# Properties

 Label 637.2.a.m.1.3 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.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: $$5.08647060876$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4507648.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 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 1.3 Root $$-1.20475$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.656184 q^{2} +0.204753 q^{3} -1.56942 q^{4} +1.35996 q^{5} -0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} +O(q^{10})$$ $$q-0.656184 q^{2} +0.204753 q^{3} -1.56942 q^{4} +1.35996 q^{5} -0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} -0.892385 q^{10} -1.90551 q^{11} -0.321345 q^{12} -1.00000 q^{13} +0.278457 q^{15} +1.60193 q^{16} -3.56633 q^{17} +1.94104 q^{18} -0.985255 q^{19} -2.13436 q^{20} +1.25037 q^{22} +1.69605 q^{23} +0.479573 q^{24} -3.15050 q^{25} +0.656184 q^{26} -1.21994 q^{27} +6.54835 q^{29} -0.182719 q^{30} -7.69550 q^{31} -5.73556 q^{32} -0.390160 q^{33} +2.34017 q^{34} +4.64247 q^{36} -2.02005 q^{37} +0.646508 q^{38} -0.204753 q^{39} +3.18530 q^{40} -9.88779 q^{41} -3.16639 q^{43} +2.99055 q^{44} -4.02287 q^{45} -1.11292 q^{46} -7.76416 q^{47} +0.328001 q^{48} +2.06731 q^{50} -0.730219 q^{51} +1.56942 q^{52} +0.354194 q^{53} +0.800502 q^{54} -2.59143 q^{55} -0.201734 q^{57} -4.29692 q^{58} -2.16385 q^{59} -0.437016 q^{60} -12.2002 q^{61} +5.04967 q^{62} +0.559715 q^{64} -1.35996 q^{65} +0.256017 q^{66} +11.3134 q^{67} +5.59709 q^{68} +0.347272 q^{69} -9.05268 q^{71} -6.92840 q^{72} +7.13619 q^{73} +1.32553 q^{74} -0.645076 q^{75} +1.54628 q^{76} +0.134356 q^{78} -5.39629 q^{79} +2.17857 q^{80} +8.62444 q^{81} +6.48821 q^{82} +2.03494 q^{83} -4.85008 q^{85} +2.07773 q^{86} +1.34080 q^{87} -4.46309 q^{88} +6.89864 q^{89} +2.63974 q^{90} -2.66182 q^{92} -1.57568 q^{93} +5.09472 q^{94} -1.33991 q^{95} -1.17437 q^{96} +14.6223 q^{97} +5.63665 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9}+O(q^{10})$$ 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 $$6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 12 q^{15} - 16 q^{17} - 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} - 6 q^{31} - 20 q^{32} - 4 q^{33} - 24 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{40} + 8 q^{41} + 2 q^{43} - 4 q^{44} - 14 q^{45} + 8 q^{46} - 30 q^{47} + 8 q^{48} + 8 q^{50} - 4 q^{51} - 4 q^{52} - 14 q^{53} + 48 q^{54} + 8 q^{55} + 4 q^{57} - 8 q^{58} - 24 q^{59} + 12 q^{60} - 28 q^{62} - 20 q^{64} + 6 q^{65} + 4 q^{66} + 16 q^{67} - 28 q^{68} + 20 q^{69} + 8 q^{71} + 28 q^{72} + 6 q^{73} - 12 q^{74} - 12 q^{75} + 16 q^{76} + 4 q^{78} - 22 q^{79} + 28 q^{80} + 46 q^{81} + 40 q^{82} - 50 q^{83} - 8 q^{85} - 16 q^{86} + 16 q^{87} - 44 q^{88} - 26 q^{89} + 40 q^{90} + 20 q^{92} + 16 q^{93} + 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 - 4 * q^10 + 4 * q^11 + 4 * q^12 - 6 * q^13 + 12 * q^15 - 16 * q^17 - 4 * q^18 - 2 * q^19 - 16 * q^20 - 12 * q^22 - 6 * q^23 - 12 * q^24 - 4 * q^25 - 20 * q^27 - 6 * q^29 - 6 * q^31 - 20 * q^32 - 4 * q^33 - 24 * q^36 - 8 * q^38 + 8 * q^39 - 4 * q^40 + 8 * q^41 + 2 * q^43 - 4 * q^44 - 14 * q^45 + 8 * q^46 - 30 * q^47 + 8 * q^48 + 8 * q^50 - 4 * q^51 - 4 * q^52 - 14 * q^53 + 48 * q^54 + 8 * q^55 + 4 * q^57 - 8 * q^58 - 24 * q^59 + 12 * q^60 - 28 * q^62 - 20 * q^64 + 6 * q^65 + 4 * q^66 + 16 * q^67 - 28 * q^68 + 20 * q^69 + 8 * q^71 + 28 * q^72 + 6 * q^73 - 12 * q^74 - 12 * q^75 + 16 * q^76 + 4 * q^78 - 22 * q^79 + 28 * q^80 + 46 * q^81 + 40 * q^82 - 50 * q^83 - 8 * q^85 - 16 * q^86 + 16 * q^87 - 44 * q^88 - 26 * q^89 + 40 * q^90 + 20 * q^92 + 16 * q^93 + 32 * q^94 - 6 * q^95 + 20 * q^96 + 14 * q^97 + 12 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.656184 −0.463992 −0.231996 0.972717i $$-0.574526\pi$$
−0.231996 + 0.972717i $$0.574526\pi$$
$$3$$ 0.204753 0.118214 0.0591072 0.998252i $$-0.481175\pi$$
0.0591072 + 0.998252i $$0.481175\pi$$
$$4$$ −1.56942 −0.784711
$$5$$ 1.35996 0.608194 0.304097 0.952641i $$-0.401645\pi$$
0.304097 + 0.952641i $$0.401645\pi$$
$$6$$ −0.134356 −0.0548505
$$7$$ 0 0
$$8$$ 2.34220 0.828092
$$9$$ −2.95808 −0.986025
$$10$$ −0.892385 −0.282197
$$11$$ −1.90551 −0.574534 −0.287267 0.957851i $$-0.592747\pi$$
−0.287267 + 0.957851i $$0.592747\pi$$
$$12$$ −0.321345 −0.0927642
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0.278457 0.0718972
$$16$$ 1.60193 0.400483
$$17$$ −3.56633 −0.864963 −0.432482 0.901643i $$-0.642362\pi$$
−0.432482 + 0.901643i $$0.642362\pi$$
$$18$$ 1.94104 0.457508
$$19$$ −0.985255 −0.226033 −0.113016 0.993593i $$-0.536051\pi$$
−0.113016 + 0.993593i $$0.536051\pi$$
$$20$$ −2.13436 −0.477256
$$21$$ 0 0
$$22$$ 1.25037 0.266579
$$23$$ 1.69605 0.353651 0.176826 0.984242i $$-0.443417\pi$$
0.176826 + 0.984242i $$0.443417\pi$$
$$24$$ 0.479573 0.0978924
$$25$$ −3.15050 −0.630100
$$26$$ 0.656184 0.128688
$$27$$ −1.21994 −0.234777
$$28$$ 0 0
$$29$$ 6.54835 1.21600 0.607999 0.793938i $$-0.291972\pi$$
0.607999 + 0.793938i $$0.291972\pi$$
$$30$$ −0.182719 −0.0333598
$$31$$ −7.69550 −1.38215 −0.691077 0.722781i $$-0.742863\pi$$
−0.691077 + 0.722781i $$0.742863\pi$$
$$32$$ −5.73556 −1.01391
$$33$$ −0.390160 −0.0679181
$$34$$ 2.34017 0.401336
$$35$$ 0 0
$$36$$ 4.64247 0.773745
$$37$$ −2.02005 −0.332095 −0.166047 0.986118i $$-0.553100\pi$$
−0.166047 + 0.986118i $$0.553100\pi$$
$$38$$ 0.646508 0.104877
$$39$$ −0.204753 −0.0327868
$$40$$ 3.18530 0.503640
$$41$$ −9.88779 −1.54421 −0.772107 0.635493i $$-0.780797\pi$$
−0.772107 + 0.635493i $$0.780797\pi$$
$$42$$ 0 0
$$43$$ −3.16639 −0.482869 −0.241435 0.970417i $$-0.577618\pi$$
−0.241435 + 0.970417i $$0.577618\pi$$
$$44$$ 2.99055 0.450843
$$45$$ −4.02287 −0.599694
$$46$$ −1.11292 −0.164091
$$47$$ −7.76416 −1.13252 −0.566260 0.824227i $$-0.691610\pi$$
−0.566260 + 0.824227i $$0.691610\pi$$
$$48$$ 0.328001 0.0473429
$$49$$ 0 0
$$50$$ 2.06731 0.292362
$$51$$ −0.730219 −0.102251
$$52$$ 1.56942 0.217640
$$53$$ 0.354194 0.0486522 0.0243261 0.999704i $$-0.492256\pi$$
0.0243261 + 0.999704i $$0.492256\pi$$
$$54$$ 0.800502 0.108935
$$55$$ −2.59143 −0.349428
$$56$$ 0 0
$$57$$ −0.201734 −0.0267203
$$58$$ −4.29692 −0.564214
$$59$$ −2.16385 −0.281709 −0.140854 0.990030i $$-0.544985\pi$$
−0.140854 + 0.990030i $$0.544985\pi$$
$$60$$ −0.437016 −0.0564186
$$61$$ −12.2002 −1.56207 −0.781037 0.624484i $$-0.785309\pi$$
−0.781037 + 0.624484i $$0.785309\pi$$
$$62$$ 5.04967 0.641308
$$63$$ 0 0
$$64$$ 0.559715 0.0699644
$$65$$ −1.35996 −0.168683
$$66$$ 0.256017 0.0315135
$$67$$ 11.3134 1.38215 0.691076 0.722782i $$-0.257137\pi$$
0.691076 + 0.722782i $$0.257137\pi$$
$$68$$ 5.59709 0.678746
$$69$$ 0.347272 0.0418067
$$70$$ 0 0
$$71$$ −9.05268 −1.07436 −0.537178 0.843469i $$-0.680510\pi$$
−0.537178 + 0.843469i $$0.680510\pi$$
$$72$$ −6.92840 −0.816520
$$73$$ 7.13619 0.835228 0.417614 0.908625i $$-0.362866\pi$$
0.417614 + 0.908625i $$0.362866\pi$$
$$74$$ 1.32553 0.154089
$$75$$ −0.645076 −0.0744869
$$76$$ 1.54628 0.177371
$$77$$ 0 0
$$78$$ 0.134356 0.0152128
$$79$$ −5.39629 −0.607130 −0.303565 0.952811i $$-0.598177\pi$$
−0.303565 + 0.952811i $$0.598177\pi$$
$$80$$ 2.17857 0.243571
$$81$$ 8.62444 0.958271
$$82$$ 6.48821 0.716503
$$83$$ 2.03494 0.223364 0.111682 0.993744i $$-0.464376\pi$$
0.111682 + 0.993744i $$0.464376\pi$$
$$84$$ 0 0
$$85$$ −4.85008 −0.526065
$$86$$ 2.07773 0.224048
$$87$$ 1.34080 0.143749
$$88$$ −4.46309 −0.475767
$$89$$ 6.89864 0.731254 0.365627 0.930761i $$-0.380855\pi$$
0.365627 + 0.930761i $$0.380855\pi$$
$$90$$ 2.63974 0.278253
$$91$$ 0 0
$$92$$ −2.66182 −0.277514
$$93$$ −1.57568 −0.163390
$$94$$ 5.09472 0.525480
$$95$$ −1.33991 −0.137472
$$96$$ −1.17437 −0.119859
$$97$$ 14.6223 1.48467 0.742334 0.670030i $$-0.233719\pi$$
0.742334 + 0.670030i $$0.233719\pi$$
$$98$$ 0 0
$$99$$ 5.63665 0.566505
$$100$$ 4.94447 0.494447
$$101$$ −2.81368 −0.279972 −0.139986 0.990153i $$-0.544706\pi$$
−0.139986 + 0.990153i $$0.544706\pi$$
$$102$$ 0.479158 0.0474437
$$103$$ −6.04988 −0.596112 −0.298056 0.954548i $$-0.596338\pi$$
−0.298056 + 0.954548i $$0.596338\pi$$
$$104$$ −2.34220 −0.229671
$$105$$ 0 0
$$106$$ −0.232416 −0.0225743
$$107$$ 17.3204 1.67442 0.837211 0.546880i $$-0.184185\pi$$
0.837211 + 0.546880i $$0.184185\pi$$
$$108$$ 1.91460 0.184232
$$109$$ 10.5762 1.01302 0.506509 0.862234i $$-0.330936\pi$$
0.506509 + 0.862234i $$0.330936\pi$$
$$110$$ 1.70045 0.162132
$$111$$ −0.413613 −0.0392584
$$112$$ 0 0
$$113$$ −2.34665 −0.220755 −0.110377 0.993890i $$-0.535206\pi$$
−0.110377 + 0.993890i $$0.535206\pi$$
$$114$$ 0.132375 0.0123980
$$115$$ 2.30657 0.215089
$$116$$ −10.2771 −0.954208
$$117$$ 2.95808 0.273474
$$118$$ 1.41988 0.130711
$$119$$ 0 0
$$120$$ 0.652201 0.0595375
$$121$$ −7.36902 −0.669911
$$122$$ 8.00557 0.724790
$$123$$ −2.02456 −0.182548
$$124$$ 12.0775 1.08459
$$125$$ −11.0844 −0.991417
$$126$$ 0 0
$$127$$ −18.1639 −1.61179 −0.805894 0.592060i $$-0.798315\pi$$
−0.805894 + 0.592060i $$0.798315\pi$$
$$128$$ 11.1038 0.981450
$$129$$ −0.648328 −0.0570821
$$130$$ 0.892385 0.0782674
$$131$$ −10.1014 −0.882560 −0.441280 0.897369i $$-0.645476\pi$$
−0.441280 + 0.897369i $$0.645476\pi$$
$$132$$ 0.612326 0.0532961
$$133$$ 0 0
$$134$$ −7.42367 −0.641308
$$135$$ −1.65907 −0.142790
$$136$$ −8.35306 −0.716269
$$137$$ 6.55601 0.560118 0.280059 0.959983i $$-0.409646\pi$$
0.280059 + 0.959983i $$0.409646\pi$$
$$138$$ −0.227875 −0.0193980
$$139$$ 13.8243 1.17256 0.586281 0.810108i $$-0.300591\pi$$
0.586281 + 0.810108i $$0.300591\pi$$
$$140$$ 0 0
$$141$$ −1.58974 −0.133880
$$142$$ 5.94023 0.498493
$$143$$ 1.90551 0.159347
$$144$$ −4.73864 −0.394887
$$145$$ 8.90551 0.739563
$$146$$ −4.68265 −0.387539
$$147$$ 0 0
$$148$$ 3.17032 0.260598
$$149$$ 8.09389 0.663078 0.331539 0.943442i $$-0.392432\pi$$
0.331539 + 0.943442i $$0.392432\pi$$
$$150$$ 0.423288 0.0345614
$$151$$ 12.0365 0.979520 0.489760 0.871857i $$-0.337084\pi$$
0.489760 + 0.871857i $$0.337084\pi$$
$$152$$ −2.30766 −0.187176
$$153$$ 10.5495 0.852876
$$154$$ 0 0
$$155$$ −10.4656 −0.840617
$$156$$ 0.321345 0.0257282
$$157$$ 18.3935 1.46796 0.733982 0.679169i $$-0.237660\pi$$
0.733982 + 0.679169i $$0.237660\pi$$
$$158$$ 3.54096 0.281704
$$159$$ 0.0725223 0.00575139
$$160$$ −7.80014 −0.616655
$$161$$ 0 0
$$162$$ −5.65922 −0.444630
$$163$$ 2.51286 0.196823 0.0984114 0.995146i $$-0.468624\pi$$
0.0984114 + 0.995146i $$0.468624\pi$$
$$164$$ 15.5181 1.21176
$$165$$ −0.530603 −0.0413074
$$166$$ −1.33530 −0.103639
$$167$$ −17.9095 −1.38588 −0.692941 0.720995i $$-0.743685\pi$$
−0.692941 + 0.720995i $$0.743685\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 3.18254 0.244090
$$171$$ 2.91446 0.222874
$$172$$ 4.96940 0.378913
$$173$$ −6.51981 −0.495692 −0.247846 0.968799i $$-0.579723\pi$$
−0.247846 + 0.968799i $$0.579723\pi$$
$$174$$ −0.879809 −0.0666982
$$175$$ 0 0
$$176$$ −3.05250 −0.230091
$$177$$ −0.443055 −0.0333020
$$178$$ −4.52678 −0.339296
$$179$$ −26.6014 −1.98828 −0.994141 0.108091i $$-0.965526\pi$$
−0.994141 + 0.108091i $$0.965526\pi$$
$$180$$ 6.31359 0.470587
$$181$$ 24.6402 1.83149 0.915746 0.401758i $$-0.131601\pi$$
0.915746 + 0.401758i $$0.131601\pi$$
$$182$$ 0 0
$$183$$ −2.49803 −0.184660
$$184$$ 3.97249 0.292856
$$185$$ −2.74720 −0.201978
$$186$$ 1.03394 0.0758119
$$187$$ 6.79569 0.496950
$$188$$ 12.1853 0.888701
$$189$$ 0 0
$$190$$ 0.879227 0.0637858
$$191$$ 3.02099 0.218591 0.109295 0.994009i $$-0.465141\pi$$
0.109295 + 0.994009i $$0.465141\pi$$
$$192$$ 0.114604 0.00827080
$$193$$ 21.2522 1.52977 0.764884 0.644168i $$-0.222796\pi$$
0.764884 + 0.644168i $$0.222796\pi$$
$$194$$ −9.59491 −0.688874
$$195$$ −0.278457 −0.0199407
$$196$$ 0 0
$$197$$ −2.72191 −0.193928 −0.0969639 0.995288i $$-0.530913\pi$$
−0.0969639 + 0.995288i $$0.530913\pi$$
$$198$$ −3.69868 −0.262854
$$199$$ 21.6670 1.53593 0.767967 0.640490i $$-0.221269\pi$$
0.767967 + 0.640490i $$0.221269\pi$$
$$200$$ −7.37910 −0.521781
$$201$$ 2.31646 0.163390
$$202$$ 1.84629 0.129905
$$203$$ 0 0
$$204$$ 1.14602 0.0802376
$$205$$ −13.4470 −0.939181
$$206$$ 3.96983 0.276591
$$207$$ −5.01705 −0.348709
$$208$$ −1.60193 −0.111074
$$209$$ 1.87741 0.129864
$$210$$ 0 0
$$211$$ 18.3885 1.26592 0.632959 0.774186i $$-0.281840\pi$$
0.632959 + 0.774186i $$0.281840\pi$$
$$212$$ −0.555879 −0.0381780
$$213$$ −1.85357 −0.127004
$$214$$ −11.3653 −0.776918
$$215$$ −4.30617 −0.293678
$$216$$ −2.85733 −0.194417
$$217$$ 0 0
$$218$$ −6.93995 −0.470033
$$219$$ 1.46116 0.0987359
$$220$$ 4.06704 0.274200
$$221$$ 3.56633 0.239898
$$222$$ 0.271406 0.0182156
$$223$$ −22.2216 −1.48807 −0.744035 0.668141i $$-0.767090\pi$$
−0.744035 + 0.668141i $$0.767090\pi$$
$$224$$ 0 0
$$225$$ 9.31943 0.621295
$$226$$ 1.53984 0.102428
$$227$$ −26.7374 −1.77463 −0.887313 0.461167i $$-0.847431\pi$$
−0.887313 + 0.461167i $$0.847431\pi$$
$$228$$ 0.316606 0.0209678
$$229$$ 16.7008 1.10362 0.551809 0.833970i $$-0.313938\pi$$
0.551809 + 0.833970i $$0.313938\pi$$
$$230$$ −1.51353 −0.0997994
$$231$$ 0 0
$$232$$ 15.3375 1.00696
$$233$$ −13.8776 −0.909149 −0.454575 0.890709i $$-0.650209\pi$$
−0.454575 + 0.890709i $$0.650209\pi$$
$$234$$ −1.94104 −0.126890
$$235$$ −10.5590 −0.688791
$$236$$ 3.39599 0.221060
$$237$$ −1.10491 −0.0717715
$$238$$ 0 0
$$239$$ 0.465845 0.0301330 0.0150665 0.999886i $$-0.495204\pi$$
0.0150665 + 0.999886i $$0.495204\pi$$
$$240$$ 0.446069 0.0287936
$$241$$ −5.69314 −0.366727 −0.183364 0.983045i $$-0.558699\pi$$
−0.183364 + 0.983045i $$0.558699\pi$$
$$242$$ 4.83543 0.310833
$$243$$ 5.42569 0.348058
$$244$$ 19.1473 1.22578
$$245$$ 0 0
$$246$$ 1.32848 0.0847010
$$247$$ 0.985255 0.0626902
$$248$$ −18.0244 −1.14455
$$249$$ 0.416662 0.0264049
$$250$$ 7.27339 0.460010
$$251$$ −17.9066 −1.13025 −0.565126 0.825005i $$-0.691172\pi$$
−0.565126 + 0.825005i $$0.691172\pi$$
$$252$$ 0 0
$$253$$ −3.23185 −0.203185
$$254$$ 11.9189 0.747857
$$255$$ −0.993070 −0.0621885
$$256$$ −8.40559 −0.525350
$$257$$ −25.9756 −1.62031 −0.810156 0.586214i $$-0.800618\pi$$
−0.810156 + 0.586214i $$0.800618\pi$$
$$258$$ 0.425423 0.0264857
$$259$$ 0 0
$$260$$ 2.13436 0.132367
$$261$$ −19.3705 −1.19901
$$262$$ 6.62835 0.409501
$$263$$ 27.8402 1.71670 0.858349 0.513067i $$-0.171491\pi$$
0.858349 + 0.513067i $$0.171491\pi$$
$$264$$ −0.913832 −0.0562425
$$265$$ 0.481690 0.0295900
$$266$$ 0 0
$$267$$ 1.41252 0.0864448
$$268$$ −17.7555 −1.08459
$$269$$ −5.90486 −0.360025 −0.180013 0.983664i $$-0.557614\pi$$
−0.180013 + 0.983664i $$0.557614\pi$$
$$270$$ 1.08865 0.0662533
$$271$$ 4.34932 0.264203 0.132101 0.991236i $$-0.457828\pi$$
0.132101 + 0.991236i $$0.457828\pi$$
$$272$$ −5.71303 −0.346403
$$273$$ 0 0
$$274$$ −4.30195 −0.259890
$$275$$ 6.00332 0.362014
$$276$$ −0.545017 −0.0328062
$$277$$ 1.08051 0.0649217 0.0324609 0.999473i $$-0.489666\pi$$
0.0324609 + 0.999473i $$0.489666\pi$$
$$278$$ −9.07129 −0.544060
$$279$$ 22.7639 1.36284
$$280$$ 0 0
$$281$$ 14.4912 0.864473 0.432236 0.901760i $$-0.357725\pi$$
0.432236 + 0.901760i $$0.357725\pi$$
$$282$$ 1.04316 0.0621193
$$283$$ −24.4979 −1.45625 −0.728123 0.685446i $$-0.759607\pi$$
−0.728123 + 0.685446i $$0.759607\pi$$
$$284$$ 14.2075 0.843059
$$285$$ −0.274351 −0.0162511
$$286$$ −1.25037 −0.0739357
$$287$$ 0 0
$$288$$ 16.9662 0.999744
$$289$$ −4.28126 −0.251839
$$290$$ −5.84365 −0.343151
$$291$$ 2.99396 0.175509
$$292$$ −11.1997 −0.655413
$$293$$ −1.16105 −0.0678296 −0.0339148 0.999425i $$-0.510797\pi$$
−0.0339148 + 0.999425i $$0.510797\pi$$
$$294$$ 0 0
$$295$$ −2.94275 −0.171334
$$296$$ −4.73136 −0.275005
$$297$$ 2.32460 0.134887
$$298$$ −5.31108 −0.307663
$$299$$ −1.69605 −0.0980852
$$300$$ 1.01240 0.0584507
$$301$$ 0 0
$$302$$ −7.89818 −0.454489
$$303$$ −0.576111 −0.0330967
$$304$$ −1.57831 −0.0905224
$$305$$ −16.5918 −0.950044
$$306$$ −6.92240 −0.395728
$$307$$ −25.3731 −1.44812 −0.724060 0.689737i $$-0.757726\pi$$
−0.724060 + 0.689737i $$0.757726\pi$$
$$308$$ 0 0
$$309$$ −1.23873 −0.0704690
$$310$$ 6.86736 0.390040
$$311$$ −24.4645 −1.38726 −0.693628 0.720333i $$-0.743989\pi$$
−0.693628 + 0.720333i $$0.743989\pi$$
$$312$$ −0.479573 −0.0271505
$$313$$ −6.61665 −0.373995 −0.186997 0.982360i $$-0.559876\pi$$
−0.186997 + 0.982360i $$0.559876\pi$$
$$314$$ −12.0695 −0.681124
$$315$$ 0 0
$$316$$ 8.46906 0.476422
$$317$$ 8.19803 0.460447 0.230224 0.973138i $$-0.426054\pi$$
0.230224 + 0.973138i $$0.426054\pi$$
$$318$$ −0.0475880 −0.00266860
$$319$$ −12.4780 −0.698632
$$320$$ 0.761192 0.0425519
$$321$$ 3.54640 0.197941
$$322$$ 0 0
$$323$$ 3.51375 0.195510
$$324$$ −13.5354 −0.751966
$$325$$ 3.15050 0.174758
$$326$$ −1.64890 −0.0913242
$$327$$ 2.16552 0.119753
$$328$$ −23.1592 −1.27875
$$329$$ 0 0
$$330$$ 0.348173 0.0191663
$$331$$ −0.724715 −0.0398339 −0.0199170 0.999802i $$-0.506340\pi$$
−0.0199170 + 0.999802i $$0.506340\pi$$
$$332$$ −3.19369 −0.175276
$$333$$ 5.97547 0.327454
$$334$$ 11.7519 0.643038
$$335$$ 15.3858 0.840616
$$336$$ 0 0
$$337$$ −8.58299 −0.467545 −0.233773 0.972291i $$-0.575107\pi$$
−0.233773 + 0.972291i $$0.575107\pi$$
$$338$$ −0.656184 −0.0356917
$$339$$ −0.480485 −0.0260964
$$340$$ 7.61183 0.412809
$$341$$ 14.6639 0.794094
$$342$$ −1.91242 −0.103412
$$343$$ 0 0
$$344$$ −7.41630 −0.399860
$$345$$ 0.472277 0.0254266
$$346$$ 4.27820 0.229997
$$347$$ −25.9250 −1.39173 −0.695864 0.718173i $$-0.744979\pi$$
−0.695864 + 0.718173i $$0.744979\pi$$
$$348$$ −2.10428 −0.112801
$$349$$ −17.1403 −0.917501 −0.458750 0.888565i $$-0.651703\pi$$
−0.458750 + 0.888565i $$0.651703\pi$$
$$350$$ 0 0
$$351$$ 1.21994 0.0651154
$$352$$ 10.9292 0.582527
$$353$$ 18.9101 1.00648 0.503242 0.864146i $$-0.332140\pi$$
0.503242 + 0.864146i $$0.332140\pi$$
$$354$$ 0.290725 0.0154519
$$355$$ −12.3113 −0.653417
$$356$$ −10.8269 −0.573823
$$357$$ 0 0
$$358$$ 17.4554 0.922547
$$359$$ 14.1049 0.744431 0.372215 0.928146i $$-0.378598\pi$$
0.372215 + 0.928146i $$0.378598\pi$$
$$360$$ −9.42236 −0.496602
$$361$$ −18.0293 −0.948909
$$362$$ −16.1685 −0.849798
$$363$$ −1.50883 −0.0791931
$$364$$ 0 0
$$365$$ 9.70495 0.507980
$$366$$ 1.63917 0.0856806
$$367$$ −0.360258 −0.0188053 −0.00940266 0.999956i $$-0.502993\pi$$
−0.00940266 + 0.999956i $$0.502993\pi$$
$$368$$ 2.71696 0.141631
$$369$$ 29.2488 1.52263
$$370$$ 1.80267 0.0937162
$$371$$ 0 0
$$372$$ 2.47291 0.128214
$$373$$ −25.1680 −1.30315 −0.651575 0.758584i $$-0.725892\pi$$
−0.651575 + 0.758584i $$0.725892\pi$$
$$374$$ −4.45923 −0.230581
$$375$$ −2.26956 −0.117200
$$376$$ −18.1852 −0.937830
$$377$$ −6.54835 −0.337257
$$378$$ 0 0
$$379$$ 31.8947 1.63832 0.819161 0.573564i $$-0.194440\pi$$
0.819161 + 0.573564i $$0.194440\pi$$
$$380$$ 2.10288 0.107876
$$381$$ −3.71913 −0.190537
$$382$$ −1.98232 −0.101424
$$383$$ −32.8172 −1.67688 −0.838440 0.544994i $$-0.816532\pi$$
−0.838440 + 0.544994i $$0.816532\pi$$
$$384$$ 2.27355 0.116022
$$385$$ 0 0
$$386$$ −13.9454 −0.709800
$$387$$ 9.36641 0.476122
$$388$$ −22.9486 −1.16504
$$389$$ −25.3087 −1.28320 −0.641600 0.767040i $$-0.721729\pi$$
−0.641600 + 0.767040i $$0.721729\pi$$
$$390$$ 0.182719 0.00925233
$$391$$ −6.04869 −0.305895
$$392$$ 0 0
$$393$$ −2.06829 −0.104331
$$394$$ 1.78607 0.0899810
$$395$$ −7.33875 −0.369253
$$396$$ −8.84629 −0.444543
$$397$$ 5.29395 0.265696 0.132848 0.991136i $$-0.457588\pi$$
0.132848 + 0.991136i $$0.457588\pi$$
$$398$$ −14.2175 −0.712661
$$399$$ 0 0
$$400$$ −5.04689 −0.252345
$$401$$ −24.6335 −1.23014 −0.615070 0.788473i $$-0.710872\pi$$
−0.615070 + 0.788473i $$0.710872\pi$$
$$402$$ −1.52002 −0.0758118
$$403$$ 7.69550 0.383340
$$404$$ 4.41586 0.219697
$$405$$ 11.7289 0.582815
$$406$$ 0 0
$$407$$ 3.84924 0.190800
$$408$$ −1.71032 −0.0846733
$$409$$ −10.2223 −0.505458 −0.252729 0.967537i $$-0.581328\pi$$
−0.252729 + 0.967537i $$0.581328\pi$$
$$410$$ 8.82372 0.435773
$$411$$ 1.34236 0.0662140
$$412$$ 9.49481 0.467776
$$413$$ 0 0
$$414$$ 3.29211 0.161798
$$415$$ 2.76745 0.135849
$$416$$ 5.73556 0.281209
$$417$$ 2.83057 0.138614
$$418$$ −1.23193 −0.0602556
$$419$$ −30.6503 −1.49736 −0.748682 0.662929i $$-0.769313\pi$$
−0.748682 + 0.662929i $$0.769313\pi$$
$$420$$ 0 0
$$421$$ 25.9764 1.26601 0.633007 0.774146i $$-0.281820\pi$$
0.633007 + 0.774146i $$0.281820\pi$$
$$422$$ −12.0662 −0.587376
$$423$$ 22.9670 1.11669
$$424$$ 0.829592 0.0402885
$$425$$ 11.2357 0.545014
$$426$$ 1.21628 0.0589290
$$427$$ 0 0
$$428$$ −27.1830 −1.31394
$$429$$ 0.390160 0.0188371
$$430$$ 2.82564 0.136264
$$431$$ 14.8818 0.716829 0.358415 0.933563i $$-0.383317\pi$$
0.358415 + 0.933563i $$0.383317\pi$$
$$432$$ −1.95426 −0.0940242
$$433$$ 40.0871 1.92647 0.963233 0.268669i $$-0.0865837\pi$$
0.963233 + 0.268669i $$0.0865837\pi$$
$$434$$ 0 0
$$435$$ 1.82343 0.0874269
$$436$$ −16.5986 −0.794927
$$437$$ −1.67104 −0.0799368
$$438$$ −0.958789 −0.0458127
$$439$$ 3.00243 0.143298 0.0716491 0.997430i $$-0.477174\pi$$
0.0716491 + 0.997430i $$0.477174\pi$$
$$440$$ −6.06963 −0.289358
$$441$$ 0 0
$$442$$ −2.34017 −0.111311
$$443$$ 1.43566 0.0682101 0.0341051 0.999418i $$-0.489142\pi$$
0.0341051 + 0.999418i $$0.489142\pi$$
$$444$$ 0.649133 0.0308065
$$445$$ 9.38189 0.444744
$$446$$ 14.5815 0.690453
$$447$$ 1.65725 0.0783853
$$448$$ 0 0
$$449$$ 19.6313 0.926460 0.463230 0.886238i $$-0.346690\pi$$
0.463230 + 0.886238i $$0.346690\pi$$
$$450$$ −6.11526 −0.288276
$$451$$ 18.8413 0.887203
$$452$$ 3.68289 0.173229
$$453$$ 2.46452 0.115793
$$454$$ 17.5447 0.823413
$$455$$ 0 0
$$456$$ −0.472501 −0.0221269
$$457$$ 10.6421 0.497815 0.248907 0.968527i $$-0.419929\pi$$
0.248907 + 0.968527i $$0.419929\pi$$
$$458$$ −10.9588 −0.512070
$$459$$ 4.35070 0.203073
$$460$$ −3.61998 −0.168782
$$461$$ 38.5930 1.79745 0.898727 0.438509i $$-0.144493\pi$$
0.898727 + 0.438509i $$0.144493\pi$$
$$462$$ 0 0
$$463$$ −27.9993 −1.30124 −0.650618 0.759405i $$-0.725490\pi$$
−0.650618 + 0.759405i $$0.725490\pi$$
$$464$$ 10.4900 0.486987
$$465$$ −2.14287 −0.0993730
$$466$$ 9.10623 0.421838
$$467$$ 25.8199 1.19480 0.597401 0.801943i $$-0.296200\pi$$
0.597401 + 0.801943i $$0.296200\pi$$
$$468$$ −4.64247 −0.214598
$$469$$ 0 0
$$470$$ 6.92862 0.319594
$$471$$ 3.76614 0.173535
$$472$$ −5.06816 −0.233281
$$473$$ 6.03359 0.277425
$$474$$ 0.725023 0.0333014
$$475$$ 3.10405 0.142423
$$476$$ 0 0
$$477$$ −1.04773 −0.0479723
$$478$$ −0.305680 −0.0139815
$$479$$ 39.4071 1.80055 0.900277 0.435317i $$-0.143364\pi$$
0.900277 + 0.435317i $$0.143364\pi$$
$$480$$ −1.59711 −0.0728976
$$481$$ 2.02005 0.0921065
$$482$$ 3.73575 0.170159
$$483$$ 0 0
$$484$$ 11.5651 0.525687
$$485$$ 19.8858 0.902966
$$486$$ −3.56025 −0.161496
$$487$$ −34.0959 −1.54503 −0.772517 0.634994i $$-0.781003\pi$$
−0.772517 + 0.634994i $$0.781003\pi$$
$$488$$ −28.5753 −1.29354
$$489$$ 0.514517 0.0232673
$$490$$ 0 0
$$491$$ −9.70414 −0.437942 −0.218971 0.975731i $$-0.570270\pi$$
−0.218971 + 0.975731i $$0.570270\pi$$
$$492$$ 3.17739 0.143248
$$493$$ −23.3536 −1.05179
$$494$$ −0.646508 −0.0290878
$$495$$ 7.66563 0.344545
$$496$$ −12.3277 −0.553529
$$497$$ 0 0
$$498$$ −0.273407 −0.0122516
$$499$$ 18.0463 0.807862 0.403931 0.914789i $$-0.367644\pi$$
0.403931 + 0.914789i $$0.367644\pi$$
$$500$$ 17.3961 0.777976
$$501$$ −3.66704 −0.163831
$$502$$ 11.7500 0.524428
$$503$$ −16.4922 −0.735352 −0.367676 0.929954i $$-0.619846\pi$$
−0.367676 + 0.929954i $$0.619846\pi$$
$$504$$ 0 0
$$505$$ −3.82650 −0.170277
$$506$$ 2.12069 0.0942761
$$507$$ 0.204753 0.00909341
$$508$$ 28.5069 1.26479
$$509$$ 19.1977 0.850921 0.425460 0.904977i $$-0.360112\pi$$
0.425460 + 0.904977i $$0.360112\pi$$
$$510$$ 0.651637 0.0288550
$$511$$ 0 0
$$512$$ −16.6921 −0.737692
$$513$$ 1.20195 0.0530673
$$514$$ 17.0448 0.751812
$$515$$ −8.22760 −0.362552
$$516$$ 1.01750 0.0447930
$$517$$ 14.7947 0.650670
$$518$$ 0 0
$$519$$ −1.33495 −0.0585979
$$520$$ −3.18530 −0.139685
$$521$$ 8.76034 0.383797 0.191899 0.981415i $$-0.438535\pi$$
0.191899 + 0.981415i $$0.438535\pi$$
$$522$$ 12.7106 0.556329
$$523$$ 11.7406 0.513381 0.256690 0.966494i $$-0.417368\pi$$
0.256690 + 0.966494i $$0.417368\pi$$
$$524$$ 15.8533 0.692555
$$525$$ 0 0
$$526$$ −18.2683 −0.796534
$$527$$ 27.4447 1.19551
$$528$$ −0.625010 −0.0272001
$$529$$ −20.1234 −0.874931
$$530$$ −0.316077 −0.0137295
$$531$$ 6.40082 0.277772
$$532$$ 0 0
$$533$$ 9.88779 0.428288
$$534$$ −0.926872 −0.0401097
$$535$$ 23.5550 1.01837
$$536$$ 26.4982 1.14455
$$537$$ −5.44673 −0.235044
$$538$$ 3.87467 0.167049
$$539$$ 0 0
$$540$$ 2.60378 0.112049
$$541$$ 16.4587 0.707614 0.353807 0.935318i $$-0.384887\pi$$
0.353807 + 0.935318i $$0.384887\pi$$
$$542$$ −2.85396 −0.122588
$$543$$ 5.04516 0.216509
$$544$$ 20.4549 0.876997
$$545$$ 14.3833 0.616112
$$546$$ 0 0
$$547$$ −8.19375 −0.350339 −0.175170 0.984538i $$-0.556047\pi$$
−0.175170 + 0.984538i $$0.556047\pi$$
$$548$$ −10.2891 −0.439531
$$549$$ 36.0891 1.54025
$$550$$ −3.93928 −0.167972
$$551$$ −6.45179 −0.274856
$$552$$ 0.813381 0.0346198
$$553$$ 0 0
$$554$$ −0.709015 −0.0301232
$$555$$ −0.562498 −0.0238767
$$556$$ −21.6962 −0.920123
$$557$$ −13.8144 −0.585335 −0.292667 0.956214i $$-0.594543\pi$$
−0.292667 + 0.956214i $$0.594543\pi$$
$$558$$ −14.9373 −0.632346
$$559$$ 3.16639 0.133924
$$560$$ 0 0
$$561$$ 1.39144 0.0587467
$$562$$ −9.50889 −0.401108
$$563$$ −9.75559 −0.411149 −0.205575 0.978641i $$-0.565906\pi$$
−0.205575 + 0.978641i $$0.565906\pi$$
$$564$$ 2.49497 0.105057
$$565$$ −3.19136 −0.134262
$$566$$ 16.0751 0.675687
$$567$$ 0 0
$$568$$ −21.2032 −0.889666
$$569$$ −1.43264 −0.0600593 −0.0300296 0.999549i $$-0.509560\pi$$
−0.0300296 + 0.999549i $$0.509560\pi$$
$$570$$ 0.180025 0.00754040
$$571$$ 16.2872 0.681598 0.340799 0.940136i $$-0.389303\pi$$
0.340799 + 0.940136i $$0.389303\pi$$
$$572$$ −2.99055 −0.125041
$$573$$ 0.618557 0.0258406
$$574$$ 0 0
$$575$$ −5.34342 −0.222836
$$576$$ −1.65568 −0.0689867
$$577$$ 4.81846 0.200595 0.100297 0.994957i $$-0.468021\pi$$
0.100297 + 0.994957i $$0.468021\pi$$
$$578$$ 2.80929 0.116851
$$579$$ 4.35146 0.180841
$$580$$ −13.9765 −0.580343
$$581$$ 0 0
$$582$$ −1.96459 −0.0814349
$$583$$ −0.674920 −0.0279523
$$584$$ 16.7144 0.691645
$$585$$ 4.02287 0.166325
$$586$$ 0.761866 0.0314724
$$587$$ −36.6215 −1.51153 −0.755765 0.654843i $$-0.772735\pi$$
−0.755765 + 0.654843i $$0.772735\pi$$
$$588$$ 0 0
$$589$$ 7.58203 0.312412
$$590$$ 1.93099 0.0794974
$$591$$ −0.557319 −0.0229251
$$592$$ −3.23599 −0.132998
$$593$$ 18.7081 0.768251 0.384125 0.923281i $$-0.374503\pi$$
0.384125 + 0.923281i $$0.374503\pi$$
$$594$$ −1.52537 −0.0625866
$$595$$ 0 0
$$596$$ −12.7027 −0.520325
$$597$$ 4.43639 0.181569
$$598$$ 1.11292 0.0455108
$$599$$ −25.4194 −1.03861 −0.519305 0.854589i $$-0.673809\pi$$
−0.519305 + 0.854589i $$0.673809\pi$$
$$600$$ −1.51090 −0.0616820
$$601$$ −17.7790 −0.725219 −0.362609 0.931941i $$-0.618114\pi$$
−0.362609 + 0.931941i $$0.618114\pi$$
$$602$$ 0 0
$$603$$ −33.4659 −1.36284
$$604$$ −18.8904 −0.768640
$$605$$ −10.0216 −0.407436
$$606$$ 0.378035 0.0153566
$$607$$ −26.0047 −1.05550 −0.527750 0.849400i $$-0.676964\pi$$
−0.527750 + 0.849400i $$0.676964\pi$$
$$608$$ 5.65098 0.229178
$$609$$ 0 0
$$610$$ 10.8873 0.440813
$$611$$ 7.76416 0.314104
$$612$$ −16.5566 −0.669261
$$613$$ 26.8230 1.08337 0.541685 0.840582i $$-0.317787\pi$$
0.541685 + 0.840582i $$0.317787\pi$$
$$614$$ 16.6494 0.671916
$$615$$ −2.75332 −0.111025
$$616$$ 0 0
$$617$$ −18.6491 −0.750784 −0.375392 0.926866i $$-0.622492\pi$$
−0.375392 + 0.926866i $$0.622492\pi$$
$$618$$ 0.812836 0.0326971
$$619$$ −13.0685 −0.525269 −0.262634 0.964895i $$-0.584591\pi$$
−0.262634 + 0.964895i $$0.584591\pi$$
$$620$$ 16.4249 0.659642
$$621$$ −2.06908 −0.0830291
$$622$$ 16.0532 0.643676
$$623$$ 0 0
$$624$$ −0.328001 −0.0131306
$$625$$ 0.678175 0.0271270
$$626$$ 4.34174 0.173531
$$627$$ 0.384407 0.0153517
$$628$$ −28.8672 −1.15193
$$629$$ 7.20418 0.287250
$$630$$ 0 0
$$631$$ −49.1745 −1.95761 −0.978804 0.204801i $$-0.934345\pi$$
−0.978804 + 0.204801i $$0.934345\pi$$
$$632$$ −12.6392 −0.502760
$$633$$ 3.76511 0.149650
$$634$$ −5.37941 −0.213644
$$635$$ −24.7023 −0.980279
$$636$$ −0.113818 −0.00451318
$$637$$ 0 0
$$638$$ 8.18784 0.324160
$$639$$ 26.7785 1.05934
$$640$$ 15.1008 0.596912
$$641$$ 4.37503 0.172803 0.0864016 0.996260i $$-0.472463\pi$$
0.0864016 + 0.996260i $$0.472463\pi$$
$$642$$ −2.32709 −0.0918429
$$643$$ 5.34651 0.210846 0.105423 0.994427i $$-0.466380\pi$$
0.105423 + 0.994427i $$0.466380\pi$$
$$644$$ 0 0
$$645$$ −0.881702 −0.0347170
$$646$$ −2.30566 −0.0907151
$$647$$ 29.7346 1.16899 0.584493 0.811398i $$-0.301293\pi$$
0.584493 + 0.811398i $$0.301293\pi$$
$$648$$ 20.2001 0.793537
$$649$$ 4.12324 0.161851
$$650$$ −2.06731 −0.0810865
$$651$$ 0 0
$$652$$ −3.94375 −0.154449
$$653$$ −2.54262 −0.0995005 −0.0497503 0.998762i $$-0.515843\pi$$
−0.0497503 + 0.998762i $$0.515843\pi$$
$$654$$ −1.42098 −0.0555646
$$655$$ −13.7375 −0.536768
$$656$$ −15.8396 −0.618432
$$657$$ −21.1094 −0.823556
$$658$$ 0 0
$$659$$ 1.87019 0.0728523 0.0364261 0.999336i $$-0.488403\pi$$
0.0364261 + 0.999336i $$0.488403\pi$$
$$660$$ 0.832740 0.0324144
$$661$$ 12.0763 0.469714 0.234857 0.972030i $$-0.424538\pi$$
0.234857 + 0.972030i $$0.424538\pi$$
$$662$$ 0.475546 0.0184826
$$663$$ 0.730219 0.0283593
$$664$$ 4.76624 0.184966
$$665$$ 0 0
$$666$$ −3.92101 −0.151936
$$667$$ 11.1063 0.430040
$$668$$ 28.1076 1.08752
$$669$$ −4.54995 −0.175911
$$670$$ −10.0959 −0.390039
$$671$$ 23.2476 0.897464
$$672$$ 0 0
$$673$$ −7.81691 −0.301320 −0.150660 0.988586i $$-0.548140\pi$$
−0.150660 + 0.988586i $$0.548140\pi$$
$$674$$ 5.63202 0.216937
$$675$$ 3.84341 0.147933
$$676$$ −1.56942 −0.0603624
$$677$$ −6.25628 −0.240448 −0.120224 0.992747i $$-0.538361\pi$$
−0.120224 + 0.992747i $$0.538361\pi$$
$$678$$ 0.315287 0.0121085
$$679$$ 0 0
$$680$$ −11.3598 −0.435630
$$681$$ −5.47458 −0.209786
$$682$$ −9.62220 −0.368453
$$683$$ 22.3656 0.855795 0.427898 0.903827i $$-0.359254\pi$$
0.427898 + 0.903827i $$0.359254\pi$$
$$684$$ −4.57402 −0.174892
$$685$$ 8.91593 0.340660
$$686$$ 0 0
$$687$$ 3.41954 0.130464
$$688$$ −5.07234 −0.193381
$$689$$ −0.354194 −0.0134937
$$690$$ −0.309901 −0.0117977
$$691$$ 3.80771 0.144852 0.0724260 0.997374i $$-0.476926\pi$$
0.0724260 + 0.997374i $$0.476926\pi$$
$$692$$ 10.2323 0.388975
$$693$$ 0 0
$$694$$ 17.0116 0.645751
$$695$$ 18.8005 0.713145
$$696$$ 3.14041 0.119037
$$697$$ 35.2632 1.33569
$$698$$ 11.2472 0.425713
$$699$$ −2.84148 −0.107475
$$700$$ 0 0
$$701$$ −8.14218 −0.307526 −0.153763 0.988108i $$-0.549139\pi$$
−0.153763 + 0.988108i $$0.549139\pi$$
$$702$$ −0.800502 −0.0302130
$$703$$ 1.99027 0.0750643
$$704$$ −1.06654 −0.0401969
$$705$$ −2.16198 −0.0814250
$$706$$ −12.4085 −0.467001
$$707$$ 0 0
$$708$$ 0.695340 0.0261325
$$709$$ −24.0319 −0.902536 −0.451268 0.892388i $$-0.649028\pi$$
−0.451268 + 0.892388i $$0.649028\pi$$
$$710$$ 8.07848 0.303180
$$711$$ 15.9626 0.598646
$$712$$ 16.1580 0.605546
$$713$$ −13.0520 −0.488800
$$714$$ 0 0
$$715$$ 2.59143 0.0969138
$$716$$ 41.7488 1.56023
$$717$$ 0.0953833 0.00356215
$$718$$ −9.25544 −0.345410
$$719$$ 32.6960 1.21936 0.609678 0.792649i $$-0.291299\pi$$
0.609678 + 0.792649i $$0.291299\pi$$
$$720$$ −6.44437 −0.240168
$$721$$ 0 0
$$722$$ 11.8305 0.440286
$$723$$ −1.16569 −0.0433525
$$724$$ −38.6709 −1.43719
$$725$$ −20.6306 −0.766201
$$726$$ 0.990071 0.0367450
$$727$$ 15.7712 0.584921 0.292460 0.956278i $$-0.405526\pi$$
0.292460 + 0.956278i $$0.405526\pi$$
$$728$$ 0 0
$$729$$ −24.7624 −0.917126
$$730$$ −6.36823 −0.235699
$$731$$ 11.2924 0.417664
$$732$$ 3.92046 0.144905
$$733$$ −23.5094 −0.868340 −0.434170 0.900831i $$-0.642958\pi$$
−0.434170 + 0.900831i $$0.642958\pi$$
$$734$$ 0.236396 0.00872552
$$735$$ 0 0
$$736$$ −9.72781 −0.358572
$$737$$ −21.5578 −0.794093
$$738$$ −19.1926 −0.706490
$$739$$ −22.5267 −0.828658 −0.414329 0.910127i $$-0.635984\pi$$
−0.414329 + 0.910127i $$0.635984\pi$$
$$740$$ 4.31151 0.158494
$$741$$ 0.201734 0.00741089
$$742$$ 0 0
$$743$$ −13.9795 −0.512857 −0.256429 0.966563i $$-0.582546\pi$$
−0.256429 + 0.966563i $$0.582546\pi$$
$$744$$ −3.69055 −0.135302
$$745$$ 11.0074 0.403280
$$746$$ 16.5149 0.604652
$$747$$ −6.01952 −0.220243
$$748$$ −10.6653 −0.389963
$$749$$ 0 0
$$750$$ 1.48925 0.0543798
$$751$$ 2.54591 0.0929017 0.0464509 0.998921i $$-0.485209\pi$$
0.0464509 + 0.998921i $$0.485209\pi$$
$$752$$ −12.4377 −0.453555
$$753$$ −3.66643 −0.133612
$$754$$ 4.29692 0.156485
$$755$$ 16.3692 0.595738
$$756$$ 0 0
$$757$$ 19.6921 0.715721 0.357861 0.933775i $$-0.383506\pi$$
0.357861 + 0.933775i $$0.383506\pi$$
$$758$$ −20.9288 −0.760168
$$759$$ −0.661732 −0.0240193
$$760$$ −3.13833 −0.113839
$$761$$ 39.0905 1.41703 0.708515 0.705696i $$-0.249365\pi$$
0.708515 + 0.705696i $$0.249365\pi$$
$$762$$ 2.44043 0.0884075
$$763$$ 0 0
$$764$$ −4.74120 −0.171531
$$765$$ 14.3469 0.518714
$$766$$ 21.5341 0.778059
$$767$$ 2.16385 0.0781320
$$768$$ −1.72107 −0.0621039
$$769$$ 21.7389 0.783926 0.391963 0.919981i $$-0.371796\pi$$
0.391963 + 0.919981i $$0.371796\pi$$
$$770$$ 0 0
$$771$$ −5.31859 −0.191544
$$772$$ −33.3537 −1.20043
$$773$$ −4.12806 −0.148476 −0.0742381 0.997241i $$-0.523652\pi$$
−0.0742381 + 0.997241i $$0.523652\pi$$
$$774$$ −6.14609 −0.220917
$$775$$ 24.2447 0.870895
$$776$$ 34.2483 1.22944
$$777$$ 0 0
$$778$$ 16.6071 0.595395
$$779$$ 9.74199 0.349043
$$780$$ 0.437016 0.0156477
$$781$$ 17.2500 0.617254
$$782$$ 3.96905 0.141933
$$783$$ −7.98857 −0.285488
$$784$$ 0 0
$$785$$ 25.0145 0.892807
$$786$$ 1.35718 0.0484089
$$787$$ −38.7863 −1.38258 −0.691291 0.722576i $$-0.742958\pi$$
−0.691291 + 0.722576i $$0.742958\pi$$
$$788$$ 4.27182 0.152177
$$789$$ 5.70037 0.202938
$$790$$ 4.81557 0.171330
$$791$$ 0 0
$$792$$ 13.2022 0.469118
$$793$$ 12.2002 0.433242
$$794$$ −3.47381 −0.123281
$$795$$ 0.0986276 0.00349796
$$796$$ −34.0047 −1.20526
$$797$$ −19.9651 −0.707201 −0.353601 0.935397i $$-0.615043\pi$$
−0.353601 + 0.935397i $$0.615043\pi$$
$$798$$ 0 0
$$799$$ 27.6896 0.979587
$$800$$ 18.0699 0.638867
$$801$$ −20.4067 −0.721035
$$802$$ 16.1641 0.570775
$$803$$ −13.5981 −0.479866
$$804$$ −3.63550 −0.128214
$$805$$ 0 0
$$806$$ −5.04967 −0.177867
$$807$$ −1.20904 −0.0425602
$$808$$ −6.59020 −0.231842
$$809$$ −30.2441 −1.06332 −0.531662 0.846956i $$-0.678432\pi$$
−0.531662 + 0.846956i $$0.678432\pi$$
$$810$$ −7.69633 −0.270421
$$811$$ −23.4099 −0.822034 −0.411017 0.911628i $$-0.634826\pi$$
−0.411017 + 0.911628i $$0.634826\pi$$
$$812$$ 0 0
$$813$$ 0.890538 0.0312325
$$814$$ −2.52581 −0.0885295
$$815$$ 3.41740 0.119706
$$816$$ −1.16976 −0.0409498
$$817$$ 3.11970 0.109144
$$818$$ 6.70768 0.234528
$$819$$ 0 0
$$820$$ 21.1041 0.736986
$$821$$ −46.5536 −1.62473 −0.812366 0.583147i $$-0.801821\pi$$
−0.812366 + 0.583147i $$0.801821\pi$$
$$822$$ −0.880838 −0.0307228
$$823$$ −5.68031 −0.198003 −0.0990017 0.995087i $$-0.531565\pi$$
−0.0990017 + 0.995087i $$0.531565\pi$$
$$824$$ −14.1700 −0.493636
$$825$$ 1.22920 0.0427953
$$826$$ 0 0
$$827$$ 48.3198 1.68024 0.840122 0.542398i $$-0.182483\pi$$
0.840122 + 0.542398i $$0.182483\pi$$
$$828$$ 7.87388 0.273636
$$829$$ 11.8545 0.411722 0.205861 0.978581i $$-0.434000\pi$$
0.205861 + 0.978581i $$0.434000\pi$$
$$830$$ −1.81595 −0.0630327
$$831$$ 0.221239 0.00767468
$$832$$ −0.559715 −0.0194046
$$833$$ 0 0
$$834$$ −1.85738 −0.0643157
$$835$$ −24.3563 −0.842884
$$836$$ −2.94646 −0.101905
$$837$$ 9.38802 0.324497
$$838$$ 20.1122 0.694765
$$839$$ 2.69386 0.0930025 0.0465013 0.998918i $$-0.485193\pi$$
0.0465013 + 0.998918i $$0.485193\pi$$
$$840$$ 0 0
$$841$$ 13.8809 0.478652
$$842$$ −17.0453 −0.587420
$$843$$ 2.96712 0.102193
$$844$$ −28.8593 −0.993380
$$845$$ 1.35996 0.0467841
$$846$$ −15.0706 −0.518137
$$847$$ 0 0
$$848$$ 0.567394 0.0194844
$$849$$ −5.01602 −0.172149
$$850$$ −7.37271 −0.252882
$$851$$ −3.42612 −0.117446
$$852$$ 2.90903 0.0996617
$$853$$ 2.48965 0.0852440 0.0426220 0.999091i $$-0.486429\pi$$
0.0426220 + 0.999091i $$0.486429\pi$$
$$854$$ 0 0
$$855$$ 3.96355 0.135551
$$856$$ 40.5677 1.38658
$$857$$ −46.2110 −1.57854 −0.789270 0.614047i $$-0.789541\pi$$
−0.789270 + 0.614047i $$0.789541\pi$$
$$858$$ −0.256017 −0.00874027
$$859$$ 10.9170 0.372482 0.186241 0.982504i $$-0.440369\pi$$
0.186241 + 0.982504i $$0.440369\pi$$
$$860$$ 6.75820 0.230453
$$861$$ 0 0
$$862$$ −9.76517 −0.332603
$$863$$ −29.2330 −0.995104 −0.497552 0.867434i $$-0.665768\pi$$
−0.497552 + 0.867434i $$0.665768\pi$$
$$864$$ 6.99701 0.238043
$$865$$ −8.86670 −0.301477
$$866$$ −26.3045 −0.893865
$$867$$ −0.876603 −0.0297710
$$868$$ 0 0
$$869$$ 10.2827 0.348817
$$870$$ −1.19651 −0.0405654
$$871$$ −11.3134 −0.383340
$$872$$ 24.7716 0.838873
$$873$$ −43.2538 −1.46392
$$874$$ 1.09651 0.0370901
$$875$$ 0 0
$$876$$ −2.29318 −0.0774792
$$877$$ −7.38110 −0.249242 −0.124621 0.992204i $$-0.539772\pi$$
−0.124621 + 0.992204i $$0.539772\pi$$
$$878$$ −1.97015 −0.0664892
$$879$$ −0.237730 −0.00801843
$$880$$ −4.15129 −0.139940
$$881$$ −16.4854 −0.555407 −0.277703 0.960667i $$-0.589573\pi$$
−0.277703 + 0.960667i $$0.589573\pi$$
$$882$$ 0 0
$$883$$ 12.4427 0.418732 0.209366 0.977837i $$-0.432860\pi$$
0.209366 + 0.977837i $$0.432860\pi$$
$$884$$ −5.59709 −0.188250
$$885$$ −0.602538 −0.0202541
$$886$$ −0.942055 −0.0316490
$$887$$ 8.11331 0.272418 0.136209 0.990680i $$-0.456508\pi$$
0.136209 + 0.990680i $$0.456508\pi$$
$$888$$ −0.968763 −0.0325095
$$889$$ 0 0
$$890$$ −6.15624 −0.206358
$$891$$ −16.4340 −0.550559
$$892$$ 34.8751 1.16771
$$893$$ 7.64968 0.255987
$$894$$ −1.08746 −0.0363702
$$895$$ −36.1769 −1.20926
$$896$$ 0 0
$$897$$ −0.347272 −0.0115951
$$898$$ −12.8818 −0.429870
$$899$$ −50.3929 −1.68070
$$900$$ −14.6261 −0.487537
$$901$$ −1.26317 −0.0420824
$$902$$ −12.3634 −0.411655
$$903$$ 0 0
$$904$$ −5.49633 −0.182805
$$905$$ 33.5098 1.11390
$$906$$ −1.61718 −0.0537272
$$907$$ −18.4796 −0.613605 −0.306803 0.951773i $$-0.599259\pi$$
−0.306803 + 0.951773i $$0.599259\pi$$
$$908$$ 41.9623 1.39257
$$909$$ 8.32308 0.276059
$$910$$ 0 0
$$911$$ 15.8660 0.525663 0.262831 0.964842i $$-0.415344\pi$$
0.262831 + 0.964842i $$0.415344\pi$$
$$912$$ −0.323165 −0.0107010
$$913$$ −3.87761 −0.128330
$$914$$ −6.98315 −0.230982
$$915$$ −3.39723 −0.112309
$$916$$ −26.2106 −0.866022
$$917$$ 0 0
$$918$$ −2.85486 −0.0942244
$$919$$ 43.4568 1.43351 0.716753 0.697327i $$-0.245627\pi$$
0.716753 + 0.697327i $$0.245627\pi$$
$$920$$ 5.40244 0.178113
$$921$$ −5.19523 −0.171189
$$922$$ −25.3241 −0.834004
$$923$$ 9.05268 0.297973
$$924$$ 0 0
$$925$$ 6.36418 0.209253
$$926$$ 18.3727 0.603763
$$927$$ 17.8960 0.587782
$$928$$ −37.5585 −1.23292
$$929$$ 25.6312 0.840931 0.420465 0.907309i $$-0.361867\pi$$
0.420465 + 0.907309i $$0.361867\pi$$
$$930$$ 1.40611 0.0461083
$$931$$ 0 0
$$932$$ 21.7798 0.713420
$$933$$ −5.00920 −0.163994
$$934$$ −16.9426 −0.554379
$$935$$ 9.24189 0.302242
$$936$$ 6.92840 0.226462
$$937$$ −23.9639 −0.782867 −0.391433 0.920206i $$-0.628021\pi$$
−0.391433 + 0.920206i $$0.628021\pi$$
$$938$$ 0 0
$$939$$ −1.35478 −0.0442116
$$940$$ 16.5715 0.540502
$$941$$ 42.2934 1.37872 0.689362 0.724417i $$-0.257891\pi$$
0.689362 + 0.724417i $$0.257891\pi$$
$$942$$ −2.47128 −0.0805186
$$943$$ −16.7702 −0.546113
$$944$$ −3.46634 −0.112820
$$945$$ 0 0
$$946$$ −3.95914 −0.128723
$$947$$ −49.5957 −1.61165 −0.805823 0.592157i $$-0.798277\pi$$
−0.805823 + 0.592157i $$0.798277\pi$$
$$948$$ 1.73407 0.0563199
$$949$$ −7.13619 −0.231650
$$950$$ −2.03683 −0.0660833
$$951$$ 1.67857 0.0544315
$$952$$ 0 0
$$953$$ 18.7156 0.606259 0.303129 0.952949i $$-0.401969\pi$$
0.303129 + 0.952949i $$0.401969\pi$$
$$954$$ 0.687505 0.0222588
$$955$$ 4.10843 0.132946
$$956$$ −0.731107 −0.0236457
$$957$$ −2.55491 −0.0825884
$$958$$ −25.8583 −0.835443
$$959$$ 0 0
$$960$$ 0.155857 0.00503025
$$961$$ 28.2208 0.910348
$$962$$ −1.32553 −0.0427367
$$963$$ −51.2349 −1.65102
$$964$$ 8.93494 0.287775
$$965$$ 28.9022 0.930395
$$966$$ 0 0
$$967$$ −34.3284 −1.10393 −0.551964 0.833868i $$-0.686121\pi$$
−0.551964 + 0.833868i $$0.686121\pi$$
$$968$$ −17.2597 −0.554748
$$969$$ 0.719451 0.0231121
$$970$$ −13.0487 −0.418969
$$971$$ −56.7118 −1.81997 −0.909984 0.414642i $$-0.863907\pi$$
−0.909984 + 0.414642i $$0.863907\pi$$
$$972$$ −8.51520 −0.273125
$$973$$ 0 0
$$974$$ 22.3732 0.716884
$$975$$ 0.645076 0.0206590
$$976$$ −19.5439 −0.625585
$$977$$ −39.8809 −1.27590 −0.637951 0.770077i $$-0.720218\pi$$
−0.637951 + 0.770077i $$0.720218\pi$$
$$978$$ −0.337618 −0.0107958
$$979$$ −13.1454 −0.420130
$$980$$ 0 0
$$981$$ −31.2853 −0.998862
$$982$$ 6.36770 0.203201
$$983$$ −41.2678 −1.31624 −0.658120 0.752913i $$-0.728648\pi$$
−0.658120 + 0.752913i $$0.728648\pi$$
$$984$$ −4.74192 −0.151167
$$985$$ −3.70169 −0.117946
$$986$$ 15.3243 0.488024
$$987$$ 0 0
$$988$$ −1.54628 −0.0491937
$$989$$ −5.37036 −0.170767
$$990$$ −5.03007 −0.159866
$$991$$ −26.1784 −0.831583 −0.415792 0.909460i $$-0.636495\pi$$
−0.415792 + 0.909460i $$0.636495\pi$$
$$992$$ 44.1380 1.40138
$$993$$ −0.148388 −0.00470894
$$994$$ 0 0
$$995$$ 29.4663 0.934145
$$996$$ −0.653918 −0.0207202
$$997$$ −8.62009 −0.273001 −0.136501 0.990640i $$-0.543586\pi$$
−0.136501 + 0.990640i $$0.543586\pi$$
$$998$$ −11.8417 −0.374842
$$999$$ 2.46434 0.0779681
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 637.2.a.m.1.3 6
3.2 odd 2 5733.2.a.bu.1.4 6
7.2 even 3 637.2.e.o.508.4 12
7.3 odd 6 637.2.e.n.79.4 12
7.4 even 3 637.2.e.o.79.4 12
7.5 odd 6 637.2.e.n.508.4 12
7.6 odd 2 637.2.a.n.1.3 yes 6
13.12 even 2 8281.2.a.cc.1.4 6
21.20 even 2 5733.2.a.br.1.4 6
91.90 odd 2 8281.2.a.cd.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.3 6 1.1 even 1 trivial
637.2.a.n.1.3 yes 6 7.6 odd 2
637.2.e.n.79.4 12 7.3 odd 6
637.2.e.n.508.4 12 7.5 odd 6
637.2.e.o.79.4 12 7.4 even 3
637.2.e.o.508.4 12 7.2 even 3
5733.2.a.br.1.4 6 21.20 even 2
5733.2.a.bu.1.4 6 3.2 odd 2
8281.2.a.cc.1.4 6 13.12 even 2
8281.2.a.cd.1.4 6 91.90 odd 2