Properties

 Label 950.2.a.m.1.2 Level $950$ Weight $2$ Character 950.1 Self dual yes Analytic conductor $7.586$ 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] = [950,2,Mod(1,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, 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("950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-1.91223$$ of defining polynomial Character $$\chi$$ $$=$$ 950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.25561 q^{3} +1.00000 q^{4} +2.25561 q^{6} +4.22547 q^{7} +1.00000 q^{8} +2.08777 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.25561 q^{3} +1.00000 q^{4} +2.25561 q^{6} +4.22547 q^{7} +1.00000 q^{8} +2.08777 q^{9} -5.13770 q^{11} +2.25561 q^{12} +3.16784 q^{13} +4.22547 q^{14} +1.00000 q^{16} -6.48108 q^{17} +2.08777 q^{18} -1.00000 q^{19} +9.53101 q^{21} -5.13770 q^{22} +7.56885 q^{23} +2.25561 q^{24} +3.16784 q^{26} -2.05763 q^{27} +4.22547 q^{28} +0.832162 q^{29} -4.51122 q^{31} +1.00000 q^{32} -11.5886 q^{33} -6.48108 q^{34} +2.08777 q^{36} +0.137699 q^{37} -1.00000 q^{38} +7.14540 q^{39} -11.6489 q^{41} +9.53101 q^{42} -2.51122 q^{43} -5.13770 q^{44} +7.56885 q^{46} -5.96216 q^{47} +2.25561 q^{48} +10.8546 q^{49} -14.6188 q^{51} +3.16784 q^{52} +0.225470 q^{53} -2.05763 q^{54} +4.22547 q^{56} -2.25561 q^{57} +0.832162 q^{58} +5.39331 q^{59} +14.4509 q^{61} -4.51122 q^{62} +8.82181 q^{63} +1.00000 q^{64} -11.5886 q^{66} -4.11021 q^{67} -6.48108 q^{68} +17.0724 q^{69} +3.82446 q^{71} +2.08777 q^{72} +4.70655 q^{73} +0.137699 q^{74} -1.00000 q^{76} -21.7092 q^{77} +7.14540 q^{78} +10.6265 q^{79} -10.9045 q^{81} -11.6489 q^{82} -12.0999 q^{83} +9.53101 q^{84} -2.51122 q^{86} +1.87703 q^{87} -5.13770 q^{88} -10.0000 q^{89} +13.3856 q^{91} +7.56885 q^{92} -10.1755 q^{93} -5.96216 q^{94} +2.25561 q^{96} +3.93972 q^{97} +10.8546 q^{98} -10.7263 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{6} + 2 q^{7} + 3 q^{8} + 13 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^6 + 2 * q^7 + 3 * q^8 + 13 * q^9 $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{6} + 2 q^{7} + 3 q^{8} + 13 q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} + 3 q^{16} - 4 q^{17} + 13 q^{18} - 3 q^{19} - 11 q^{21} + 2 q^{22} + 14 q^{23} + 2 q^{24} - 2 q^{26} - 7 q^{27} + 2 q^{28} + 14 q^{29} - 4 q^{31} + 3 q^{32} + 4 q^{33} - 4 q^{34} + 13 q^{36} - 17 q^{37} - 3 q^{38} + 29 q^{39} - 8 q^{41} - 11 q^{42} + 2 q^{43} + 2 q^{44} + 14 q^{46} + 13 q^{47} + 2 q^{48} + 25 q^{49} - 11 q^{51} - 2 q^{52} - 10 q^{53} - 7 q^{54} + 2 q^{56} - 2 q^{57} + 14 q^{58} - 6 q^{59} + 22 q^{61} - 4 q^{62} + 2 q^{63} + 3 q^{64} + 4 q^{66} - 4 q^{68} + 8 q^{69} - 2 q^{71} + 13 q^{72} - 12 q^{73} - 17 q^{74} - 3 q^{76} - 50 q^{77} + 29 q^{78} + 24 q^{79} - q^{81} - 8 q^{82} + 12 q^{83} - 11 q^{84} + 2 q^{86} - 21 q^{87} + 2 q^{88} - 30 q^{89} - 7 q^{91} + 14 q^{92} - 44 q^{93} + 13 q^{94} + 2 q^{96} + 25 q^{98} + 24 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^6 + 2 * q^7 + 3 * q^8 + 13 * q^9 + 2 * q^11 + 2 * q^12 - 2 * q^13 + 2 * q^14 + 3 * q^16 - 4 * q^17 + 13 * q^18 - 3 * q^19 - 11 * q^21 + 2 * q^22 + 14 * q^23 + 2 * q^24 - 2 * q^26 - 7 * q^27 + 2 * q^28 + 14 * q^29 - 4 * q^31 + 3 * q^32 + 4 * q^33 - 4 * q^34 + 13 * q^36 - 17 * q^37 - 3 * q^38 + 29 * q^39 - 8 * q^41 - 11 * q^42 + 2 * q^43 + 2 * q^44 + 14 * q^46 + 13 * q^47 + 2 * q^48 + 25 * q^49 - 11 * q^51 - 2 * q^52 - 10 * q^53 - 7 * q^54 + 2 * q^56 - 2 * q^57 + 14 * q^58 - 6 * q^59 + 22 * q^61 - 4 * q^62 + 2 * q^63 + 3 * q^64 + 4 * q^66 - 4 * q^68 + 8 * q^69 - 2 * q^71 + 13 * q^72 - 12 * q^73 - 17 * q^74 - 3 * q^76 - 50 * q^77 + 29 * q^78 + 24 * q^79 - q^81 - 8 * q^82 + 12 * q^83 - 11 * q^84 + 2 * q^86 - 21 * q^87 + 2 * q^88 - 30 * q^89 - 7 * q^91 + 14 * q^92 - 44 * q^93 + 13 * q^94 + 2 * q^96 + 25 * q^98 + 24 * q^99

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.25561 1.30228 0.651138 0.758959i $$-0.274292\pi$$
0.651138 + 0.758959i $$0.274292\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.25561 0.920848
$$7$$ 4.22547 1.59708 0.798539 0.601943i $$-0.205607\pi$$
0.798539 + 0.601943i $$0.205607\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 2.08777 0.695924
$$10$$ 0 0
$$11$$ −5.13770 −1.54907 −0.774537 0.632528i $$-0.782017\pi$$
−0.774537 + 0.632528i $$0.782017\pi$$
$$12$$ 2.25561 0.651138
$$13$$ 3.16784 0.878600 0.439300 0.898340i $$-0.355226\pi$$
0.439300 + 0.898340i $$0.355226\pi$$
$$14$$ 4.22547 1.12930
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.48108 −1.57189 −0.785946 0.618295i $$-0.787824\pi$$
−0.785946 + 0.618295i $$0.787824\pi$$
$$18$$ 2.08777 0.492092
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 9.53101 2.07984
$$22$$ −5.13770 −1.09536
$$23$$ 7.56885 1.57821 0.789107 0.614256i $$-0.210544\pi$$
0.789107 + 0.614256i $$0.210544\pi$$
$$24$$ 2.25561 0.460424
$$25$$ 0 0
$$26$$ 3.16784 0.621264
$$27$$ −2.05763 −0.395991
$$28$$ 4.22547 0.798539
$$29$$ 0.832162 0.154529 0.0772643 0.997011i $$-0.475381\pi$$
0.0772643 + 0.997011i $$0.475381\pi$$
$$30$$ 0 0
$$31$$ −4.51122 −0.810239 −0.405119 0.914264i $$-0.632770\pi$$
−0.405119 + 0.914264i $$0.632770\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −11.5886 −2.01732
$$34$$ −6.48108 −1.11150
$$35$$ 0 0
$$36$$ 2.08777 0.347962
$$37$$ 0.137699 0.0226376 0.0113188 0.999936i $$-0.496397\pi$$
0.0113188 + 0.999936i $$0.496397\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 7.14540 1.14418
$$40$$ 0 0
$$41$$ −11.6489 −1.81926 −0.909628 0.415425i $$-0.863633\pi$$
−0.909628 + 0.415425i $$0.863633\pi$$
$$42$$ 9.53101 1.47067
$$43$$ −2.51122 −0.382957 −0.191479 0.981497i $$-0.561328\pi$$
−0.191479 + 0.981497i $$0.561328\pi$$
$$44$$ −5.13770 −0.774537
$$45$$ 0 0
$$46$$ 7.56885 1.11597
$$47$$ −5.96216 −0.869670 −0.434835 0.900510i $$-0.643193\pi$$
−0.434835 + 0.900510i $$0.643193\pi$$
$$48$$ 2.25561 0.325569
$$49$$ 10.8546 1.55066
$$50$$ 0 0
$$51$$ −14.6188 −2.04704
$$52$$ 3.16784 0.439300
$$53$$ 0.225470 0.0309707 0.0154853 0.999880i $$-0.495071\pi$$
0.0154853 + 0.999880i $$0.495071\pi$$
$$54$$ −2.05763 −0.280008
$$55$$ 0 0
$$56$$ 4.22547 0.564652
$$57$$ −2.25561 −0.298763
$$58$$ 0.832162 0.109268
$$59$$ 5.39331 0.702149 0.351074 0.936348i $$-0.385816\pi$$
0.351074 + 0.936348i $$0.385816\pi$$
$$60$$ 0 0
$$61$$ 14.4509 1.85025 0.925127 0.379659i $$-0.123959\pi$$
0.925127 + 0.379659i $$0.123959\pi$$
$$62$$ −4.51122 −0.572925
$$63$$ 8.82181 1.11144
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −11.5886 −1.42646
$$67$$ −4.11021 −0.502142 −0.251071 0.967969i $$-0.580783\pi$$
−0.251071 + 0.967969i $$0.580783\pi$$
$$68$$ −6.48108 −0.785946
$$69$$ 17.0724 2.05527
$$70$$ 0 0
$$71$$ 3.82446 0.453880 0.226940 0.973909i $$-0.427128\pi$$
0.226940 + 0.973909i $$0.427128\pi$$
$$72$$ 2.08777 0.246046
$$73$$ 4.70655 0.550860 0.275430 0.961321i $$-0.411180\pi$$
0.275430 + 0.961321i $$0.411180\pi$$
$$74$$ 0.137699 0.0160072
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −21.7092 −2.47399
$$78$$ 7.14540 0.809058
$$79$$ 10.6265 1.19557 0.597786 0.801655i $$-0.296047\pi$$
0.597786 + 0.801655i $$0.296047\pi$$
$$80$$ 0 0
$$81$$ −10.9045 −1.21161
$$82$$ −11.6489 −1.28641
$$83$$ −12.0999 −1.32813 −0.664066 0.747674i $$-0.731171\pi$$
−0.664066 + 0.747674i $$0.731171\pi$$
$$84$$ 9.53101 1.03992
$$85$$ 0 0
$$86$$ −2.51122 −0.270792
$$87$$ 1.87703 0.201239
$$88$$ −5.13770 −0.547681
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 13.3856 1.40319
$$92$$ 7.56885 0.789107
$$93$$ −10.1755 −1.05515
$$94$$ −5.96216 −0.614950
$$95$$ 0 0
$$96$$ 2.25561 0.230212
$$97$$ 3.93972 0.400018 0.200009 0.979794i $$-0.435903\pi$$
0.200009 + 0.979794i $$0.435903\pi$$
$$98$$ 10.8546 1.09648
$$99$$ −10.7263 −1.07804
$$100$$ 0 0
$$101$$ 3.19798 0.318211 0.159105 0.987262i $$-0.449139\pi$$
0.159105 + 0.987262i $$0.449139\pi$$
$$102$$ −14.6188 −1.44747
$$103$$ −10.6868 −1.05300 −0.526499 0.850176i $$-0.676496\pi$$
−0.526499 + 0.850176i $$0.676496\pi$$
$$104$$ 3.16784 0.310632
$$105$$ 0 0
$$106$$ 0.225470 0.0218996
$$107$$ 10.8168 1.04570 0.522848 0.852426i $$-0.324870\pi$$
0.522848 + 0.852426i $$0.324870\pi$$
$$108$$ −2.05763 −0.197996
$$109$$ 9.24791 0.885789 0.442894 0.896574i $$-0.353952\pi$$
0.442894 + 0.896574i $$0.353952\pi$$
$$110$$ 0 0
$$111$$ 0.310596 0.0294804
$$112$$ 4.22547 0.399269
$$113$$ 17.6489 1.66027 0.830135 0.557562i $$-0.188263\pi$$
0.830135 + 0.557562i $$0.188263\pi$$
$$114$$ −2.25561 −0.211257
$$115$$ 0 0
$$116$$ 0.832162 0.0772643
$$117$$ 6.61372 0.611439
$$118$$ 5.39331 0.496494
$$119$$ −27.3856 −2.51043
$$120$$ 0 0
$$121$$ 15.3960 1.39963
$$122$$ 14.4509 1.30833
$$123$$ −26.2754 −2.36917
$$124$$ −4.51122 −0.405119
$$125$$ 0 0
$$126$$ 8.82181 0.785910
$$127$$ −12.7866 −1.13463 −0.567314 0.823501i $$-0.692018\pi$$
−0.567314 + 0.823501i $$0.692018\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −5.66432 −0.498716
$$130$$ 0 0
$$131$$ −2.11526 −0.184811 −0.0924057 0.995721i $$-0.529456\pi$$
−0.0924057 + 0.995721i $$0.529456\pi$$
$$132$$ −11.5886 −1.00866
$$133$$ −4.22547 −0.366395
$$134$$ −4.11021 −0.355068
$$135$$ 0 0
$$136$$ −6.48108 −0.555748
$$137$$ −5.36317 −0.458206 −0.229103 0.973402i $$-0.573579\pi$$
−0.229103 + 0.973402i $$0.573579\pi$$
$$138$$ 17.0724 1.45330
$$139$$ −10.1601 −0.861771 −0.430886 0.902407i $$-0.641799\pi$$
−0.430886 + 0.902407i $$0.641799\pi$$
$$140$$ 0 0
$$141$$ −13.4483 −1.13255
$$142$$ 3.82446 0.320941
$$143$$ −16.2754 −1.36102
$$144$$ 2.08777 0.173981
$$145$$ 0 0
$$146$$ 4.70655 0.389517
$$147$$ 24.4837 2.01938
$$148$$ 0.137699 0.0113188
$$149$$ −5.93972 −0.486601 −0.243301 0.969951i $$-0.578230\pi$$
−0.243301 + 0.969951i $$0.578230\pi$$
$$150$$ 0 0
$$151$$ −15.2978 −1.24492 −0.622460 0.782652i $$-0.713867\pi$$
−0.622460 + 0.782652i $$0.713867\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −13.5310 −1.09392
$$154$$ −21.7092 −1.74938
$$155$$ 0 0
$$156$$ 7.14540 0.572090
$$157$$ −12.7866 −1.02048 −0.510242 0.860031i $$-0.670444\pi$$
−0.510242 + 0.860031i $$0.670444\pi$$
$$158$$ 10.6265 0.845397
$$159$$ 0.508572 0.0403324
$$160$$ 0 0
$$161$$ 31.9819 2.52053
$$162$$ −10.9045 −0.856740
$$163$$ −11.4734 −0.898664 −0.449332 0.893365i $$-0.648338\pi$$
−0.449332 + 0.893365i $$0.648338\pi$$
$$164$$ −11.6489 −0.909628
$$165$$ 0 0
$$166$$ −12.0999 −0.939131
$$167$$ 19.4131 1.50223 0.751115 0.660171i $$-0.229516\pi$$
0.751115 + 0.660171i $$0.229516\pi$$
$$168$$ 9.53101 0.735333
$$169$$ −2.96480 −0.228062
$$170$$ 0 0
$$171$$ −2.08777 −0.159656
$$172$$ −2.51122 −0.191479
$$173$$ 9.78662 0.744063 0.372031 0.928220i $$-0.378661\pi$$
0.372031 + 0.928220i $$0.378661\pi$$
$$174$$ 1.87703 0.142297
$$175$$ 0 0
$$176$$ −5.13770 −0.387269
$$177$$ 12.1652 0.914392
$$178$$ −10.0000 −0.749532
$$179$$ −6.82446 −0.510084 −0.255042 0.966930i $$-0.582089\pi$$
−0.255042 + 0.966930i $$0.582089\pi$$
$$180$$ 0 0
$$181$$ −0.137699 −0.0102351 −0.00511755 0.999987i $$-0.501629\pi$$
−0.00511755 + 0.999987i $$0.501629\pi$$
$$182$$ 13.3856 0.992207
$$183$$ 32.5957 2.40954
$$184$$ 7.56885 0.557983
$$185$$ 0 0
$$186$$ −10.1755 −0.746107
$$187$$ 33.2978 2.43498
$$188$$ −5.96216 −0.434835
$$189$$ −8.69446 −0.632429
$$190$$ 0 0
$$191$$ 19.3779 1.40214 0.701068 0.713095i $$-0.252707\pi$$
0.701068 + 0.713095i $$0.252707\pi$$
$$192$$ 2.25561 0.162785
$$193$$ −5.42851 −0.390752 −0.195376 0.980728i $$-0.562593\pi$$
−0.195376 + 0.980728i $$0.562593\pi$$
$$194$$ 3.93972 0.282856
$$195$$ 0 0
$$196$$ 10.8546 0.775328
$$197$$ −15.6489 −1.11494 −0.557470 0.830197i $$-0.688228\pi$$
−0.557470 + 0.830197i $$0.688228\pi$$
$$198$$ −10.7263 −0.762288
$$199$$ 18.0499 1.27953 0.639763 0.768572i $$-0.279033\pi$$
0.639763 + 0.768572i $$0.279033\pi$$
$$200$$ 0 0
$$201$$ −9.27102 −0.653927
$$202$$ 3.19798 0.225009
$$203$$ 3.51628 0.246794
$$204$$ −14.6188 −1.02352
$$205$$ 0 0
$$206$$ −10.6868 −0.744582
$$207$$ 15.8020 1.09832
$$208$$ 3.16784 0.219650
$$209$$ 5.13770 0.355382
$$210$$ 0 0
$$211$$ −7.50857 −0.516911 −0.258456 0.966023i $$-0.583214\pi$$
−0.258456 + 0.966023i $$0.583214\pi$$
$$212$$ 0.225470 0.0154853
$$213$$ 8.62648 0.591077
$$214$$ 10.8168 0.739418
$$215$$ 0 0
$$216$$ −2.05763 −0.140004
$$217$$ −19.0620 −1.29401
$$218$$ 9.24791 0.626347
$$219$$ 10.6161 0.717372
$$220$$ 0 0
$$221$$ −20.5310 −1.38106
$$222$$ 0.310596 0.0208458
$$223$$ 27.8091 1.86223 0.931116 0.364723i $$-0.118836\pi$$
0.931116 + 0.364723i $$0.118836\pi$$
$$224$$ 4.22547 0.282326
$$225$$ 0 0
$$226$$ 17.6489 1.17399
$$227$$ 8.91223 0.591525 0.295763 0.955261i $$-0.404426\pi$$
0.295763 + 0.955261i $$0.404426\pi$$
$$228$$ −2.25561 −0.149381
$$229$$ −13.6489 −0.901946 −0.450973 0.892538i $$-0.648923\pi$$
−0.450973 + 0.892538i $$0.648923\pi$$
$$230$$ 0 0
$$231$$ −48.9674 −3.22182
$$232$$ 0.832162 0.0546341
$$233$$ 23.2754 1.52482 0.762411 0.647093i $$-0.224015\pi$$
0.762411 + 0.647093i $$0.224015\pi$$
$$234$$ 6.61372 0.432352
$$235$$ 0 0
$$236$$ 5.39331 0.351074
$$237$$ 23.9692 1.55697
$$238$$ −27.3856 −1.77515
$$239$$ 3.72898 0.241208 0.120604 0.992701i $$-0.461517\pi$$
0.120604 + 0.992701i $$0.461517\pi$$
$$240$$ 0 0
$$241$$ −1.48878 −0.0959009 −0.0479505 0.998850i $$-0.515269\pi$$
−0.0479505 + 0.998850i $$0.515269\pi$$
$$242$$ 15.3960 0.989689
$$243$$ −18.4234 −1.18186
$$244$$ 14.4509 0.925127
$$245$$ 0 0
$$246$$ −26.2754 −1.67526
$$247$$ −3.16784 −0.201565
$$248$$ −4.51122 −0.286463
$$249$$ −27.2925 −1.72959
$$250$$ 0 0
$$251$$ −8.78662 −0.554606 −0.277303 0.960782i $$-0.589441\pi$$
−0.277303 + 0.960782i $$0.589441\pi$$
$$252$$ 8.82181 0.555722
$$253$$ −38.8865 −2.44477
$$254$$ −12.7866 −0.802304
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.74175 0.171025 0.0855127 0.996337i $$-0.472747\pi$$
0.0855127 + 0.996337i $$0.472747\pi$$
$$258$$ −5.66432 −0.352645
$$259$$ 0.581844 0.0361540
$$260$$ 0 0
$$261$$ 1.73736 0.107540
$$262$$ −2.11526 −0.130681
$$263$$ 2.74704 0.169390 0.0846948 0.996407i $$-0.473008\pi$$
0.0846948 + 0.996407i $$0.473008\pi$$
$$264$$ −11.5886 −0.713231
$$265$$ 0 0
$$266$$ −4.22547 −0.259080
$$267$$ −22.5561 −1.38041
$$268$$ −4.11021 −0.251071
$$269$$ −9.74175 −0.593965 −0.296982 0.954883i $$-0.595980\pi$$
−0.296982 + 0.954883i $$0.595980\pi$$
$$270$$ 0 0
$$271$$ 26.3555 1.60098 0.800490 0.599346i $$-0.204573\pi$$
0.800490 + 0.599346i $$0.204573\pi$$
$$272$$ −6.48108 −0.392973
$$273$$ 30.1927 1.82734
$$274$$ −5.36317 −0.324001
$$275$$ 0 0
$$276$$ 17.0724 1.02764
$$277$$ −0.962158 −0.0578104 −0.0289052 0.999582i $$-0.509202\pi$$
−0.0289052 + 0.999582i $$0.509202\pi$$
$$278$$ −10.1601 −0.609364
$$279$$ −9.41839 −0.563864
$$280$$ 0 0
$$281$$ −14.6714 −0.875219 −0.437610 0.899165i $$-0.644175\pi$$
−0.437610 + 0.899165i $$0.644175\pi$$
$$282$$ −13.4483 −0.800834
$$283$$ −26.1601 −1.55506 −0.777529 0.628847i $$-0.783527\pi$$
−0.777529 + 0.628847i $$0.783527\pi$$
$$284$$ 3.82446 0.226940
$$285$$ 0 0
$$286$$ −16.2754 −0.962384
$$287$$ −49.2221 −2.90549
$$288$$ 2.08777 0.123023
$$289$$ 25.0044 1.47085
$$290$$ 0 0
$$291$$ 8.88647 0.520934
$$292$$ 4.70655 0.275430
$$293$$ −22.4657 −1.31246 −0.656229 0.754562i $$-0.727850\pi$$
−0.656229 + 0.754562i $$0.727850\pi$$
$$294$$ 24.4837 1.42792
$$295$$ 0 0
$$296$$ 0.137699 0.00800360
$$297$$ 10.5715 0.613420
$$298$$ −5.93972 −0.344079
$$299$$ 23.9769 1.38662
$$300$$ 0 0
$$301$$ −10.6111 −0.611612
$$302$$ −15.2978 −0.880291
$$303$$ 7.21338 0.414398
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −13.5310 −0.773516
$$307$$ 17.2204 0.982821 0.491410 0.870928i $$-0.336482\pi$$
0.491410 + 0.870928i $$0.336482\pi$$
$$308$$ −21.7092 −1.23700
$$309$$ −24.1051 −1.37129
$$310$$ 0 0
$$311$$ −7.87439 −0.446516 −0.223258 0.974759i $$-0.571669\pi$$
−0.223258 + 0.974759i $$0.571669\pi$$
$$312$$ 7.14540 0.404529
$$313$$ −25.1678 −1.42257 −0.711285 0.702904i $$-0.751887\pi$$
−0.711285 + 0.702904i $$0.751887\pi$$
$$314$$ −12.7866 −0.721590
$$315$$ 0 0
$$316$$ 10.6265 0.597786
$$317$$ −23.9045 −1.34261 −0.671306 0.741180i $$-0.734266\pi$$
−0.671306 + 0.741180i $$0.734266\pi$$
$$318$$ 0.508572 0.0285193
$$319$$ −4.27540 −0.239376
$$320$$ 0 0
$$321$$ 24.3984 1.36178
$$322$$ 31.9819 1.78228
$$323$$ 6.48108 0.360617
$$324$$ −10.9045 −0.605807
$$325$$ 0 0
$$326$$ −11.4734 −0.635451
$$327$$ 20.8597 1.15354
$$328$$ −11.6489 −0.643204
$$329$$ −25.1929 −1.38893
$$330$$ 0 0
$$331$$ 30.7565 1.69053 0.845264 0.534348i $$-0.179443\pi$$
0.845264 + 0.534348i $$0.179443\pi$$
$$332$$ −12.0999 −0.664066
$$333$$ 0.287484 0.0157540
$$334$$ 19.4131 1.06224
$$335$$ 0 0
$$336$$ 9.53101 0.519959
$$337$$ 13.4734 0.733942 0.366971 0.930232i $$-0.380395\pi$$
0.366971 + 0.930232i $$0.380395\pi$$
$$338$$ −2.96480 −0.161264
$$339$$ 39.8091 2.16213
$$340$$ 0 0
$$341$$ 23.1773 1.25512
$$342$$ −2.08777 −0.112894
$$343$$ 16.2875 0.879441
$$344$$ −2.51122 −0.135396
$$345$$ 0 0
$$346$$ 9.78662 0.526132
$$347$$ 28.9468 1.55394 0.776971 0.629536i $$-0.216755\pi$$
0.776971 + 0.629536i $$0.216755\pi$$
$$348$$ 1.87703 0.100619
$$349$$ 27.9243 1.49475 0.747377 0.664400i $$-0.231313\pi$$
0.747377 + 0.664400i $$0.231313\pi$$
$$350$$ 0 0
$$351$$ −6.51825 −0.347918
$$352$$ −5.13770 −0.273840
$$353$$ 28.3099 1.50678 0.753392 0.657571i $$-0.228416\pi$$
0.753392 + 0.657571i $$0.228416\pi$$
$$354$$ 12.1652 0.646573
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ −61.7712 −3.26928
$$358$$ −6.82446 −0.360684
$$359$$ 2.60163 0.137309 0.0686545 0.997640i $$-0.478129\pi$$
0.0686545 + 0.997640i $$0.478129\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −0.137699 −0.00723731
$$363$$ 34.7272 1.82271
$$364$$ 13.3856 0.701596
$$365$$ 0 0
$$366$$ 32.5957 1.70380
$$367$$ 11.6489 0.608069 0.304034 0.952661i $$-0.401666\pi$$
0.304034 + 0.952661i $$0.401666\pi$$
$$368$$ 7.56885 0.394554
$$369$$ −24.3203 −1.26606
$$370$$ 0 0
$$371$$ 0.952717 0.0494626
$$372$$ −10.1755 −0.527577
$$373$$ −12.8064 −0.663091 −0.331545 0.943439i $$-0.607570\pi$$
−0.331545 + 0.943439i $$0.607570\pi$$
$$374$$ 33.2978 1.72179
$$375$$ 0 0
$$376$$ −5.96216 −0.307475
$$377$$ 2.63615 0.135769
$$378$$ −8.69446 −0.447195
$$379$$ −20.9122 −1.07419 −0.537095 0.843522i $$-0.680478\pi$$
−0.537095 + 0.843522i $$0.680478\pi$$
$$380$$ 0 0
$$381$$ −28.8416 −1.47760
$$382$$ 19.3779 0.991460
$$383$$ 10.3511 0.528916 0.264458 0.964397i $$-0.414807\pi$$
0.264458 + 0.964397i $$0.414807\pi$$
$$384$$ 2.25561 0.115106
$$385$$ 0 0
$$386$$ −5.42851 −0.276304
$$387$$ −5.24285 −0.266509
$$388$$ 3.93972 0.200009
$$389$$ 6.56620 0.332920 0.166460 0.986048i $$-0.446766\pi$$
0.166460 + 0.986048i $$0.446766\pi$$
$$390$$ 0 0
$$391$$ −49.0543 −2.48078
$$392$$ 10.8546 0.548240
$$393$$ −4.77121 −0.240676
$$394$$ −15.6489 −0.788381
$$395$$ 0 0
$$396$$ −10.7263 −0.539019
$$397$$ 23.5284 1.18085 0.590427 0.807091i $$-0.298959\pi$$
0.590427 + 0.807091i $$0.298959\pi$$
$$398$$ 18.0499 0.904761
$$399$$ −9.53101 −0.477147
$$400$$ 0 0
$$401$$ 6.22041 0.310633 0.155316 0.987865i $$-0.450360\pi$$
0.155316 + 0.987865i $$0.450360\pi$$
$$402$$ −9.27102 −0.462396
$$403$$ −14.2908 −0.711876
$$404$$ 3.19798 0.159105
$$405$$ 0 0
$$406$$ 3.51628 0.174510
$$407$$ −0.707457 −0.0350674
$$408$$ −14.6188 −0.723737
$$409$$ −34.4905 −1.70545 −0.852723 0.522363i $$-0.825051\pi$$
−0.852723 + 0.522363i $$0.825051\pi$$
$$410$$ 0 0
$$411$$ −12.0972 −0.596711
$$412$$ −10.6868 −0.526499
$$413$$ 22.7893 1.12139
$$414$$ 15.8020 0.776627
$$415$$ 0 0
$$416$$ 3.16784 0.155316
$$417$$ −22.9173 −1.12226
$$418$$ 5.13770 0.251293
$$419$$ −10.6265 −0.519138 −0.259569 0.965725i $$-0.583580\pi$$
−0.259569 + 0.965725i $$0.583580\pi$$
$$420$$ 0 0
$$421$$ −1.98021 −0.0965095 −0.0482548 0.998835i $$-0.515366\pi$$
−0.0482548 + 0.998835i $$0.515366\pi$$
$$422$$ −7.50857 −0.365512
$$423$$ −12.4476 −0.605224
$$424$$ 0.225470 0.0109498
$$425$$ 0 0
$$426$$ 8.62648 0.417954
$$427$$ 61.0620 2.95500
$$428$$ 10.8168 0.522848
$$429$$ −36.7109 −1.77242
$$430$$ 0 0
$$431$$ 15.0774 0.726254 0.363127 0.931740i $$-0.381709\pi$$
0.363127 + 0.931740i $$0.381709\pi$$
$$432$$ −2.05763 −0.0989979
$$433$$ 13.5337 0.650386 0.325193 0.945648i $$-0.394571\pi$$
0.325193 + 0.945648i $$0.394571\pi$$
$$434$$ −19.0620 −0.915006
$$435$$ 0 0
$$436$$ 9.24791 0.442894
$$437$$ −7.56885 −0.362067
$$438$$ 10.6161 0.507258
$$439$$ 39.1773 1.86983 0.934915 0.354872i $$-0.115476\pi$$
0.934915 + 0.354872i $$0.115476\pi$$
$$440$$ 0 0
$$441$$ 22.6619 1.07914
$$442$$ −20.5310 −0.976560
$$443$$ 19.3132 0.917600 0.458800 0.888540i $$-0.348279\pi$$
0.458800 + 0.888540i $$0.348279\pi$$
$$444$$ 0.310596 0.0147402
$$445$$ 0 0
$$446$$ 27.8091 1.31680
$$447$$ −13.3977 −0.633689
$$448$$ 4.22547 0.199635
$$449$$ 41.4131 1.95440 0.977202 0.212310i $$-0.0680985\pi$$
0.977202 + 0.212310i $$0.0680985\pi$$
$$450$$ 0 0
$$451$$ 59.8486 2.81816
$$452$$ 17.6489 0.830135
$$453$$ −34.5059 −1.62123
$$454$$ 8.91223 0.418272
$$455$$ 0 0
$$456$$ −2.25561 −0.105629
$$457$$ −37.8392 −1.77004 −0.885021 0.465551i $$-0.845856\pi$$
−0.885021 + 0.465551i $$0.845856\pi$$
$$458$$ −13.6489 −0.637772
$$459$$ 13.3357 0.622456
$$460$$ 0 0
$$461$$ 1.58864 0.0739903 0.0369952 0.999315i $$-0.488221\pi$$
0.0369952 + 0.999315i $$0.488221\pi$$
$$462$$ −48.9674 −2.27817
$$463$$ 40.3581 1.87560 0.937800 0.347175i $$-0.112859\pi$$
0.937800 + 0.347175i $$0.112859\pi$$
$$464$$ 0.832162 0.0386322
$$465$$ 0 0
$$466$$ 23.2754 1.07821
$$467$$ −39.5130 −1.82844 −0.914221 0.405217i $$-0.867196\pi$$
−0.914221 + 0.405217i $$0.867196\pi$$
$$468$$ 6.61372 0.305719
$$469$$ −17.3676 −0.801959
$$470$$ 0 0
$$471$$ −28.8416 −1.32895
$$472$$ 5.39331 0.248247
$$473$$ 12.9019 0.593229
$$474$$ 23.9692 1.10094
$$475$$ 0 0
$$476$$ −27.3856 −1.25522
$$477$$ 0.470730 0.0215532
$$478$$ 3.72898 0.170560
$$479$$ 7.61107 0.347759 0.173879 0.984767i $$-0.444370\pi$$
0.173879 + 0.984767i $$0.444370\pi$$
$$480$$ 0 0
$$481$$ 0.436209 0.0198894
$$482$$ −1.48878 −0.0678122
$$483$$ 72.1388 3.28243
$$484$$ 15.3960 0.699816
$$485$$ 0 0
$$486$$ −18.4234 −0.835705
$$487$$ −20.3907 −0.923989 −0.461995 0.886883i $$-0.652866\pi$$
−0.461995 + 0.886883i $$0.652866\pi$$
$$488$$ 14.4509 0.654163
$$489$$ −25.8794 −1.17031
$$490$$ 0 0
$$491$$ 25.0224 1.12925 0.564623 0.825349i $$-0.309021\pi$$
0.564623 + 0.825349i $$0.309021\pi$$
$$492$$ −26.2754 −1.18459
$$493$$ −5.39331 −0.242902
$$494$$ −3.16784 −0.142528
$$495$$ 0 0
$$496$$ −4.51122 −0.202560
$$497$$ 16.1601 0.724881
$$498$$ −27.2925 −1.22301
$$499$$ 22.6111 1.01221 0.506105 0.862472i $$-0.331085\pi$$
0.506105 + 0.862472i $$0.331085\pi$$
$$500$$ 0 0
$$501$$ 43.7884 1.95632
$$502$$ −8.78662 −0.392166
$$503$$ 11.5035 0.512916 0.256458 0.966555i $$-0.417444\pi$$
0.256458 + 0.966555i $$0.417444\pi$$
$$504$$ 8.82181 0.392955
$$505$$ 0 0
$$506$$ −38.8865 −1.72871
$$507$$ −6.68744 −0.296999
$$508$$ −12.7866 −0.567314
$$509$$ 9.11526 0.404027 0.202013 0.979383i $$-0.435252\pi$$
0.202013 + 0.979383i $$0.435252\pi$$
$$510$$ 0 0
$$511$$ 19.8874 0.879766
$$512$$ 1.00000 0.0441942
$$513$$ 2.05763 0.0908467
$$514$$ 2.74175 0.120933
$$515$$ 0 0
$$516$$ −5.66432 −0.249358
$$517$$ 30.6318 1.34718
$$518$$ 0.581844 0.0255648
$$519$$ 22.0748 0.968975
$$520$$ 0 0
$$521$$ 15.4888 0.678576 0.339288 0.940683i $$-0.389814\pi$$
0.339288 + 0.940683i $$0.389814\pi$$
$$522$$ 1.73736 0.0760423
$$523$$ 4.34073 0.189807 0.0949035 0.995486i $$-0.469746\pi$$
0.0949035 + 0.995486i $$0.469746\pi$$
$$524$$ −2.11526 −0.0924057
$$525$$ 0 0
$$526$$ 2.74704 0.119776
$$527$$ 29.2376 1.27361
$$528$$ −11.5886 −0.504331
$$529$$ 34.2875 1.49076
$$530$$ 0 0
$$531$$ 11.2600 0.488642
$$532$$ −4.22547 −0.183197
$$533$$ −36.9019 −1.59840
$$534$$ −22.5561 −0.976097
$$535$$ 0 0
$$536$$ −4.11021 −0.177534
$$537$$ −15.3933 −0.664270
$$538$$ −9.74175 −0.419996
$$539$$ −55.7677 −2.40208
$$540$$ 0 0
$$541$$ 19.5491 0.840480 0.420240 0.907413i $$-0.361946\pi$$
0.420240 + 0.907413i $$0.361946\pi$$
$$542$$ 26.3555 1.13206
$$543$$ −0.310596 −0.0133289
$$544$$ −6.48108 −0.277874
$$545$$ 0 0
$$546$$ 30.1927 1.29213
$$547$$ −41.8038 −1.78740 −0.893700 0.448665i $$-0.851900\pi$$
−0.893700 + 0.448665i $$0.851900\pi$$
$$548$$ −5.36317 −0.229103
$$549$$ 30.1703 1.28763
$$550$$ 0 0
$$551$$ −0.832162 −0.0354513
$$552$$ 17.0724 0.726648
$$553$$ 44.9019 1.90942
$$554$$ −0.962158 −0.0408782
$$555$$ 0 0
$$556$$ −10.1601 −0.430886
$$557$$ −9.76418 −0.413722 −0.206861 0.978370i $$-0.566325\pi$$
−0.206861 + 0.978370i $$0.566325\pi$$
$$558$$ −9.41839 −0.398712
$$559$$ −7.95513 −0.336466
$$560$$ 0 0
$$561$$ 75.1069 3.17102
$$562$$ −14.6714 −0.618874
$$563$$ 11.4509 0.482600 0.241300 0.970451i $$-0.422426\pi$$
0.241300 + 0.970451i $$0.422426\pi$$
$$564$$ −13.4483 −0.566275
$$565$$ 0 0
$$566$$ −26.1601 −1.09959
$$567$$ −46.0767 −1.93504
$$568$$ 3.82446 0.160471
$$569$$ −13.4338 −0.563174 −0.281587 0.959536i $$-0.590861\pi$$
−0.281587 + 0.959536i $$0.590861\pi$$
$$570$$ 0 0
$$571$$ 37.6335 1.57491 0.787457 0.616370i $$-0.211397\pi$$
0.787457 + 0.616370i $$0.211397\pi$$
$$572$$ −16.2754 −0.680509
$$573$$ 43.7090 1.82597
$$574$$ −49.2221 −2.05449
$$575$$ 0 0
$$576$$ 2.08777 0.0869905
$$577$$ 1.56885 0.0653121 0.0326560 0.999467i $$-0.489603\pi$$
0.0326560 + 0.999467i $$0.489603\pi$$
$$578$$ 25.0044 1.04005
$$579$$ −12.2446 −0.508868
$$580$$ 0 0
$$581$$ −51.1276 −2.12113
$$582$$ 8.88647 0.368356
$$583$$ −1.15840 −0.0479759
$$584$$ 4.70655 0.194758
$$585$$ 0 0
$$586$$ −22.4657 −0.928048
$$587$$ 25.8693 1.06774 0.533871 0.845566i $$-0.320737\pi$$
0.533871 + 0.845566i $$0.320737\pi$$
$$588$$ 24.4837 1.00969
$$589$$ 4.51122 0.185881
$$590$$ 0 0
$$591$$ −35.2978 −1.45196
$$592$$ 0.137699 0.00565940
$$593$$ −28.9243 −1.18778 −0.593890 0.804547i $$-0.702408\pi$$
−0.593890 + 0.804547i $$0.702408\pi$$
$$594$$ 10.5715 0.433754
$$595$$ 0 0
$$596$$ −5.93972 −0.243301
$$597$$ 40.7136 1.66630
$$598$$ 23.9769 0.980488
$$599$$ 41.3581 1.68985 0.844923 0.534887i $$-0.179646\pi$$
0.844923 + 0.534887i $$0.179646\pi$$
$$600$$ 0 0
$$601$$ 2.68147 0.109379 0.0546897 0.998503i $$-0.482583\pi$$
0.0546897 + 0.998503i $$0.482583\pi$$
$$602$$ −10.6111 −0.432475
$$603$$ −8.58117 −0.349452
$$604$$ −15.2978 −0.622460
$$605$$ 0 0
$$606$$ 7.21338 0.293024
$$607$$ 3.31324 0.134480 0.0672401 0.997737i $$-0.478581\pi$$
0.0672401 + 0.997737i $$0.478581\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 7.93134 0.321394
$$610$$ 0 0
$$611$$ −18.8871 −0.764092
$$612$$ −13.5310 −0.546959
$$613$$ −10.0603 −0.406331 −0.203165 0.979144i $$-0.565123\pi$$
−0.203165 + 0.979144i $$0.565123\pi$$
$$614$$ 17.2204 0.694959
$$615$$ 0 0
$$616$$ −21.7092 −0.874688
$$617$$ −27.6265 −1.11220 −0.556100 0.831115i $$-0.687703\pi$$
−0.556100 + 0.831115i $$0.687703\pi$$
$$618$$ −24.1051 −0.969651
$$619$$ 16.6714 0.670078 0.335039 0.942204i $$-0.391250\pi$$
0.335039 + 0.942204i $$0.391250\pi$$
$$620$$ 0 0
$$621$$ −15.5739 −0.624959
$$622$$ −7.87439 −0.315734
$$623$$ −42.2547 −1.69290
$$624$$ 7.14540 0.286045
$$625$$ 0 0
$$626$$ −25.1678 −1.00591
$$627$$ 11.5886 0.462806
$$628$$ −12.7866 −0.510242
$$629$$ −0.892439 −0.0355839
$$630$$ 0 0
$$631$$ −0.709194 −0.0282326 −0.0141163 0.999900i $$-0.504494\pi$$
−0.0141163 + 0.999900i $$0.504494\pi$$
$$632$$ 10.6265 0.422699
$$633$$ −16.9364 −0.673162
$$634$$ −23.9045 −0.949370
$$635$$ 0 0
$$636$$ 0.508572 0.0201662
$$637$$ 34.3856 1.36241
$$638$$ −4.27540 −0.169265
$$639$$ 7.98459 0.315866
$$640$$ 0 0
$$641$$ −2.43553 −0.0961978 −0.0480989 0.998843i $$-0.515316\pi$$
−0.0480989 + 0.998843i $$0.515316\pi$$
$$642$$ 24.3984 0.962927
$$643$$ 7.70390 0.303812 0.151906 0.988395i $$-0.451459\pi$$
0.151906 + 0.988395i $$0.451459\pi$$
$$644$$ 31.9819 1.26027
$$645$$ 0 0
$$646$$ 6.48108 0.254995
$$647$$ −10.0499 −0.395103 −0.197552 0.980292i $$-0.563299\pi$$
−0.197552 + 0.980292i $$0.563299\pi$$
$$648$$ −10.9045 −0.428370
$$649$$ −27.7092 −1.08768
$$650$$ 0 0
$$651$$ −42.9964 −1.68516
$$652$$ −11.4734 −0.449332
$$653$$ −7.09283 −0.277564 −0.138782 0.990323i $$-0.544319\pi$$
−0.138782 + 0.990323i $$0.544319\pi$$
$$654$$ 20.8597 0.815677
$$655$$ 0 0
$$656$$ −11.6489 −0.454814
$$657$$ 9.82620 0.383356
$$658$$ −25.1929 −0.982122
$$659$$ −1.16346 −0.0453218 −0.0226609 0.999743i $$-0.507214\pi$$
−0.0226609 + 0.999743i $$0.507214\pi$$
$$660$$ 0 0
$$661$$ 1.79432 0.0697909 0.0348955 0.999391i $$-0.488890\pi$$
0.0348955 + 0.999391i $$0.488890\pi$$
$$662$$ 30.7565 1.19538
$$663$$ −46.3099 −1.79853
$$664$$ −12.0999 −0.469566
$$665$$ 0 0
$$666$$ 0.287484 0.0111398
$$667$$ 6.29851 0.243879
$$668$$ 19.4131 0.751115
$$669$$ 62.7263 2.42514
$$670$$ 0 0
$$671$$ −74.2446 −2.86618
$$672$$ 9.53101 0.367667
$$673$$ −25.2824 −0.974566 −0.487283 0.873244i $$-0.662012\pi$$
−0.487283 + 0.873244i $$0.662012\pi$$
$$674$$ 13.4734 0.518975
$$675$$ 0 0
$$676$$ −2.96480 −0.114031
$$677$$ −4.89682 −0.188200 −0.0941001 0.995563i $$-0.529997\pi$$
−0.0941001 + 0.995563i $$0.529997\pi$$
$$678$$ 39.8091 1.52886
$$679$$ 16.6472 0.638860
$$680$$ 0 0
$$681$$ 20.1025 0.770330
$$682$$ 23.1773 0.887504
$$683$$ 10.1980 0.390215 0.195107 0.980782i $$-0.437494\pi$$
0.195107 + 0.980782i $$0.437494\pi$$
$$684$$ −2.08777 −0.0798279
$$685$$ 0 0
$$686$$ 16.2875 0.621859
$$687$$ −30.7866 −1.17458
$$688$$ −2.51122 −0.0957393
$$689$$ 0.714253 0.0272109
$$690$$ 0 0
$$691$$ −39.9846 −1.52109 −0.760543 0.649288i $$-0.775067\pi$$
−0.760543 + 0.649288i $$0.775067\pi$$
$$692$$ 9.78662 0.372031
$$693$$ −45.3238 −1.72171
$$694$$ 28.9468 1.09880
$$695$$ 0 0
$$696$$ 1.87703 0.0711487
$$697$$ 75.4975 2.85967
$$698$$ 27.9243 1.05695
$$699$$ 52.5002 1.98574
$$700$$ 0 0
$$701$$ 4.91729 0.185723 0.0928617 0.995679i $$-0.470399\pi$$
0.0928617 + 0.995679i $$0.470399\pi$$
$$702$$ −6.51825 −0.246015
$$703$$ −0.137699 −0.00519342
$$704$$ −5.13770 −0.193634
$$705$$ 0 0
$$706$$ 28.3099 1.06546
$$707$$ 13.5130 0.508207
$$708$$ 12.1652 0.457196
$$709$$ −36.8865 −1.38530 −0.692650 0.721274i $$-0.743557\pi$$
−0.692650 + 0.721274i $$0.743557\pi$$
$$710$$ 0 0
$$711$$ 22.1857 0.832027
$$712$$ −10.0000 −0.374766
$$713$$ −34.1447 −1.27873
$$714$$ −61.7712 −2.31173
$$715$$ 0 0
$$716$$ −6.82446 −0.255042
$$717$$ 8.41113 0.314119
$$718$$ 2.60163 0.0970921
$$719$$ −23.8891 −0.890914 −0.445457 0.895303i $$-0.646959\pi$$
−0.445457 + 0.895303i $$0.646959\pi$$
$$720$$ 0 0
$$721$$ −45.1566 −1.68172
$$722$$ 1.00000 0.0372161
$$723$$ −3.35811 −0.124889
$$724$$ −0.137699 −0.00511755
$$725$$ 0 0
$$726$$ 34.7272 1.28885
$$727$$ 19.6894 0.730240 0.365120 0.930961i $$-0.381028\pi$$
0.365120 + 0.930961i $$0.381028\pi$$
$$728$$ 13.3856 0.496104
$$729$$ −8.84251 −0.327500
$$730$$ 0 0
$$731$$ 16.2754 0.601967
$$732$$ 32.5957 1.20477
$$733$$ 1.76418 0.0651615 0.0325808 0.999469i $$-0.489627\pi$$
0.0325808 + 0.999469i $$0.489627\pi$$
$$734$$ 11.6489 0.429969
$$735$$ 0 0
$$736$$ 7.56885 0.278991
$$737$$ 21.1170 0.777855
$$738$$ −24.3203 −0.895241
$$739$$ 28.1755 1.03645 0.518227 0.855243i $$-0.326592\pi$$
0.518227 + 0.855243i $$0.326592\pi$$
$$740$$ 0 0
$$741$$ −7.14540 −0.262493
$$742$$ 0.952717 0.0349753
$$743$$ −5.42851 −0.199153 −0.0995763 0.995030i $$-0.531749\pi$$
−0.0995763 + 0.995030i $$0.531749\pi$$
$$744$$ −10.1755 −0.373053
$$745$$ 0 0
$$746$$ −12.8064 −0.468876
$$747$$ −25.2617 −0.924278
$$748$$ 33.2978 1.21749
$$749$$ 45.7059 1.67006
$$750$$ 0 0
$$751$$ 25.8640 0.943792 0.471896 0.881654i $$-0.343570\pi$$
0.471896 + 0.881654i $$0.343570\pi$$
$$752$$ −5.96216 −0.217418
$$753$$ −19.8192 −0.722251
$$754$$ 2.63615 0.0960031
$$755$$ 0 0
$$756$$ −8.69446 −0.316215
$$757$$ −3.76947 −0.137004 −0.0685019 0.997651i $$-0.521822\pi$$
−0.0685019 + 0.997651i $$0.521822\pi$$
$$758$$ −20.9122 −0.759566
$$759$$ −87.7127 −3.18377
$$760$$ 0 0
$$761$$ 18.3605 0.665568 0.332784 0.943003i $$-0.392012\pi$$
0.332784 + 0.943003i $$0.392012\pi$$
$$762$$ −28.8416 −1.04482
$$763$$ 39.0767 1.41467
$$764$$ 19.3779 0.701068
$$765$$ 0 0
$$766$$ 10.3511 0.374000
$$767$$ 17.0851 0.616908
$$768$$ 2.25561 0.0813923
$$769$$ −38.1300 −1.37500 −0.687501 0.726183i $$-0.741292\pi$$
−0.687501 + 0.726183i $$0.741292\pi$$
$$770$$ 0 0
$$771$$ 6.18431 0.222722
$$772$$ −5.42851 −0.195376
$$773$$ −29.1575 −1.04872 −0.524361 0.851496i $$-0.675696\pi$$
−0.524361 + 0.851496i $$0.675696\pi$$
$$774$$ −5.24285 −0.188450
$$775$$ 0 0
$$776$$ 3.93972 0.141428
$$777$$ 1.31241 0.0470825
$$778$$ 6.56620 0.235410
$$779$$ 11.6489 0.417366
$$780$$ 0 0
$$781$$ −19.6489 −0.703094
$$782$$ −49.0543 −1.75418
$$783$$ −1.71228 −0.0611920
$$784$$ 10.8546 0.387664
$$785$$ 0 0
$$786$$ −4.77121 −0.170183
$$787$$ −47.0165 −1.67596 −0.837978 0.545704i $$-0.816262\pi$$
−0.837978 + 0.545704i $$0.816262\pi$$
$$788$$ −15.6489 −0.557470
$$789$$ 6.19624 0.220592
$$790$$ 0 0
$$791$$ 74.5750 2.65158
$$792$$ −10.7263 −0.381144
$$793$$ 45.7782 1.62563
$$794$$ 23.5284 0.834990
$$795$$ 0 0
$$796$$ 18.0499 0.639763
$$797$$ −42.6359 −1.51024 −0.755121 0.655586i $$-0.772422\pi$$
−0.755121 + 0.655586i $$0.772422\pi$$
$$798$$ −9.53101 −0.337394
$$799$$ 38.6412 1.36703
$$800$$ 0 0
$$801$$ −20.8777 −0.737678
$$802$$ 6.22041 0.219650
$$803$$ −24.1808 −0.853323
$$804$$ −9.27102 −0.326964
$$805$$ 0 0
$$806$$ −14.2908 −0.503372
$$807$$ −21.9736 −0.773506
$$808$$ 3.19798 0.112504
$$809$$ 49.4630 1.73903 0.869514 0.493909i $$-0.164432\pi$$
0.869514 + 0.493909i $$0.164432\pi$$
$$810$$ 0 0
$$811$$ −16.7816 −0.589280 −0.294640 0.955608i $$-0.595200\pi$$
−0.294640 + 0.955608i $$0.595200\pi$$
$$812$$ 3.51628 0.123397
$$813$$ 59.4476 2.08492
$$814$$ −0.707457 −0.0247964
$$815$$ 0 0
$$816$$ −14.6188 −0.511760
$$817$$ 2.51122 0.0878564
$$818$$ −34.4905 −1.20593
$$819$$ 27.9461 0.976515
$$820$$ 0 0
$$821$$ 11.5337 0.402527 0.201264 0.979537i $$-0.435495\pi$$
0.201264 + 0.979537i $$0.435495\pi$$
$$822$$ −12.0972 −0.421939
$$823$$ 32.6309 1.13744 0.568720 0.822531i $$-0.307439\pi$$
0.568720 + 0.822531i $$0.307439\pi$$
$$824$$ −10.6868 −0.372291
$$825$$ 0 0
$$826$$ 22.7893 0.792940
$$827$$ 6.64650 0.231122 0.115561 0.993300i $$-0.463133\pi$$
0.115561 + 0.993300i $$0.463133\pi$$
$$828$$ 15.8020 0.549158
$$829$$ −30.9217 −1.07395 −0.536977 0.843597i $$-0.680434\pi$$
−0.536977 + 0.843597i $$0.680434\pi$$
$$830$$ 0 0
$$831$$ −2.17025 −0.0752852
$$832$$ 3.16784 0.109825
$$833$$ −70.3495 −2.43747
$$834$$ −22.9173 −0.793561
$$835$$ 0 0
$$836$$ 5.13770 0.177691
$$837$$ 9.28243 0.320848
$$838$$ −10.6265 −0.367086
$$839$$ −48.7109 −1.68169 −0.840844 0.541277i $$-0.817941\pi$$
−0.840844 + 0.541277i $$0.817941\pi$$
$$840$$ 0 0
$$841$$ −28.3075 −0.976121
$$842$$ −1.98021 −0.0682426
$$843$$ −33.0928 −1.13978
$$844$$ −7.50857 −0.258456
$$845$$ 0 0
$$846$$ −12.4476 −0.427958
$$847$$ 65.0551 2.23532
$$848$$ 0.225470 0.00774267
$$849$$ −59.0070 −2.02512
$$850$$ 0 0
$$851$$ 1.04222 0.0357270
$$852$$ 8.62648 0.295538
$$853$$ 46.1447 1.57997 0.789983 0.613129i $$-0.210089\pi$$
0.789983 + 0.613129i $$0.210089\pi$$
$$854$$ 61.0620 2.08950
$$855$$ 0 0
$$856$$ 10.8168 0.369709
$$857$$ −8.40607 −0.287146 −0.143573 0.989640i $$-0.545859\pi$$
−0.143573 + 0.989640i $$0.545859\pi$$
$$858$$ −36.7109 −1.25329
$$859$$ −8.49581 −0.289873 −0.144937 0.989441i $$-0.546298\pi$$
−0.144937 + 0.989441i $$0.546298\pi$$
$$860$$ 0 0
$$861$$ −111.026 −3.78375
$$862$$ 15.0774 0.513539
$$863$$ 3.37352 0.114836 0.0574179 0.998350i $$-0.481713\pi$$
0.0574179 + 0.998350i $$0.481713\pi$$
$$864$$ −2.05763 −0.0700021
$$865$$ 0 0
$$866$$ 13.5337 0.459892
$$867$$ 56.4001 1.91545
$$868$$ −19.0620 −0.647007
$$869$$ −54.5957 −1.85203
$$870$$ 0 0
$$871$$ −13.0205 −0.441182
$$872$$ 9.24791 0.313174
$$873$$ 8.22524 0.278382
$$874$$ −7.56885 −0.256020
$$875$$ 0 0
$$876$$ 10.6161 0.358686
$$877$$ −13.2780 −0.448368 −0.224184 0.974547i $$-0.571972\pi$$
−0.224184 + 0.974547i $$0.571972\pi$$
$$878$$ 39.1773 1.32217
$$879$$ −50.6738 −1.70918
$$880$$ 0 0
$$881$$ −26.6489 −0.897825 −0.448912 0.893576i $$-0.648188\pi$$
−0.448912 + 0.893576i $$0.648188\pi$$
$$882$$ 22.6619 0.763066
$$883$$ −47.7884 −1.60821 −0.804103 0.594490i $$-0.797354\pi$$
−0.804103 + 0.594490i $$0.797354\pi$$
$$884$$ −20.5310 −0.690532
$$885$$ 0 0
$$886$$ 19.3132 0.648841
$$887$$ −36.3304 −1.21985 −0.609927 0.792457i $$-0.708801\pi$$
−0.609927 + 0.792457i $$0.708801\pi$$
$$888$$ 0.310596 0.0104229
$$889$$ −54.0295 −1.81209
$$890$$ 0 0
$$891$$ 56.0242 1.87688
$$892$$ 27.8091 0.931116
$$893$$ 5.96216 0.199516
$$894$$ −13.3977 −0.448086
$$895$$ 0 0
$$896$$ 4.22547 0.141163
$$897$$ 54.0825 1.80576
$$898$$ 41.4131 1.38197
$$899$$ −3.75406 −0.125205
$$900$$ 0 0
$$901$$ −1.46129 −0.0486826
$$902$$ 59.8486 1.99274
$$903$$ −23.9344 −0.796488
$$904$$ 17.6489 0.586994
$$905$$ 0 0
$$906$$ −34.5059 −1.14638
$$907$$ −39.3873 −1.30784 −0.653918 0.756566i $$-0.726876\pi$$
−0.653918 + 0.756566i $$0.726876\pi$$
$$908$$ 8.91223 0.295763
$$909$$ 6.67664 0.221450
$$910$$ 0 0
$$911$$ −20.5561 −0.681054 −0.340527 0.940235i $$-0.610605\pi$$
−0.340527 + 0.940235i $$0.610605\pi$$
$$912$$ −2.25561 −0.0746907
$$913$$ 62.1654 2.05738
$$914$$ −37.8392 −1.25161
$$915$$ 0 0
$$916$$ −13.6489 −0.450973
$$917$$ −8.93799 −0.295158
$$918$$ 13.3357 0.440143
$$919$$ 12.4054 0.409216 0.204608 0.978844i $$-0.434408\pi$$
0.204608 + 0.978844i $$0.434408\pi$$
$$920$$ 0 0
$$921$$ 38.8425 1.27990
$$922$$ 1.58864 0.0523190
$$923$$ 12.1153 0.398779
$$924$$ −48.9674 −1.61091
$$925$$ 0 0
$$926$$ 40.3581 1.32625
$$927$$ −22.3115 −0.732806
$$928$$ 0.832162 0.0273171
$$929$$ −31.3330 −1.02800 −0.514002 0.857789i $$-0.671838\pi$$
−0.514002 + 0.857789i $$0.671838\pi$$
$$930$$ 0 0
$$931$$ −10.8546 −0.355745
$$932$$ 23.2754 0.762411
$$933$$ −17.7615 −0.581487
$$934$$ −39.5130 −1.29290
$$935$$ 0 0
$$936$$ 6.61372 0.216176
$$937$$ 7.27299 0.237598 0.118799 0.992918i $$-0.462096\pi$$
0.118799 + 0.992918i $$0.462096\pi$$
$$938$$ −17.3676 −0.567071
$$939$$ −56.7688 −1.85258
$$940$$ 0 0
$$941$$ 6.72128 0.219107 0.109554 0.993981i $$-0.465058\pi$$
0.109554 + 0.993981i $$0.465058\pi$$
$$942$$ −28.8416 −0.939710
$$943$$ −88.1689 −2.87117
$$944$$ 5.39331 0.175537
$$945$$ 0 0
$$946$$ 12.9019 0.419476
$$947$$ 10.0757 0.327416 0.163708 0.986509i $$-0.447655\pi$$
0.163708 + 0.986509i $$0.447655\pi$$
$$948$$ 23.9692 0.778483
$$949$$ 14.9096 0.483986
$$950$$ 0 0
$$951$$ −53.9193 −1.74845
$$952$$ −27.3856 −0.887573
$$953$$ −8.14473 −0.263834 −0.131917 0.991261i $$-0.542113\pi$$
−0.131917 + 0.991261i $$0.542113\pi$$
$$954$$ 0.470730 0.0152404
$$955$$ 0 0
$$956$$ 3.72898 0.120604
$$957$$ −9.64363 −0.311734
$$958$$ 7.61107 0.245903
$$959$$ −22.6619 −0.731791
$$960$$ 0 0
$$961$$ −10.6489 −0.343513
$$962$$ 0.436209 0.0140639
$$963$$ 22.5829 0.727724
$$964$$ −1.48878 −0.0479505
$$965$$ 0 0
$$966$$ 72.1388 2.32103
$$967$$ 47.1773 1.51712 0.758560 0.651604i $$-0.225903\pi$$
0.758560 + 0.651604i $$0.225903\pi$$
$$968$$ 15.3960 0.494845
$$969$$ 14.6188 0.469623
$$970$$ 0 0
$$971$$ 0.230528 0.00739801 0.00369901 0.999993i $$-0.498823\pi$$
0.00369901 + 0.999993i $$0.498823\pi$$
$$972$$ −18.4234 −0.590932
$$973$$ −42.9313 −1.37632
$$974$$ −20.3907 −0.653359
$$975$$ 0 0
$$976$$ 14.4509 0.462563
$$977$$ −5.80905 −0.185848 −0.0929240 0.995673i $$-0.529621\pi$$
−0.0929240 + 0.995673i $$0.529621\pi$$
$$978$$ −25.8794 −0.827533
$$979$$ 51.3770 1.64202
$$980$$ 0 0
$$981$$ 19.3075 0.616441
$$982$$ 25.0224 0.798498
$$983$$ −22.3511 −0.712889 −0.356444 0.934317i $$-0.616011\pi$$
−0.356444 + 0.934317i $$0.616011\pi$$
$$984$$ −26.2754 −0.837629
$$985$$ 0 0
$$986$$ −5.39331 −0.171758
$$987$$ −56.8254 −1.80877
$$988$$ −3.16784 −0.100782
$$989$$ −19.0070 −0.604388
$$990$$ 0 0
$$991$$ −34.9415 −1.10995 −0.554976 0.831866i $$-0.687273\pi$$
−0.554976 + 0.831866i $$0.687273\pi$$
$$992$$ −4.51122 −0.143231
$$993$$ 69.3746 2.20154
$$994$$ 16.1601 0.512568
$$995$$ 0 0
$$996$$ −27.2925 −0.864797
$$997$$ −9.35282 −0.296207 −0.148103 0.988972i $$-0.547317\pi$$
−0.148103 + 0.988972i $$0.547317\pi$$
$$998$$ 22.6111 0.715741
$$999$$ −0.283334 −0.00896430
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 950.2.a.m.1.2 yes 3
3.2 odd 2 8550.2.a.cj.1.3 3
4.3 odd 2 7600.2.a.bm.1.2 3
5.2 odd 4 950.2.b.g.799.5 6
5.3 odd 4 950.2.b.g.799.2 6
5.4 even 2 950.2.a.k.1.2 3
15.14 odd 2 8550.2.a.co.1.1 3
20.19 odd 2 7600.2.a.cb.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.2 3 5.4 even 2
950.2.a.m.1.2 yes 3 1.1 even 1 trivial
950.2.b.g.799.2 6 5.3 odd 4
950.2.b.g.799.5 6 5.2 odd 4
7600.2.a.bm.1.2 3 4.3 odd 2
7600.2.a.cb.1.2 3 20.19 odd 2
8550.2.a.cj.1.3 3 3.2 odd 2
8550.2.a.co.1.1 3 15.14 odd 2