25$$ 1.00000 0.200000
$$26$$ 0.937254 0.183811
$$27$$ −1.00000 −0.192450
$$28$$ 8.22876 1.55509
$$29$$ 8.93725 1.65961 0.829803 0.558056i $$-0.188453\pi$$
0.829803 + 0.558056i $$0.188453\pi$$
$$30$$ 2.64575 0.483046
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ −13.2288 −2.33854
$$33$$ −0.354249 −0.0616668
$$34$$ 10.5830 1.81497
$$35$$ 1.64575 0.278183
$$36$$ 5.00000 0.833333
$$37$$ 3.64575 0.599358 0.299679 0.954040i $$-0.403120\pi$$
0.299679 + 0.954040i $$0.403120\pi$$
$$38$$ 2.64575 0.429198
$$39$$ 0.354249 0.0567252
$$40$$ −7.93725 −1.25499
$$41$$ −9.64575 −1.50641 −0.753207 0.657784i $$-0.771494\pi$$
−0.753207 + 0.657784i $$0.771494\pi$$
$$42$$ 4.35425 0.671875
$$43$$ 5.64575 0.860969 0.430485 0.902598i $$-0.358343\pi$$
0.430485 + 0.902598i $$0.358343\pi$$
$$44$$ 1.77124 0.267025
$$45$$ 1.00000 0.149071
$$46$$ −24.5830 −3.62457
$$47$$ −1.29150 −0.188385 −0.0941925 0.995554i $$-0.530027\pi$$
−0.0941925 + 0.995554i $$0.530027\pi$$
$$48$$ −11.0000 −1.58771
$$49$$ −4.29150 −0.613072
$$50$$ −2.64575 −0.374166
$$51$$ 4.00000 0.560112
$$52$$ −1.77124 −0.245627
$$53$$ 11.2915 1.55101 0.775504 0.631343i $$-0.217496\pi$$
0.775504 + 0.631343i $$0.217496\pi$$
$$54$$ 2.64575 0.360041
$$55$$ 0.354249 0.0477669
$$56$$ −13.0627 −1.74558
$$57$$ 1.00000 0.132453
$$58$$ −23.6458 −3.10484
$$59$$ 11.2915 1.47003 0.735014 0.678052i $$-0.237175\pi$$
0.735014 + 0.678052i $$0.237175\pi$$
$$60$$ −5.00000 −0.645497
$$61$$ −11.2915 −1.44573 −0.722864 0.690990i $$-0.757175\pi$$
−0.722864 + 0.690990i $$0.757175\pi$$
$$62$$ −15.8745 −2.01606
$$63$$ 1.64575 0.207345
$$64$$ 13.0000 1.62500
$$65$$ −0.354249 −0.0439391
$$66$$ 0.937254 0.115368
$$67$$ −6.58301 −0.804242 −0.402121 0.915587i $$-0.631727\pi$$
−0.402121 + 0.915587i $$0.631727\pi$$
$$68$$ −20.0000 −2.42536
$$69$$ −9.29150 −1.11857
$$70$$ −4.35425 −0.520432
$$71$$ 7.29150 0.865342 0.432671 0.901552i $$-0.357571\pi$$
0.432671 + 0.901552i $$0.357571\pi$$
$$72$$ −7.93725 −0.935414
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −9.64575 −1.12130
$$75$$ −1.00000 −0.115470
$$76$$ −5.00000 −0.573539
$$77$$ 0.583005 0.0664396
$$78$$ −0.937254 −0.106123
$$79$$ 6.58301 0.740646 0.370323 0.928903i $$-0.379247\pi$$
0.370323 + 0.928903i $$0.379247\pi$$
$$80$$ 11.0000 1.22984
$$81$$ 1.00000 0.111111
$$82$$ 25.5203 2.81824
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ −8.22876 −0.897831
$$85$$ −4.00000 −0.433861
$$86$$ −14.9373 −1.61073
$$87$$ −8.93725 −0.958174
$$88$$ −2.81176 −0.299735
$$89$$ −16.9373 −1.79535 −0.897673 0.440663i $$-0.854743\pi$$
−0.897673 + 0.440663i $$0.854743\pi$$
$$90$$ −2.64575 −0.278887
$$91$$ −0.583005 −0.0611156
$$92$$ 46.4575 4.84353
$$93$$ −6.00000 −0.622171
$$94$$ 3.41699 0.352436
$$95$$ −1.00000 −0.102598
$$96$$ 13.2288 1.35015
$$97$$ −2.93725 −0.298233 −0.149116 0.988820i $$-0.547643\pi$$
−0.149116 + 0.988820i $$0.547643\pi$$
$$98$$ 11.3542 1.14695
$$99$$ 0.354249 0.0356033
$$100$$ 5.00000 0.500000
$$101$$ 9.29150 0.924539 0.462270 0.886739i $$-0.347035\pi$$
0.462270 + 0.886739i $$0.347035\pi$$
$$102$$ −10.5830 −1.04787
$$103$$ −10.5830 −1.04277 −0.521387 0.853320i $$-0.674585\pi$$
−0.521387 + 0.853320i $$0.674585\pi$$
$$104$$ 2.81176 0.275716
$$105$$ −1.64575 −0.160609
$$106$$ −29.8745 −2.90167
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ 5.29150 0.506834 0.253417 0.967357i $$-0.418446\pi$$
0.253417 + 0.967357i $$0.418446\pi$$
$$110$$ −0.937254 −0.0893637
$$111$$ −3.64575 −0.346039
$$112$$ 18.1033 1.71060
$$113$$ −4.00000 −0.376288 −0.188144 0.982141i $$-0.560247\pi$$
−0.188144 + 0.982141i $$0.560247\pi$$
$$114$$ −2.64575 −0.247797
$$115$$ 9.29150 0.866437
$$116$$ 44.6863 4.14902
$$117$$ −0.354249 −0.0327503
$$118$$ −29.8745 −2.75017
$$119$$ −6.58301 −0.603463
$$120$$ 7.93725 0.724569
$$121$$ −10.8745 −0.988592
$$122$$ 29.8745 2.70471
$$123$$ 9.64575 0.869728
$$124$$ 30.0000 2.69408
$$125$$ 1.00000 0.0894427
$$126$$ −4.35425 −0.387907
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −7.93725 −0.701561
$$129$$ −5.64575 −0.497081
$$130$$ 0.937254 0.0822026
$$131$$ −14.2288 −1.24317 −0.621586 0.783346i $$-0.713511\pi$$
−0.621586 + 0.783346i $$0.713511\pi$$
$$132$$ −1.77124 −0.154167
$$133$$ −1.64575 −0.142705
$$134$$ 17.4170 1.50460
$$135$$ −1.00000 −0.0860663
$$136$$ 31.7490 2.72246
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 24.5830 2.09264
$$139$$ 17.8745 1.51610 0.758048 0.652199i $$-0.226153\pi$$
0.758048 + 0.652199i $$0.226153\pi$$
$$140$$ 8.22876 0.695457
$$141$$ 1.29150 0.108764
$$142$$ −19.2915 −1.61891
$$143$$ −0.125492 −0.0104942
$$144$$ 11.0000 0.916667
$$145$$ 8.93725 0.742199
$$146$$ −26.4575 −2.18964
$$147$$ 4.29150 0.353957
$$148$$ 18.2288 1.49839
$$149$$ −20.5830 −1.68623 −0.843113 0.537737i $$-0.819279\pi$$
−0.843113 + 0.537737i $$0.819279\pi$$
$$150$$ 2.64575 0.216025
$$151$$ −8.58301 −0.698475 −0.349238 0.937034i $$-0.613559\pi$$
−0.349238 + 0.937034i $$0.613559\pi$$
$$152$$ 7.93725 0.643796
$$153$$ −4.00000 −0.323381
$$154$$ −1.54249 −0.124297
$$155$$ 6.00000 0.481932
$$156$$ 1.77124 0.141813
$$157$$ −13.2915 −1.06078 −0.530389 0.847755i $$-0.677954\pi$$
−0.530389 + 0.847755i $$0.677954\pi$$
$$158$$ −17.4170 −1.38562
$$159$$ −11.2915 −0.895474
$$160$$ −13.2288 −1.04583
$$161$$ 15.2915 1.20514
$$162$$ −2.64575 −0.207870
$$163$$ −2.35425 −0.184399 −0.0921995 0.995741i $$-0.529390\pi$$
−0.0921995 + 0.995741i $$0.529390\pi$$
$$164$$ −48.2288 −3.76603
$$165$$ −0.354249 −0.0275782
$$166$$ −15.8745 −1.23210
$$167$$ −21.2915 −1.64759 −0.823793 0.566891i $$-0.808146\pi$$
−0.823793 + 0.566891i $$0.808146\pi$$
$$168$$ 13.0627 1.00781
$$169$$ −12.8745 −0.990347
$$170$$ 10.5830 0.811679
$$171$$ −1.00000 −0.0764719
$$172$$ 28.2288 2.15242
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 23.6458 1.79258
$$175$$ 1.64575 0.124407
$$176$$ 3.89674 0.293727
$$177$$ −11.2915 −0.848721
$$178$$ 44.8118 3.35878
$$179$$ −7.29150 −0.544992 −0.272496 0.962157i $$-0.587849\pi$$
−0.272496 + 0.962157i $$0.587849\pi$$
$$180$$ 5.00000 0.372678
$$181$$ 17.2915 1.28527 0.642634 0.766174i $$-0.277842\pi$$
0.642634 + 0.766174i $$0.277842\pi$$
$$182$$ 1.54249 0.114337
$$183$$ 11.2915 0.834692
$$184$$ −73.7490 −5.43685
$$185$$ 3.64575 0.268041
$$186$$ 15.8745 1.16398
$$187$$ −1.41699 −0.103621
$$188$$ −6.45751 −0.470963
$$189$$ −1.64575 −0.119711
$$190$$ 2.64575 0.191943
$$191$$ 6.22876 0.450697 0.225349 0.974278i $$-0.427648\pi$$
0.225349 + 0.974278i $$0.427648\pi$$
$$192$$ −13.0000 −0.938194
$$193$$ −11.6458 −0.838280 −0.419140 0.907922i $$-0.637668\pi$$
−0.419140 + 0.907922i $$0.637668\pi$$
$$194$$ 7.77124 0.557943
$$195$$ 0.354249 0.0253683
$$196$$ −21.4575 −1.53268
$$197$$ −25.1660 −1.79300 −0.896502 0.443040i $$-0.853900\pi$$
−0.896502 + 0.443040i $$0.853900\pi$$
$$198$$ −0.937254 −0.0666077
$$199$$ 7.29150 0.516881 0.258440 0.966027i $$-0.416791\pi$$
0.258440 + 0.966027i $$0.416791\pi$$
$$200$$ −7.93725 −0.561249
$$201$$ 6.58301 0.464329
$$202$$ −24.5830 −1.72965
$$203$$ 14.7085 1.03233
$$204$$ 20.0000 1.40028
$$205$$ −9.64575 −0.673688
$$206$$ 28.0000 1.95085
$$207$$ 9.29150 0.645804
$$208$$ −3.89674 −0.270190
$$209$$ −0.354249 −0.0245039
$$210$$ 4.35425 0.300472
$$211$$ 2.58301 0.177821 0.0889107 0.996040i $$-0.471661\pi$$
0.0889107 + 0.996040i $$0.471661\pi$$
$$212$$ 56.4575 3.87752
$$213$$ −7.29150 −0.499606
$$214$$ 0 0
$$215$$ 5.64575 0.385037
$$216$$ 7.93725 0.540062
$$217$$ 9.87451 0.670325
$$218$$ −14.0000 −0.948200
$$219$$ −10.0000 −0.675737
$$220$$ 1.77124 0.119417
$$221$$ 1.41699 0.0953174
$$222$$ 9.64575 0.647380
$$223$$ 2.58301 0.172971 0.0864854 0.996253i $$-0.472436\pi$$
0.0864854 + 0.996253i $$0.472436\pi$$
$$224$$ −21.7712 −1.45465
$$225$$ 1.00000 0.0666667
$$226$$ 10.5830 0.703971
$$227$$ −10.7085 −0.710748 −0.355374 0.934724i $$-0.615646\pi$$
−0.355374 + 0.934724i $$0.615646\pi$$
$$228$$ 5.00000 0.331133
$$229$$ 8.70850 0.575474 0.287737 0.957710i $$-0.407097\pi$$
0.287737 + 0.957710i $$0.407097\pi$$
$$230$$ −24.5830 −1.62096
$$231$$ −0.583005 −0.0383589
$$232$$ −70.9373 −4.65726
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0.937254 0.0612702
$$235$$ −1.29150 −0.0842483
$$236$$ 56.4575 3.67507
$$237$$ −6.58301 −0.427612
$$238$$ 17.4170 1.12898
$$239$$ 15.6458 1.01204 0.506020 0.862522i $$-0.331116\pi$$
0.506020 + 0.862522i $$0.331116\pi$$
$$240$$ −11.0000 −0.710047
$$241$$ 16.5830 1.06821 0.534103 0.845420i $$-0.320650\pi$$
0.534103 + 0.845420i $$0.320650\pi$$
$$242$$ 28.7712 1.84949
$$243$$ −1.00000 −0.0641500
$$244$$ −56.4575 −3.61432
$$245$$ −4.29150 −0.274174
$$246$$ −25.5203 −1.62711
$$247$$ 0.354249 0.0225403
$$248$$ −47.6235 −3.02410
$$249$$ −6.00000 −0.380235
$$250$$ −2.64575 −0.167332
$$251$$ −16.3542 −1.03227 −0.516136 0.856507i $$-0.672630\pi$$
−0.516136 + 0.856507i $$0.672630\pi$$
$$252$$ 8.22876 0.518363
$$253$$ 3.29150 0.206935
$$254$$ −10.5830 −0.664037
$$255$$ 4.00000 0.250490
$$256$$ −5.00000 −0.312500
$$257$$ −5.41699 −0.337903 −0.168951 0.985624i $$-0.554038\pi$$
−0.168951 + 0.985624i $$0.554038\pi$$
$$258$$ 14.9373 0.929953
$$259$$ 6.00000 0.372822
$$260$$ −1.77124 −0.109848
$$261$$ 8.93725 0.553202
$$262$$ 37.6458 2.32576
$$263$$ 4.58301 0.282600 0.141300 0.989967i $$-0.454872\pi$$
0.141300 + 0.989967i $$0.454872\pi$$
$$264$$ 2.81176 0.173052
$$265$$ 11.2915 0.693631
$$266$$ 4.35425 0.266976
$$267$$ 16.9373 1.03654
$$268$$ −32.9150 −2.01061
$$269$$ 4.22876 0.257832 0.128916 0.991656i $$-0.458850\pi$$
0.128916 + 0.991656i $$0.458850\pi$$
$$270$$ 2.64575 0.161015
$$271$$ −4.70850 −0.286021 −0.143010 0.989721i $$-0.545678\pi$$
−0.143010 + 0.989721i $$0.545678\pi$$
$$272$$ −44.0000 −2.66789
$$273$$ 0.583005 0.0352851
$$274$$ −15.8745 −0.959014
$$275$$ 0.354249 0.0213620
$$276$$ −46.4575 −2.79641
$$277$$ 0.583005 0.0350294 0.0175147 0.999847i $$-0.494425\pi$$
0.0175147 + 0.999847i $$0.494425\pi$$
$$278$$ −47.2915 −2.83636
$$279$$ 6.00000 0.359211
$$280$$ −13.0627 −0.780648
$$281$$ −32.2288 −1.92261 −0.961303 0.275493i $$-0.911159\pi$$
−0.961303 + 0.275493i $$0.911159\pi$$
$$282$$ −3.41699 −0.203479
$$283$$ −11.5203 −0.684808 −0.342404 0.939553i $$-0.611241\pi$$
−0.342404 + 0.939553i $$0.611241\pi$$
$$284$$ 36.4575 2.16336
$$285$$ 1.00000 0.0592349
$$286$$ 0.332021 0.0196328
$$287$$ −15.8745 −0.937043
$$288$$ −13.2288 −0.779512
$$289$$ −1.00000 −0.0588235
$$290$$ −23.6458 −1.38853
$$291$$ 2.93725 0.172185
$$292$$ 50.0000 2.92603
$$293$$ 5.41699 0.316464 0.158232 0.987402i $$-0.449421\pi$$
0.158232 + 0.987402i $$0.449421\pi$$
$$294$$ −11.3542 −0.662193
$$295$$ 11.2915 0.657417
$$296$$ −28.9373 −1.68194
$$297$$ −0.354249 −0.0205556
$$298$$ 54.4575 3.15464
$$299$$ −3.29150 −0.190353
$$300$$ −5.00000 −0.288675
$$301$$ 9.29150 0.535553
$$302$$ 22.7085 1.30673
$$303$$ −9.29150 −0.533783
$$304$$ −11.0000 −0.630893
$$305$$ −11.2915 −0.646550
$$306$$ 10.5830 0.604990
$$307$$ −24.4575 −1.39586 −0.697932 0.716164i $$-0.745896\pi$$
−0.697932 + 0.716164i $$0.745896\pi$$
$$308$$ 2.91503 0.166099
$$309$$ 10.5830 0.602046
$$310$$ −15.8745 −0.901611
$$311$$ 22.9373 1.30065 0.650326 0.759655i $$-0.274632\pi$$
0.650326 + 0.759655i $$0.274632\pi$$
$$312$$ −2.81176 −0.159185
$$313$$ 17.2915 0.977374 0.488687 0.872459i $$-0.337476\pi$$
0.488687 + 0.872459i $$0.337476\pi$$
$$314$$ 35.1660 1.98453
$$315$$ 1.64575 0.0927276
$$316$$ 32.9150 1.85161
$$317$$ 20.4575 1.14901 0.574504 0.818502i $$-0.305195\pi$$
0.574504 + 0.818502i $$0.305195\pi$$
$$318$$ 29.8745 1.67528
$$319$$ 3.16601 0.177263
$$320$$ 13.0000 0.726722
$$321$$ 0 0
$$322$$ −40.4575 −2.25461
$$323$$ 4.00000 0.222566
$$324$$ 5.00000 0.277778
$$325$$ −0.354249 −0.0196502
$$326$$ 6.22876 0.344979
$$327$$ −5.29150 −0.292621
$$328$$ 76.5608 4.22736
$$329$$ −2.12549 −0.117182
$$330$$ 0.937254 0.0515941
$$331$$ 11.4170 0.627535 0.313767 0.949500i $$-0.398409\pi$$
0.313767 + 0.949500i $$0.398409\pi$$
$$332$$ 30.0000 1.64646
$$333$$ 3.64575 0.199786
$$334$$ 56.3320 3.08235
$$335$$ −6.58301 −0.359668
$$336$$ −18.1033 −0.987614
$$337$$ 14.9373 0.813684 0.406842 0.913499i $$-0.366630\pi$$
0.406842 + 0.913499i $$0.366630\pi$$
$$338$$ 34.0627 1.85277
$$339$$ 4.00000 0.217250
$$340$$ −20.0000 −1.08465
$$341$$ 2.12549 0.115102
$$342$$ 2.64575 0.143066
$$343$$ −18.5830 −1.00339
$$344$$ −44.8118 −2.41609
$$345$$ −9.29150 −0.500238
$$346$$ 0 0
$$347$$ 26.0000 1.39575 0.697877 0.716218i $$-0.254128\pi$$
0.697877 + 0.716218i $$0.254128\pi$$
$$348$$ −44.6863 −2.39544
$$349$$ 23.1660 1.24005 0.620024 0.784583i $$-0.287123\pi$$
0.620024 + 0.784583i $$0.287123\pi$$
$$350$$ −4.35425 −0.232744
$$351$$ 0.354249 0.0189084
$$352$$ −4.68627 −0.249779
$$353$$ 0.583005 0.0310302 0.0155151 0.999880i $$-0.495061\pi$$
0.0155151 + 0.999880i $$0.495061\pi$$
$$354$$ 29.8745 1.58781
$$355$$ 7.29150 0.386993
$$356$$ −84.6863 −4.48836
$$357$$ 6.58301 0.348410
$$358$$ 19.2915 1.01959
$$359$$ −36.1033 −1.90546 −0.952729 0.303822i $$-0.901737\pi$$
−0.952729 + 0.303822i $$0.901737\pi$$
$$360$$ −7.93725 −0.418330
$$361$$ 1.00000 0.0526316
$$362$$ −45.7490 −2.40451
$$363$$ 10.8745 0.570764
$$364$$ −2.91503 −0.152789
$$365$$ 10.0000 0.523424
$$366$$ −29.8745 −1.56157
$$367$$ 1.64575 0.0859075 0.0429538 0.999077i $$-0.486323\pi$$
0.0429538 + 0.999077i $$0.486323\pi$$
$$368$$ 102.207 5.32788
$$369$$ −9.64575 −0.502138
$$370$$ −9.64575 −0.501459
$$371$$ 18.5830 0.964782
$$372$$ −30.0000 −1.55543
$$373$$ 5.06275 0.262139 0.131070 0.991373i $$-0.458159\pi$$
0.131070 + 0.991373i $$0.458159\pi$$
$$374$$ 3.74902 0.193857
$$375$$ −1.00000 −0.0516398
$$376$$ 10.2510 0.528654
$$377$$ −3.16601 −0.163058
$$378$$ 4.35425 0.223958
$$379$$ 10.0000 0.513665 0.256833 0.966456i $$-0.417321\pi$$
0.256833 + 0.966456i $$0.417321\pi$$
$$380$$ −5.00000 −0.256495
$$381$$ −4.00000 −0.204926
$$382$$ −16.4797 −0.843177
$$383$$ −18.5830 −0.949547 −0.474774 0.880108i $$-0.657470\pi$$
−0.474774 + 0.880108i $$0.657470\pi$$
$$384$$ 7.93725 0.405046
$$385$$ 0.583005 0.0297127
$$386$$ 30.8118 1.56828
$$387$$ 5.64575 0.286990
$$388$$ −14.6863 −0.745582
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ −0.937254 −0.0474597
$$391$$ −37.1660 −1.87957
$$392$$ 34.0627 1.72043
$$393$$ 14.2288 0.717746
$$394$$ 66.5830 3.35440
$$395$$ 6.58301 0.331227
$$396$$ 1.77124 0.0890083
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −19.2915 −0.966996
$$399$$ 1.64575 0.0823906
$$400$$ 11.0000 0.550000
$$401$$ −4.93725 −0.246555 −0.123277 0.992372i $$-0.539340\pi$$
−0.123277 + 0.992372i $$0.539340\pi$$
$$402$$ −17.4170 −0.868681
$$403$$ −2.12549 −0.105878
$$404$$ 46.4575 2.31135
$$405$$ 1.00000 0.0496904
$$406$$ −38.9150 −1.93132
$$407$$ 1.29150 0.0640174
$$408$$ −31.7490 −1.57181
$$409$$ −6.70850 −0.331714 −0.165857 0.986150i $$-0.553039\pi$$
−0.165857 + 0.986150i $$0.553039\pi$$
$$410$$ 25.5203 1.26036
$$411$$ −6.00000 −0.295958
$$412$$ −52.9150 −2.60694
$$413$$ 18.5830 0.914410
$$414$$ −24.5830 −1.20819
$$415$$ 6.00000 0.294528
$$416$$ 4.68627 0.229763
$$417$$ −17.8745 −0.875318
$$418$$ 0.937254 0.0458426
$$419$$ 38.9373 1.90221 0.951105 0.308869i $$-0.0999504\pi$$
0.951105 + 0.308869i $$0.0999504\pi$$
$$420$$ −8.22876 −0.401522
$$421$$ 28.5830 1.39305 0.696525 0.717532i $$-0.254728\pi$$
0.696525 + 0.717532i $$0.254728\pi$$
$$422$$ −6.83399 −0.332673
$$423$$ −1.29150 −0.0627950
$$424$$ −89.6235 −4.35250
$$425$$ −4.00000 −0.194029
$$426$$ 19.2915 0.934676
$$427$$ −18.5830 −0.899295
$$428$$ 0 0
$$429$$ 0.125492 0.00605882
$$430$$ −14.9373 −0.720338
$$431$$ 3.29150 0.158546 0.0792731 0.996853i $$-0.474740\pi$$
0.0792731 + 0.996853i $$0.474740\pi$$
$$432$$ −11.0000 −0.529238
$$433$$ −27.6458 −1.32857 −0.664285 0.747479i $$-0.731264\pi$$
−0.664285 + 0.747479i $$0.731264\pi$$
$$434$$ −26.1255 −1.25406
$$435$$ −8.93725 −0.428509
$$436$$ 26.4575 1.26709
$$437$$ −9.29150 −0.444473
$$438$$ 26.4575 1.26419
$$439$$ −1.41699 −0.0676295 −0.0338147 0.999428i $$-0.510766\pi$$
−0.0338147 + 0.999428i $$0.510766\pi$$
$$440$$ −2.81176 −0.134045
$$441$$ −4.29150 −0.204357
$$442$$ −3.74902 −0.178322
$$443$$ 26.7085 1.26896 0.634480 0.772940i $$-0.281214\pi$$
0.634480 + 0.772940i $$0.281214\pi$$
$$444$$ −18.2288 −0.865099
$$445$$ −16.9373 −0.802903
$$446$$ −6.83399 −0.323599
$$447$$ 20.5830 0.973543
$$448$$ 21.3948 1.01081
$$449$$ −36.2288 −1.70974 −0.854870 0.518842i $$-0.826363\pi$$
−0.854870 + 0.518842i $$0.826363\pi$$
$$450$$ −2.64575 −0.124722
$$451$$ −3.41699 −0.160900
$$452$$ −20.0000 −0.940721
$$453$$ 8.58301 0.403265
$$454$$ 28.3320 1.32969
$$455$$ −0.583005 −0.0273317
$$456$$ −7.93725 −0.371696
$$457$$ 0.125492 0.00587027 0.00293514 0.999996i $$-0.499066\pi$$
0.00293514 + 0.999996i $$0.499066\pi$$
$$458$$ −23.0405 −1.07661
$$459$$ 4.00000 0.186704
$$460$$ 46.4575 2.16609
$$461$$ −37.7490 −1.75815 −0.879073 0.476686i $$-0.841838\pi$$
−0.879073 + 0.476686i $$0.841838\pi$$
$$462$$ 1.54249 0.0717630
$$463$$ −35.5203 −1.65077 −0.825383 0.564573i $$-0.809041\pi$$
−0.825383 + 0.564573i $$0.809041\pi$$
$$464$$ 98.3098 4.56392
$$465$$ −6.00000 −0.278243
$$466$$ 47.6235 2.20612
$$467$$ 19.8745 0.919683 0.459841 0.888001i $$-0.347906\pi$$
0.459841 + 0.888001i $$0.347906\pi$$
$$468$$ −1.77124 −0.0818758
$$469$$ −10.8340 −0.500267
$$470$$ 3.41699 0.157614
$$471$$ 13.2915 0.612440
$$472$$ −89.6235 −4.12526
$$473$$ 2.00000 0.0919601
$$474$$ 17.4170 0.799989
$$475$$ −1.00000 −0.0458831
$$476$$ −32.9150 −1.50866
$$477$$ 11.2915 0.517002
$$478$$ −41.3948 −1.89335
$$479$$ 4.35425 0.198951 0.0994754 0.995040i $$-0.468284\pi$$
0.0994754 + 0.995040i $$0.468284\pi$$
$$480$$ 13.2288 0.603807
$$481$$ −1.29150 −0.0588875
$$482$$ −43.8745 −1.99843
$$483$$ −15.2915 −0.695787
$$484$$ −54.3725 −2.47148
$$485$$ −2.93725 −0.133374
$$486$$ 2.64575 0.120014
$$487$$ 17.8745 0.809971 0.404986 0.914323i $$-0.367277\pi$$
0.404986 + 0.914323i $$0.367277\pi$$
$$488$$ 89.6235 4.05707
$$489$$ 2.35425 0.106463
$$490$$ 11.3542 0.512933
$$491$$ −5.77124 −0.260453 −0.130226 0.991484i $$-0.541570\pi$$
−0.130226 + 0.991484i $$0.541570\pi$$
$$492$$ 48.2288 2.17432
$$493$$ −35.7490 −1.61005
$$494$$ −0.937254 −0.0421690
$$495$$ 0.354249 0.0159223
$$496$$ 66.0000 2.96349
$$497$$ 12.0000 0.538274
$$498$$ 15.8745 0.711354
$$499$$ −31.0405 −1.38956 −0.694782 0.719220i $$-0.744499\pi$$
−0.694782 + 0.719220i $$0.744499\pi$$
$$500$$ 5.00000 0.223607
$$501$$ 21.2915 0.951234
$$502$$ 43.2693 1.93120
$$503$$ −2.00000 −0.0891756 −0.0445878 0.999005i $$-0.514197\pi$$
−0.0445878 + 0.999005i $$0.514197\pi$$
$$504$$ −13.0627 −0.581861
$$505$$ 9.29150 0.413466
$$506$$ −8.70850 −0.387140
$$507$$ 12.8745 0.571777
$$508$$ 20.0000 0.887357
$$509$$ −34.1033 −1.51160 −0.755800 0.654802i $$-0.772752\pi$$
−0.755800 + 0.654802i $$0.772752\pi$$
$$510$$ −10.5830 −0.468623
$$511$$ 16.4575 0.728038
$$512$$ 29.1033 1.28619
$$513$$ 1.00000 0.0441511
$$514$$ 14.3320 0.632158
$$515$$ −10.5830 −0.466343
$$516$$ −28.2288 −1.24270
$$517$$ −0.457513 −0.0201214
$$518$$ −15.8745 −0.697486
$$519$$ 0 0
$$520$$ 2.81176 0.123304
$$521$$ 22.1033 0.968362 0.484181 0.874968i $$-0.339118\pi$$
0.484181 + 0.874968i $$0.339118\pi$$
$$522$$ −23.6458 −1.03495
$$523$$ −23.2915 −1.01847 −0.509233 0.860629i $$-0.670071\pi$$
−0.509233 + 0.860629i $$0.670071\pi$$
$$524$$ −71.1438 −3.10793
$$525$$ −1.64575 −0.0718265
$$526$$ −12.1255 −0.528697
$$527$$ −24.0000 −1.04546
$$528$$ −3.89674 −0.169584
$$529$$ 63.3320 2.75357
$$530$$ −29.8745 −1.29767
$$531$$ 11.2915 0.490009
$$532$$ −8.22876 −0.356762
$$533$$ 3.41699 0.148006
$$534$$ −44.8118 −1.93919
$$535$$ 0 0
$$536$$ 52.2510 2.25690
$$537$$ 7.29150 0.314652
$$538$$ −11.1882 −0.482359
$$539$$ −1.52026 −0.0654822
$$540$$ −5.00000 −0.215166
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 12.4575 0.535096
$$543$$ −17.2915 −0.742049
$$544$$ 52.9150 2.26871
$$545$$ 5.29150 0.226663
$$546$$ −1.54249 −0.0660123
$$547$$ 1.87451 0.0801482 0.0400741 0.999197i $$-0.487241\pi$$
0.0400741 + 0.999197i $$0.487241\pi$$
$$548$$ 30.0000 1.28154
$$549$$ −11.2915 −0.481910
$$550$$ −0.937254 −0.0399646
$$551$$ −8.93725 −0.380740
$$552$$ 73.7490 3.13897
$$553$$ 10.8340 0.460708
$$554$$ −1.54249 −0.0655340
$$555$$ −3.64575 −0.154754
$$556$$ 89.3725 3.79024
$$557$$ −11.4170 −0.483754 −0.241877 0.970307i $$-0.577763\pi$$
−0.241877 + 0.970307i $$0.577763\pi$$
$$558$$ −15.8745 −0.672022
$$559$$ −2.00000 −0.0845910
$$560$$ 18.1033 0.765003
$$561$$ 1.41699 0.0598256
$$562$$ 85.2693 3.59687
$$563$$ 8.12549 0.342449 0.171224 0.985232i $$-0.445228\pi$$
0.171224 + 0.985232i $$0.445228\pi$$
$$564$$ 6.45751 0.271910
$$565$$ −4.00000 −0.168281
$$566$$ 30.4797 1.28116
$$567$$ 1.64575 0.0691151
$$568$$ −57.8745 −2.42836
$$569$$ −7.06275 −0.296086 −0.148043 0.988981i $$-0.547297\pi$$
−0.148043 + 0.988981i $$0.547297\pi$$
$$570$$ −2.64575 −0.110818
$$571$$ 16.7085 0.699229 0.349614 0.936894i $$-0.386313\pi$$
0.349614 + 0.936894i $$0.386313\pi$$
$$572$$ −0.627461 −0.0262354
$$573$$ −6.22876 −0.260210
$$574$$ 42.0000 1.75305
$$575$$ 9.29150 0.387482
$$576$$ 13.0000 0.541667
$$577$$ −13.2915 −0.553332 −0.276666 0.960966i $$-0.589230\pi$$
−0.276666 + 0.960966i $$0.589230\pi$$
$$578$$ 2.64575 0.110049
$$579$$ 11.6458 0.483981
$$580$$ 44.6863 1.85550
$$581$$ 9.87451 0.409664
$$582$$ −7.77124 −0.322128
$$583$$ 4.00000 0.165663
$$584$$ −79.3725 −3.28446
$$585$$ −0.354249 −0.0146464
$$586$$ −14.3320 −0.592050
$$587$$ −33.2915 −1.37409 −0.687044 0.726616i $$-0.741092\pi$$
−0.687044 + 0.726616i $$0.741092\pi$$
$$588$$ 21.4575 0.884893
$$589$$ −6.00000 −0.247226
$$590$$ −29.8745 −1.22991
$$591$$ 25.1660 1.03519
$$592$$ 40.1033 1.64823
$$593$$ 3.41699 0.140319 0.0701596 0.997536i $$-0.477649\pi$$
0.0701596 + 0.997536i $$0.477649\pi$$
$$594$$ 0.937254 0.0384560
$$595$$ −6.58301 −0.269877
$$596$$ −102.915 −4.21556
$$597$$ −7.29150 −0.298421
$$598$$ 8.70850 0.356117
$$599$$ 9.41699 0.384768 0.192384 0.981320i $$-0.438378\pi$$
0.192384 + 0.981320i $$0.438378\pi$$
$$600$$ 7.93725 0.324037
$$601$$ 22.7085 0.926299 0.463149 0.886280i $$-0.346719\pi$$
0.463149 + 0.886280i $$0.346719\pi$$
$$602$$ −24.5830 −1.00193
$$603$$ −6.58301 −0.268081
$$604$$ −42.9150 −1.74619
$$605$$ −10.8745 −0.442112
$$606$$ 24.5830 0.998616
$$607$$ −43.0405 −1.74696 −0.873480 0.486859i $$-0.838142\pi$$
−0.873480 + 0.486859i $$0.838142\pi$$
$$608$$ 13.2288 0.536497
$$609$$ −14.7085 −0.596018
$$610$$ 29.8745 1.20958
$$611$$ 0.457513 0.0185090
$$612$$ −20.0000 −0.808452
$$613$$ −30.4575 −1.23017 −0.615084 0.788462i $$-0.710878\pi$$
−0.615084 + 0.788462i $$0.710878\pi$$
$$614$$ 64.7085 2.61142
$$615$$ 9.64575 0.388954
$$616$$ −4.62746 −0.186446
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ −28.0000 −1.12633
$$619$$ 23.2915 0.936165 0.468082 0.883685i $$-0.344945\pi$$
0.468082 + 0.883685i $$0.344945\pi$$
$$620$$ 30.0000 1.20483
$$621$$ −9.29150 −0.372855
$$622$$ −60.6863 −2.43330
$$623$$ −27.8745 −1.11677
$$624$$ 3.89674 0.155994
$$625$$ 1.00000 0.0400000
$$626$$ −45.7490 −1.82850
$$627$$ 0.354249 0.0141473
$$628$$ −66.4575 −2.65194
$$629$$ −14.5830 −0.581462
$$630$$ −4.35425 −0.173477
$$631$$ −14.5830 −0.580540 −0.290270 0.956945i $$-0.593745\pi$$
−0.290270 + 0.956945i $$0.593745\pi$$
$$632$$ −52.2510 −2.07843
$$633$$ −2.58301 −0.102665
$$634$$ −54.1255 −2.14960
$$635$$ 4.00000 0.158735
$$636$$ −56.4575 −2.23869
$$637$$ 1.52026 0.0602349
$$638$$ −8.37648 −0.331628
$$639$$ 7.29150 0.288447
$$640$$ −7.93725 −0.313748
$$641$$ −16.9373 −0.668981 −0.334491 0.942399i $$-0.608564\pi$$
−0.334491 + 0.942399i $$0.608564\pi$$
$$642$$ 0 0
$$643$$ −13.6458 −0.538136 −0.269068 0.963121i $$-0.586716\pi$$
−0.269068 + 0.963121i $$0.586716\pi$$
$$644$$ 76.4575 3.01285
$$645$$ −5.64575 −0.222301
$$646$$ −10.5830 −0.416383
$$647$$ −3.41699 −0.134336 −0.0671680 0.997742i $$-0.521396\pi$$
−0.0671680 + 0.997742i $$0.521396\pi$$
$$648$$ −7.93725 −0.311805
$$649$$ 4.00000 0.157014
$$650$$ 0.937254 0.0367621
$$651$$ −9.87451 −0.387012
$$652$$ −11.7712 −0.460997
$$653$$ 9.41699 0.368515 0.184258 0.982878i $$-0.441012\pi$$
0.184258 + 0.982878i $$0.441012\pi$$
$$654$$ 14.0000 0.547443
$$655$$ −14.2288 −0.555964
$$656$$ −106.103 −4.14264
$$657$$ 10.0000 0.390137
$$658$$ 5.62352 0.219228
$$659$$ 25.4170 0.990106 0.495053 0.868863i $$-0.335149\pi$$
0.495053 + 0.868863i $$0.335149\pi$$
$$660$$ −1.77124 −0.0689456
$$661$$ 1.29150 0.0502336 0.0251168 0.999685i $$-0.492004\pi$$
0.0251168 + 0.999685i $$0.492004\pi$$
$$662$$ −30.2065 −1.17401
$$663$$ −1.41699 −0.0550315
$$664$$ −47.6235 −1.84815
$$665$$ −1.64575 −0.0638195
$$666$$ −9.64575 −0.373765
$$667$$ 83.0405 3.21534
$$668$$ −106.458 −4.11896
$$669$$ −2.58301 −0.0998648
$$670$$ 17.4170 0.672877
$$671$$ −4.00000 −0.154418
$$672$$ 21.7712 0.839844
$$673$$ −13.0627 −0.503532 −0.251766 0.967788i $$-0.581011\pi$$
−0.251766 + 0.967788i $$0.581011\pi$$
$$674$$ −39.5203 −1.52226
$$675$$ −1.00000 −0.0384900
$$676$$ −64.3725 −2.47587
$$677$$ 40.4575 1.55491 0.777454 0.628939i $$-0.216511\pi$$
0.777454 + 0.628939i $$0.216511\pi$$
$$678$$ −10.5830 −0.406438
$$679$$ −4.83399 −0.185511
$$680$$ 31.7490 1.21752
$$681$$ 10.7085 0.410351
$$682$$ −5.62352 −0.215336
$$683$$ 19.7490 0.755675 0.377838 0.925872i $$-0.376668\pi$$
0.377838 + 0.925872i $$0.376668\pi$$
$$684$$ −5.00000 −0.191180
$$685$$ 6.00000 0.229248
$$686$$ 49.1660 1.87717
$$687$$ −8.70850 −0.332250
$$688$$ 62.1033 2.36766
$$689$$ −4.00000 −0.152388
$$690$$ 24.5830 0.935859
$$691$$ −43.0405 −1.63734 −0.818669 0.574265i $$-0.805288\pi$$
−0.818669 + 0.574265i $$0.805288\pi$$
$$692$$ 0 0
$$693$$ 0.583005 0.0221465
$$694$$ −68.7895 −2.61122
$$695$$ 17.8745 0.678019
$$696$$ 70.9373 2.68887
$$697$$ 38.5830 1.46144
$$698$$ −61.2915 −2.31992
$$699$$ 18.0000 0.680823
$$700$$ 8.22876 0.311018
$$701$$ 16.5830 0.626331 0.313166 0.949698i $$-0.398610\pi$$
0.313166 + 0.949698i $$0.398610\pi$$
$$702$$ −0.937254 −0.0353744
$$703$$ −3.64575 −0.137502
$$704$$ 4.60523 0.173566
$$705$$ 1.29150 0.0486408
$$706$$ −1.54249 −0.0580523
$$707$$ 15.2915 0.575096
$$708$$ −56.4575 −2.12180
$$709$$ −4.70850 −0.176831 −0.0884157 0.996084i $$-0.528180\pi$$
−0.0884157 + 0.996084i $$0.528180\pi$$
$$710$$ −19.2915 −0.723997
$$711$$ 6.58301 0.246882
$$712$$ 134.435 5.03818
$$713$$ 55.7490 2.08782
$$714$$ −17.4170 −0.651815
$$715$$ −0.125492 −0.00469314
$$716$$ −36.4575 −1.36248
$$717$$ −15.6458 −0.584301
$$718$$ 95.5203 3.56478
$$719$$ 32.8118 1.22367 0.611836 0.790985i $$-0.290431\pi$$
0.611836 + 0.790985i $$0.290431\pi$$
$$720$$ 11.0000 0.409946
$$721$$ −17.4170 −0.648643
$$722$$ −2.64575 −0.0984647
$$723$$ −16.5830 −0.616729
$$724$$ 86.4575 3.21317
$$725$$ 8.93725 0.331921
$$726$$ −28.7712 −1.06780
$$727$$ 14.1033 0.523061 0.261531 0.965195i $$-0.415773\pi$$
0.261531 + 0.965195i $$0.415773\pi$$
$$728$$ 4.62746 0.171505
$$729$$ 1.00000 0.0370370
$$730$$ −26.4575 −0.979236
$$731$$ −22.5830 −0.835263
$$732$$ 56.4575 2.08673
$$733$$ 7.41699 0.273953 0.136976 0.990574i $$-0.456262\pi$$
0.136976 + 0.990574i $$0.456262\pi$$
$$734$$ −4.35425 −0.160718
$$735$$ 4.29150 0.158294
$$736$$ −122.915 −4.53071
$$737$$ −2.33202 −0.0859011
$$738$$ 25.5203 0.939414
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 18.2288 0.670102
$$741$$ −0.354249 −0.0130137
$$742$$ −49.1660 −1.80494
$$743$$ −10.5830 −0.388253 −0.194126 0.980977i $$-0.562187\pi$$
−0.194126 + 0.980977i $$0.562187\pi$$
$$744$$ 47.6235 1.74596
$$745$$ −20.5830 −0.754103
$$746$$ −13.3948 −0.490417
$$747$$ 6.00000 0.219529
$$748$$ −7.08497 −0.259052
$$749$$ 0 0
$$750$$ 2.64575 0.0966092
$$751$$ −44.5830 −1.62686 −0.813428 0.581665i $$-0.802401\pi$$
−0.813428 + 0.581665i $$0.802401\pi$$
$$752$$ −14.2065 −0.518059
$$753$$ 16.3542 0.595982
$$754$$ 8.37648 0.305053
$$755$$ −8.58301 −0.312368
$$756$$ −8.22876 −0.299277
$$757$$ −23.8745 −0.867734 −0.433867 0.900977i $$-0.642851\pi$$
−0.433867 + 0.900977i $$0.642851\pi$$
$$758$$ −26.4575 −0.960980
$$759$$ −3.29150 −0.119474
$$760$$ 7.93725 0.287914
$$761$$ 25.7490 0.933401 0.466701 0.884415i $$-0.345443\pi$$
0.466701 + 0.884415i $$0.345443\pi$$
$$762$$ 10.5830 0.383382
$$763$$ 8.70850 0.315269
$$764$$ 31.1438 1.12674
$$765$$ −4.00000 −0.144620
$$766$$ 49.1660 1.77644
$$767$$ −4.00000 −0.144432
$$768$$ 5.00000 0.180422
$$769$$ −17.7490 −0.640046 −0.320023 0.947410i $$-0.603691\pi$$
−0.320023 + 0.947410i $$0.603691\pi$$
$$770$$ −1.54249 −0.0555874
$$771$$ 5.41699 0.195088
$$772$$ −58.2288 −2.09570
$$773$$ −8.70850 −0.313223 −0.156611 0.987660i $$-0.550057\pi$$
−0.156611 + 0.987660i $$0.550057\pi$$
$$774$$ −14.9373 −0.536909
$$775$$ 6.00000 0.215526
$$776$$ 23.3137 0.836914
$$777$$ −6.00000 −0.215249
$$778$$ 15.8745 0.569129
$$779$$ 9.64575 0.345595
$$780$$ 1.77124 0.0634207
$$781$$ 2.58301 0.0924272
$$782$$ 98.3320 3.51635
$$783$$ −8.93725 −0.319391
$$784$$ −47.2065 −1.68595
$$785$$ −13.2915 −0.474394
$$786$$ −37.6458 −1.34278
$$787$$ 37.8745 1.35008 0.675040 0.737781i $$-0.264126\pi$$
0.675040 + 0.737781i $$0.264126\pi$$
$$788$$ −125.830 −4.48251
$$789$$ −4.58301 −0.163159
$$790$$ −17.4170 −0.619669
$$791$$ −6.58301 −0.234065
$$792$$ −2.81176 −0.0999116
$$793$$ 4.00000 0.142044
$$794$$ 5.29150 0.187788
$$795$$ −11.2915 −0.400468
$$796$$ 36.4575 1.29220
$$797$$ 40.0000 1.41687 0.708436 0.705775i $$-0.249401\pi$$
0.708436 + 0.705775i $$0.249401\pi$$
$$798$$ −4.35425 −0.154139
$$799$$ 5.16601 0.182760
$$800$$ −13.2288 −0.467707
$$801$$ −16.9373 −0.598448
$$802$$ 13.0627 0.461262
$$803$$ 3.54249 0.125012
$$804$$ 32.9150 1.16082
$$805$$ 15.2915 0.538955
$$806$$ 5.62352 0.198080
$$807$$ −4.22876 −0.148859
$$808$$ −73.7490 −2.59448
$$809$$ −19.4170 −0.682665 −0.341333 0.939943i $$-0.610878\pi$$
−0.341333 + 0.939943i $$0.610878\pi$$
$$810$$ −2.64575 −0.0929622
$$811$$ 38.3320 1.34602 0.673010 0.739634i $$-0.265001\pi$$
0.673010 + 0.739634i $$0.265001\pi$$
$$812$$ 73.5425 2.58084
$$813$$ 4.70850 0.165134
$$814$$ −3.41699 −0.119766
$$815$$ −2.35425 −0.0824657
$$816$$ 44.0000 1.54031
$$817$$ −5.64575 −0.197520
$$818$$ 17.7490 0.620580
$$819$$ −0.583005 −0.0203719
$$820$$ −48.2288 −1.68422
$$821$$ 47.6235 1.66207 0.831036 0.556218i $$-0.187748\pi$$
0.831036 + 0.556218i $$0.187748\pi$$
$$822$$ 15.8745 0.553687
$$823$$ 4.22876 0.147405 0.0737026 0.997280i $$-0.476518\pi$$
0.0737026 + 0.997280i $$0.476518\pi$$
$$824$$ 84.0000 2.92628
$$825$$ −0.354249 −0.0123334
$$826$$ −49.1660 −1.71070
$$827$$ 30.4575 1.05911 0.529556 0.848275i $$-0.322359\pi$$
0.529556 + 0.848275i $$0.322359\pi$$
$$828$$ 46.4575 1.61451
$$829$$ 2.70850 0.0940700 0.0470350 0.998893i $$-0.485023\pi$$
0.0470350 + 0.998893i $$0.485023\pi$$
$$830$$ −15.8745 −0.551012
$$831$$ −0.583005 −0.0202242
$$832$$ −4.60523 −0.159658
$$833$$ 17.1660 0.594767
$$834$$ 47.2915 1.63757
$$835$$ −21.2915 −0.736823
$$836$$ −1.77124 −0.0612597
$$837$$ −6.00000 −0.207390
$$838$$ −103.018 −3.55871
$$839$$ −12.4575 −0.430081 −0.215041 0.976605i $$-0.568988\pi$$
−0.215041 + 0.976605i $$0.568988\pi$$
$$840$$ 13.0627 0.450708
$$841$$ 50.8745 1.75429
$$842$$ −75.6235 −2.60616
$$843$$ 32.2288 1.11002
$$844$$ 12.9150 0.444554
$$845$$ −12.8745 −0.442897
$$846$$ 3.41699 0.117479
$$847$$ −17.8967 −0.614939
$$848$$ 124.207 4.26527
$$849$$ 11.5203 0.395374
$$850$$ 10.5830 0.362994
$$851$$ 33.8745 1.16120
$$852$$ −36.4575 −1.24901
$$853$$ 14.7085 0.503609 0.251805 0.967778i $$-0.418976\pi$$
0.251805 + 0.967778i $$0.418976\pi$$
$$854$$ 49.1660 1.68243
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ 31.0405 1.06032 0.530162 0.847896i $$-0.322131\pi$$
0.530162 + 0.847896i $$0.322131\pi$$
$$858$$ −0.332021 −0.0113350
$$859$$ −34.3320 −1.17139 −0.585697 0.810530i $$-0.699179\pi$$
−0.585697 + 0.810530i $$0.699179\pi$$
$$860$$ 28.2288 0.962593
$$861$$ 15.8745 0.541002
$$862$$ −8.70850 −0.296613
$$863$$ −20.0000 −0.680808 −0.340404 0.940279i $$-0.610564\pi$$
−0.340404 + 0.940279i $$0.610564\pi$$
$$864$$ 13.2288 0.450051
$$865$$ 0 0
$$866$$ 73.1438 2.48553
$$867$$ 1.00000 0.0339618
$$868$$ 49.3725 1.67581
$$869$$ 2.33202 0.0791084
$$870$$ 23.6458 0.801666
$$871$$ 2.33202 0.0790175
$$872$$ −42.0000 −1.42230
$$873$$ −2.93725 −0.0994110
$$874$$ 24.5830 0.831533
$$875$$ 1.64575 0.0556365
$$876$$ −50.0000 −1.68934
$$877$$ −25.0627 −0.846309 −0.423154 0.906058i $$-0.639077\pi$$
−0.423154 + 0.906058i $$0.639077\pi$$
$$878$$ 3.74902 0.126523
$$879$$ −5.41699 −0.182711
$$880$$ 3.89674 0.131359
$$881$$ −0.125492 −0.00422794 −0.00211397 0.999998i $$-0.500673\pi$$
−0.00211397 + 0.999998i $$0.500673\pi$$
$$882$$ 11.3542 0.382317
$$883$$ 10.8118 0.363845 0.181922 0.983313i $$-0.441768\pi$$
0.181922 + 0.983313i $$0.441768\pi$$
$$884$$ 7.08497 0.238293
$$885$$ −11.2915 −0.379560
$$886$$ −70.6640 −2.37400
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 28.9373 0.971071
$$889$$ 6.58301 0.220787
$$890$$ 44.8118 1.50209
$$891$$ 0.354249 0.0118678
$$892$$ 12.9150 0.432427
$$893$$ 1.29150 0.0432185
$$894$$ −54.4575 −1.82133
$$895$$ −7.29150 −0.243728
$$896$$ −13.0627 −0.436396
$$897$$ 3.29150 0.109900
$$898$$ 95.8523 3.19863
$$899$$ 53.6235 1.78844
$$900$$ 5.00000 0.166667
$$901$$ −45.1660 −1.50470
$$902$$ 9.04052 0.301016
$$903$$ −9.29150 −0.309202
$$904$$ 31.7490 1.05596
$$905$$ 17.2915 0.574789
$$906$$ −22.7085 −0.754439
$$907$$ −47.0405 −1.56195 −0.780977 0.624559i $$-0.785279\pi$$
−0.780977 + 0.624559i $$0.785279\pi$$
$$908$$ −53.5425 −1.77687
$$909$$ 9.29150 0.308180
$$910$$ 1.54249 0.0511329
$$911$$ 26.5830 0.880734 0.440367 0.897818i $$-0.354848\pi$$
0.440367 + 0.897818i $$0.354848\pi$$
$$912$$ 11.0000 0.364246
$$913$$ 2.12549 0.0703435
$$914$$ −0.332021 −0.0109823
$$915$$ 11.2915 0.373286
$$916$$ 43.5425 1.43868
$$917$$ −23.4170 −0.773297
$$918$$ −10.5830 −0.349291
$$919$$ 14.8340 0.489328 0.244664 0.969608i $$-0.421322\pi$$
0.244664 + 0.969608i $$0.421322\pi$$
$$920$$ −73.7490 −2.43143
$$921$$ 24.4575 0.805902
$$922$$ 99.8745 3.28919
$$923$$ −2.58301 −0.0850207
$$924$$ −2.91503 −0.0958973
$$925$$ 3.64575 0.119872
$$926$$ 93.9778 3.08830
$$927$$ −10.5830 −0.347591
$$928$$ −118.229 −3.88105
$$929$$ −11.8745 −0.389590 −0.194795 0.980844i $$-0.562404\pi$$
−0.194795 + 0.980844i $$0.562404\pi$$
$$930$$ 15.8745 0.520546
$$931$$ 4.29150 0.140648
$$932$$ −90.0000 −2.94805
$$933$$ −22.9373 −0.750932
$$934$$ −52.5830 −1.72057
$$935$$ −1.41699 −0.0463407
$$936$$ 2.81176 0.0919053
$$937$$ 20.1255 0.657471 0.328736 0.944422i $$-0.393378\pi$$
0.328736 + 0.944422i $$0.393378\pi$$
$$938$$ 28.6640 0.935914
$$939$$ −17.2915 −0.564287
$$940$$ −6.45751 −0.210621
$$941$$ −11.7712 −0.383732 −0.191866 0.981421i $$-0.561454\pi$$
−0.191866 + 0.981421i $$0.561454\pi$$
$$942$$ −35.1660 −1.14577
$$943$$ −89.6235 −2.91854
$$944$$ 124.207 4.04258
$$945$$ −1.64575 −0.0535363
$$946$$ −5.29150 −0.172042
$$947$$ 11.4170 0.371002 0.185501 0.982644i $$-0.440609\pi$$
0.185501 + 0.982644i $$0.440609\pi$$
$$948$$ −32.9150 −1.06903
$$949$$ −3.54249 −0.114994
$$950$$ 2.64575 0.0858395
$$951$$ −20.4575 −0.663380
$$952$$ 52.2510 1.69346
$$953$$ −10.5830 −0.342817 −0.171409 0.985200i $$-0.554832\pi$$
−0.171409 + 0.985200i $$0.554832\pi$$
$$954$$ −29.8745 −0.967223
$$955$$ 6.22876 0.201558
$$956$$ 78.2288 2.53010
$$957$$ −3.16601 −0.102343
$$958$$ −11.5203 −0.372203
$$959$$ 9.87451 0.318864
$$960$$ −13.0000 −0.419573
$$961$$ 5.00000 0.161290
$$962$$ 3.41699 0.110168
$$963$$ 0 0
$$964$$ 82.9150 2.67051
$$965$$ −11.6458 −0.374890
$$966$$ 40.4575 1.30170
$$967$$ −41.6458 −1.33924 −0.669619 0.742705i $$-0.733542\pi$$
−0.669619 + 0.742705i $$0.733542\pi$$
$$968$$ 86.3137 2.77423
$$969$$ −4.00000 −0.128499
$$970$$ 7.77124 0.249520
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ −5.00000 −0.160375
$$973$$ 29.4170 0.943066
$$974$$ −47.2915 −1.51532
$$975$$ 0.354249 0.0113450
$$976$$ −124.207 −3.97575
$$977$$ −51.0405 −1.63293 −0.816465 0.577394i $$-0.804070\pi$$
−0.816465 + 0.577394i $$0.804070\pi$$
$$978$$ −6.22876 −0.199174
$$979$$ −6.00000 −0.191761
$$980$$ −21.4575 −0.685435
$$981$$ 5.29150 0.168945
$$982$$ 15.2693 0.487262
$$983$$ 34.4575 1.09902 0.549512 0.835486i $$-0.314814\pi$$
0.549512 + 0.835486i $$0.314814\pi$$
$$984$$ −76.5608 −2.44067
$$985$$ −25.1660 −0.801856
$$986$$ 94.5830 3.01214
$$987$$ 2.12549 0.0676552
$$988$$ 1.77124 0.0563508
$$989$$ 52.4575 1.66805
$$990$$ −0.937254 −0.0297879
$$991$$ −35.7490 −1.13560 −0.567802 0.823165i $$-0.692206\pi$$
−0.567802 + 0.823165i $$0.692206\pi$$
$$992$$ −79.3725 −2.52008
$$993$$ −11.4170 −0.362307
$$994$$ −31.7490 −1.00702
$$995$$ 7.29150 0.231156
$$996$$ −30.0000 −0.950586
$$997$$ 26.7085 0.845867 0.422933 0.906161i $$-0.361000\pi$$
0.422933 + 0.906161i $$0.361000\pi$$
$$998$$ 82.1255 2.59964
$$999$$ −3.64575 −0.115346
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 285.2.a.d.1.1 2
3.2 odd 2 855.2.a.g.1.2 2
4.3 odd 2 4560.2.a.bo.1.1 2
5.2 odd 4 1425.2.c.i.799.2 4
5.3 odd 4 1425.2.c.i.799.3 4
5.4 even 2 1425.2.a.p.1.2 2
15.14 odd 2 4275.2.a.u.1.1 2
19.18 odd 2 5415.2.a.s.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.d.1.1 2 1.1 even 1 trivial
855.2.a.g.1.2 2 3.2 odd 2
1425.2.a.p.1.2 2 5.4 even 2
1425.2.c.i.799.2 4 5.2 odd 4
1425.2.c.i.799.3 4 5.3 odd 4
4275.2.a.u.1.1 2 15.14 odd 2
4560.2.a.bo.1.1 2 4.3 odd 2
5415.2.a.s.1.2 2 19.18 odd 2