# Properties

 Label 1183.2.a.p.1.5 Level $1183$ Weight $2$ Character 1183.1 Self dual yes Analytic conductor $9.446$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7674048.1 Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-1.10939$$ of defining polynomial Character $$\chi$$ $$=$$ 1183.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.10939 q^{2} +2.26165 q^{3} +2.44952 q^{4} +3.60178 q^{5} +4.77070 q^{6} -1.00000 q^{7} +0.948212 q^{8} +2.11505 q^{9} +O(q^{10})$$ $$q+2.10939 q^{2} +2.26165 q^{3} +2.44952 q^{4} +3.60178 q^{5} +4.77070 q^{6} -1.00000 q^{7} +0.948212 q^{8} +2.11505 q^{9} +7.59755 q^{10} -0.886384 q^{11} +5.53995 q^{12} -2.10939 q^{14} +8.14596 q^{15} -2.89889 q^{16} -4.96016 q^{17} +4.46147 q^{18} -2.37878 q^{19} +8.82263 q^{20} -2.26165 q^{21} -1.86973 q^{22} -3.85851 q^{23} +2.14452 q^{24} +7.97282 q^{25} -2.00144 q^{27} -2.44952 q^{28} +1.28197 q^{29} +17.1830 q^{30} -8.46921 q^{31} -8.01131 q^{32} -2.00469 q^{33} -10.4629 q^{34} -3.60178 q^{35} +5.18087 q^{36} +9.63812 q^{37} -5.01776 q^{38} +3.41525 q^{40} +12.0841 q^{41} -4.77070 q^{42} -3.64250 q^{43} -2.17122 q^{44} +7.61796 q^{45} -8.13910 q^{46} -2.98229 q^{47} -6.55628 q^{48} +1.00000 q^{49} +16.8178 q^{50} -11.2181 q^{51} +4.92032 q^{53} -4.22181 q^{54} -3.19256 q^{55} -0.948212 q^{56} -5.37995 q^{57} +2.70418 q^{58} +7.32746 q^{59} +19.9537 q^{60} -1.53926 q^{61} -17.8648 q^{62} -2.11505 q^{63} -11.1012 q^{64} -4.22867 q^{66} +8.42649 q^{67} -12.1500 q^{68} -8.72660 q^{69} -7.59755 q^{70} +6.44888 q^{71} +2.00552 q^{72} +7.14859 q^{73} +20.3305 q^{74} +18.0317 q^{75} -5.82686 q^{76} +0.886384 q^{77} +0.757551 q^{79} -10.4412 q^{80} -10.8717 q^{81} +25.4901 q^{82} +4.76766 q^{83} -5.53995 q^{84} -17.8654 q^{85} -7.68344 q^{86} +2.89937 q^{87} -0.840480 q^{88} -3.61884 q^{89} +16.0692 q^{90} -9.45150 q^{92} -19.1544 q^{93} -6.29081 q^{94} -8.56783 q^{95} -18.1188 q^{96} +0.463300 q^{97} +2.10939 q^{98} -1.87475 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10})$$ 6 * q + 4 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^6 - 6 * q^7 + 12 * q^8 + 4 * q^9 $$6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9} + 12 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{14} + 20 q^{15} + 8 q^{16} - 4 q^{17} - 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 10 q^{25} + 6 q^{27} - 4 q^{28} - 8 q^{29} + 8 q^{30} - 14 q^{31} + 8 q^{32} + 16 q^{33} - 2 q^{34} - 6 q^{35} - 10 q^{36} + 12 q^{37} - 2 q^{38} + 46 q^{40} + 28 q^{41} - 4 q^{42} + 2 q^{43} - 20 q^{44} + 16 q^{45} + 20 q^{46} + 14 q^{47} + 2 q^{48} + 6 q^{49} + 32 q^{50} - 26 q^{51} - 22 q^{53} + 14 q^{54} + 6 q^{55} - 12 q^{56} + 4 q^{58} - 2 q^{59} - 14 q^{61} - 4 q^{62} - 4 q^{63} + 26 q^{64} - 26 q^{66} + 24 q^{67} + 8 q^{68} + 4 q^{69} - 12 q^{70} + 4 q^{71} + 8 q^{72} + 36 q^{73} - 6 q^{74} + 46 q^{75} - 26 q^{76} - 4 q^{77} - 28 q^{79} + 36 q^{80} - 2 q^{81} + 14 q^{82} + 26 q^{83} - 2 q^{84} - 20 q^{85} - 24 q^{86} + 2 q^{87} - 14 q^{88} + 42 q^{89} - 12 q^{90} + 12 q^{92} - 4 q^{94} - 22 q^{95} - 42 q^{96} + 24 q^{97} + 4 q^{98} + 16 q^{99}+O(q^{100})$$ 6 * q + 4 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^6 - 6 * q^7 + 12 * q^8 + 4 * q^9 + 12 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^14 + 20 * q^15 + 8 * q^16 - 4 * q^17 - 16 * q^18 + 2 * q^19 + 26 * q^20 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 10 * q^25 + 6 * q^27 - 4 * q^28 - 8 * q^29 + 8 * q^30 - 14 * q^31 + 8 * q^32 + 16 * q^33 - 2 * q^34 - 6 * q^35 - 10 * q^36 + 12 * q^37 - 2 * q^38 + 46 * q^40 + 28 * q^41 - 4 * q^42 + 2 * q^43 - 20 * q^44 + 16 * q^45 + 20 * q^46 + 14 * q^47 + 2 * q^48 + 6 * q^49 + 32 * q^50 - 26 * q^51 - 22 * q^53 + 14 * q^54 + 6 * q^55 - 12 * q^56 + 4 * q^58 - 2 * q^59 - 14 * q^61 - 4 * q^62 - 4 * q^63 + 26 * q^64 - 26 * q^66 + 24 * q^67 + 8 * q^68 + 4 * q^69 - 12 * q^70 + 4 * q^71 + 8 * q^72 + 36 * q^73 - 6 * q^74 + 46 * q^75 - 26 * q^76 - 4 * q^77 - 28 * q^79 + 36 * q^80 - 2 * q^81 + 14 * q^82 + 26 * q^83 - 2 * q^84 - 20 * q^85 - 24 * q^86 + 2 * q^87 - 14 * q^88 + 42 * q^89 - 12 * q^90 + 12 * q^92 - 4 * q^94 - 22 * q^95 - 42 * q^96 + 24 * q^97 + 4 * q^98 + 16 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.10939 1.49156 0.745781 0.666191i $$-0.232076\pi$$
0.745781 + 0.666191i $$0.232076\pi$$
$$3$$ 2.26165 1.30576 0.652882 0.757460i $$-0.273560\pi$$
0.652882 + 0.757460i $$0.273560\pi$$
$$4$$ 2.44952 1.22476
$$5$$ 3.60178 1.61076 0.805382 0.592756i $$-0.201960\pi$$
0.805382 + 0.592756i $$0.201960\pi$$
$$6$$ 4.77070 1.94763
$$7$$ −1.00000 −0.377964
$$8$$ 0.948212 0.335243
$$9$$ 2.11505 0.705018
$$10$$ 7.59755 2.40256
$$11$$ −0.886384 −0.267255 −0.133627 0.991032i $$-0.542663\pi$$
−0.133627 + 0.991032i $$0.542663\pi$$
$$12$$ 5.53995 1.59925
$$13$$ 0 0
$$14$$ −2.10939 −0.563758
$$15$$ 8.14596 2.10328
$$16$$ −2.89889 −0.724723
$$17$$ −4.96016 −1.20302 −0.601508 0.798867i $$-0.705433\pi$$
−0.601508 + 0.798867i $$0.705433\pi$$
$$18$$ 4.46147 1.05158
$$19$$ −2.37878 −0.545729 −0.272864 0.962053i $$-0.587971\pi$$
−0.272864 + 0.962053i $$0.587971\pi$$
$$20$$ 8.82263 1.97280
$$21$$ −2.26165 −0.493532
$$22$$ −1.86973 −0.398628
$$23$$ −3.85851 −0.804555 −0.402278 0.915518i $$-0.631781\pi$$
−0.402278 + 0.915518i $$0.631781\pi$$
$$24$$ 2.14452 0.437749
$$25$$ 7.97282 1.59456
$$26$$ 0 0
$$27$$ −2.00144 −0.385177
$$28$$ −2.44952 −0.462916
$$29$$ 1.28197 0.238056 0.119028 0.992891i $$-0.462022\pi$$
0.119028 + 0.992891i $$0.462022\pi$$
$$30$$ 17.1830 3.13717
$$31$$ −8.46921 −1.52111 −0.760557 0.649271i $$-0.775074\pi$$
−0.760557 + 0.649271i $$0.775074\pi$$
$$32$$ −8.01131 −1.41621
$$33$$ −2.00469 −0.348972
$$34$$ −10.4629 −1.79437
$$35$$ −3.60178 −0.608812
$$36$$ 5.18087 0.863478
$$37$$ 9.63812 1.58450 0.792249 0.610198i $$-0.208910\pi$$
0.792249 + 0.610198i $$0.208910\pi$$
$$38$$ −5.01776 −0.813988
$$39$$ 0 0
$$40$$ 3.41525 0.539998
$$41$$ 12.0841 1.88722 0.943612 0.331053i $$-0.107404\pi$$
0.943612 + 0.331053i $$0.107404\pi$$
$$42$$ −4.77070 −0.736134
$$43$$ −3.64250 −0.555476 −0.277738 0.960657i $$-0.589585\pi$$
−0.277738 + 0.960657i $$0.589585\pi$$
$$44$$ −2.17122 −0.327323
$$45$$ 7.61796 1.13562
$$46$$ −8.13910 −1.20004
$$47$$ −2.98229 −0.435012 −0.217506 0.976059i $$-0.569792\pi$$
−0.217506 + 0.976059i $$0.569792\pi$$
$$48$$ −6.55628 −0.946317
$$49$$ 1.00000 0.142857
$$50$$ 16.8178 2.37839
$$51$$ −11.2181 −1.57085
$$52$$ 0 0
$$53$$ 4.92032 0.675858 0.337929 0.941172i $$-0.390274\pi$$
0.337929 + 0.941172i $$0.390274\pi$$
$$54$$ −4.22181 −0.574516
$$55$$ −3.19256 −0.430485
$$56$$ −0.948212 −0.126710
$$57$$ −5.37995 −0.712592
$$58$$ 2.70418 0.355076
$$59$$ 7.32746 0.953954 0.476977 0.878916i $$-0.341732\pi$$
0.476977 + 0.878916i $$0.341732\pi$$
$$60$$ 19.9537 2.57601
$$61$$ −1.53926 −0.197082 −0.0985412 0.995133i $$-0.531418\pi$$
−0.0985412 + 0.995133i $$0.531418\pi$$
$$62$$ −17.8648 −2.26884
$$63$$ −2.11505 −0.266472
$$64$$ −11.1012 −1.38765
$$65$$ 0 0
$$66$$ −4.22867 −0.520513
$$67$$ 8.42649 1.02946 0.514730 0.857352i $$-0.327892\pi$$
0.514730 + 0.857352i $$0.327892\pi$$
$$68$$ −12.1500 −1.47341
$$69$$ −8.72660 −1.05056
$$70$$ −7.59755 −0.908081
$$71$$ 6.44888 0.765342 0.382671 0.923885i $$-0.375004\pi$$
0.382671 + 0.923885i $$0.375004\pi$$
$$72$$ 2.00552 0.236353
$$73$$ 7.14859 0.836679 0.418340 0.908291i $$-0.362612\pi$$
0.418340 + 0.908291i $$0.362612\pi$$
$$74$$ 20.3305 2.36338
$$75$$ 18.0317 2.08212
$$76$$ −5.82686 −0.668386
$$77$$ 0.886384 0.101013
$$78$$ 0 0
$$79$$ 0.757551 0.0852311 0.0426156 0.999092i $$-0.486431\pi$$
0.0426156 + 0.999092i $$0.486431\pi$$
$$80$$ −10.4412 −1.16736
$$81$$ −10.8717 −1.20797
$$82$$ 25.4901 2.81491
$$83$$ 4.76766 0.523319 0.261659 0.965160i $$-0.415730\pi$$
0.261659 + 0.965160i $$0.415730\pi$$
$$84$$ −5.53995 −0.604458
$$85$$ −17.8654 −1.93778
$$86$$ −7.68344 −0.828527
$$87$$ 2.89937 0.310845
$$88$$ −0.840480 −0.0895955
$$89$$ −3.61884 −0.383596 −0.191798 0.981434i $$-0.561432\pi$$
−0.191798 + 0.981434i $$0.561432\pi$$
$$90$$ 16.0692 1.69385
$$91$$ 0 0
$$92$$ −9.45150 −0.985387
$$93$$ −19.1544 −1.98622
$$94$$ −6.29081 −0.648848
$$95$$ −8.56783 −0.879040
$$96$$ −18.1188 −1.84924
$$97$$ 0.463300 0.0470409 0.0235205 0.999723i $$-0.492513\pi$$
0.0235205 + 0.999723i $$0.492513\pi$$
$$98$$ 2.10939 0.213080
$$99$$ −1.87475 −0.188419
$$100$$ 19.5296 1.95296
$$101$$ 5.82303 0.579413 0.289707 0.957115i $$-0.406442\pi$$
0.289707 + 0.957115i $$0.406442\pi$$
$$102$$ −23.6634 −2.34303
$$103$$ 8.23888 0.811801 0.405901 0.913917i $$-0.366958\pi$$
0.405901 + 0.913917i $$0.366958\pi$$
$$104$$ 0 0
$$105$$ −8.14596 −0.794964
$$106$$ 10.3789 1.00809
$$107$$ −3.83260 −0.370511 −0.185256 0.982690i $$-0.559311\pi$$
−0.185256 + 0.982690i $$0.559311\pi$$
$$108$$ −4.90256 −0.471749
$$109$$ 10.4180 0.997867 0.498934 0.866640i $$-0.333725\pi$$
0.498934 + 0.866640i $$0.333725\pi$$
$$110$$ −6.73435 −0.642095
$$111$$ 21.7980 2.06898
$$112$$ 2.89889 0.273920
$$113$$ −4.91011 −0.461904 −0.230952 0.972965i $$-0.574184\pi$$
−0.230952 + 0.972965i $$0.574184\pi$$
$$114$$ −11.3484 −1.06288
$$115$$ −13.8975 −1.29595
$$116$$ 3.14022 0.291562
$$117$$ 0 0
$$118$$ 15.4565 1.42288
$$119$$ 4.96016 0.454697
$$120$$ 7.72409 0.705110
$$121$$ −10.2143 −0.928575
$$122$$ −3.24690 −0.293961
$$123$$ 27.3301 2.46427
$$124$$ −20.7455 −1.86300
$$125$$ 10.7074 0.957702
$$126$$ −4.46147 −0.397459
$$127$$ −12.3102 −1.09235 −0.546175 0.837671i $$-0.683917\pi$$
−0.546175 + 0.837671i $$0.683917\pi$$
$$128$$ −7.39409 −0.653551
$$129$$ −8.23805 −0.725320
$$130$$ 0 0
$$131$$ −8.20265 −0.716669 −0.358335 0.933593i $$-0.616655\pi$$
−0.358335 + 0.933593i $$0.616655\pi$$
$$132$$ −4.91053 −0.427406
$$133$$ 2.37878 0.206266
$$134$$ 17.7747 1.53550
$$135$$ −7.20874 −0.620429
$$136$$ −4.70328 −0.403303
$$137$$ 7.45555 0.636971 0.318485 0.947928i $$-0.396826\pi$$
0.318485 + 0.947928i $$0.396826\pi$$
$$138$$ −18.4078 −1.56697
$$139$$ 16.6806 1.41483 0.707413 0.706800i $$-0.249862\pi$$
0.707413 + 0.706800i $$0.249862\pi$$
$$140$$ −8.82263 −0.745648
$$141$$ −6.74490 −0.568023
$$142$$ 13.6032 1.14156
$$143$$ 0 0
$$144$$ −6.13131 −0.510943
$$145$$ 4.61738 0.383453
$$146$$ 15.0792 1.24796
$$147$$ 2.26165 0.186538
$$148$$ 23.6088 1.94063
$$149$$ 2.52163 0.206580 0.103290 0.994651i $$-0.467063\pi$$
0.103290 + 0.994651i $$0.467063\pi$$
$$150$$ 38.0359 3.10562
$$151$$ 15.8972 1.29370 0.646849 0.762618i $$-0.276086\pi$$
0.646849 + 0.762618i $$0.276086\pi$$
$$152$$ −2.25558 −0.182952
$$153$$ −10.4910 −0.848148
$$154$$ 1.86973 0.150667
$$155$$ −30.5042 −2.45016
$$156$$ 0 0
$$157$$ 12.9831 1.03616 0.518082 0.855331i $$-0.326646\pi$$
0.518082 + 0.855331i $$0.326646\pi$$
$$158$$ 1.59797 0.127128
$$159$$ 11.1280 0.882511
$$160$$ −28.8550 −2.28119
$$161$$ 3.85851 0.304093
$$162$$ −22.9327 −1.80176
$$163$$ −2.31948 −0.181676 −0.0908378 0.995866i $$-0.528954\pi$$
−0.0908378 + 0.995866i $$0.528954\pi$$
$$164$$ 29.6003 2.31140
$$165$$ −7.22045 −0.562111
$$166$$ 10.0569 0.780563
$$167$$ 13.7918 1.06724 0.533622 0.845723i $$-0.320831\pi$$
0.533622 + 0.845723i $$0.320831\pi$$
$$168$$ −2.14452 −0.165453
$$169$$ 0 0
$$170$$ −37.6851 −2.89031
$$171$$ −5.03124 −0.384748
$$172$$ −8.92237 −0.680324
$$173$$ −3.68432 −0.280113 −0.140057 0.990143i $$-0.544728\pi$$
−0.140057 + 0.990143i $$0.544728\pi$$
$$174$$ 6.11590 0.463645
$$175$$ −7.97282 −0.602688
$$176$$ 2.56953 0.193686
$$177$$ 16.5721 1.24564
$$178$$ −7.63353 −0.572157
$$179$$ −5.89277 −0.440446 −0.220223 0.975450i $$-0.570678\pi$$
−0.220223 + 0.975450i $$0.570678\pi$$
$$180$$ 18.6603 1.39086
$$181$$ 2.11543 0.157239 0.0786193 0.996905i $$-0.474949\pi$$
0.0786193 + 0.996905i $$0.474949\pi$$
$$182$$ 0 0
$$183$$ −3.48127 −0.257343
$$184$$ −3.65869 −0.269722
$$185$$ 34.7144 2.55225
$$186$$ −40.4040 −2.96257
$$187$$ 4.39661 0.321512
$$188$$ −7.30518 −0.532785
$$189$$ 2.00144 0.145583
$$190$$ −18.0729 −1.31114
$$191$$ −11.3667 −0.822462 −0.411231 0.911531i $$-0.634901\pi$$
−0.411231 + 0.911531i $$0.634901\pi$$
$$192$$ −25.1070 −1.81194
$$193$$ −14.0894 −1.01417 −0.507087 0.861895i $$-0.669278\pi$$
−0.507087 + 0.861895i $$0.669278\pi$$
$$194$$ 0.977279 0.0701645
$$195$$ 0 0
$$196$$ 2.44952 0.174966
$$197$$ −22.9571 −1.63563 −0.817814 0.575482i $$-0.804814\pi$$
−0.817814 + 0.575482i $$0.804814\pi$$
$$198$$ −3.95458 −0.281040
$$199$$ −3.14985 −0.223287 −0.111643 0.993748i $$-0.535611\pi$$
−0.111643 + 0.993748i $$0.535611\pi$$
$$200$$ 7.55992 0.534567
$$201$$ 19.0578 1.34423
$$202$$ 12.2830 0.864231
$$203$$ −1.28197 −0.0899768
$$204$$ −27.4791 −1.92392
$$205$$ 43.5244 3.03987
$$206$$ 17.3790 1.21085
$$207$$ −8.16096 −0.567226
$$208$$ 0 0
$$209$$ 2.10851 0.145849
$$210$$ −17.1830 −1.18574
$$211$$ −14.8638 −1.02327 −0.511634 0.859203i $$-0.670960\pi$$
−0.511634 + 0.859203i $$0.670960\pi$$
$$212$$ 12.0524 0.827764
$$213$$ 14.5851 0.999355
$$214$$ −8.08444 −0.552641
$$215$$ −13.1195 −0.894741
$$216$$ −1.89779 −0.129128
$$217$$ 8.46921 0.574927
$$218$$ 21.9757 1.48838
$$219$$ 16.1676 1.09251
$$220$$ −7.82024 −0.527241
$$221$$ 0 0
$$222$$ 45.9805 3.08601
$$223$$ 4.38089 0.293366 0.146683 0.989184i $$-0.453140\pi$$
0.146683 + 0.989184i $$0.453140\pi$$
$$224$$ 8.01131 0.535278
$$225$$ 16.8629 1.12420
$$226$$ −10.3573 −0.688959
$$227$$ −13.5663 −0.900428 −0.450214 0.892921i $$-0.648652\pi$$
−0.450214 + 0.892921i $$0.648652\pi$$
$$228$$ −13.1783 −0.872754
$$229$$ −16.5180 −1.09154 −0.545770 0.837935i $$-0.683763\pi$$
−0.545770 + 0.837935i $$0.683763\pi$$
$$230$$ −29.3152 −1.93299
$$231$$ 2.00469 0.131899
$$232$$ 1.21558 0.0798068
$$233$$ −16.5026 −1.08112 −0.540561 0.841305i $$-0.681788\pi$$
−0.540561 + 0.841305i $$0.681788\pi$$
$$234$$ 0 0
$$235$$ −10.7416 −0.700702
$$236$$ 17.9488 1.16836
$$237$$ 1.71331 0.111292
$$238$$ 10.4629 0.678210
$$239$$ −30.4210 −1.96777 −0.983886 0.178796i $$-0.942780\pi$$
−0.983886 + 0.178796i $$0.942780\pi$$
$$240$$ −23.6143 −1.52429
$$241$$ −29.5143 −1.90119 −0.950593 0.310440i $$-0.899524\pi$$
−0.950593 + 0.310440i $$0.899524\pi$$
$$242$$ −21.5460 −1.38503
$$243$$ −18.5837 −1.19214
$$244$$ −3.77046 −0.241379
$$245$$ 3.60178 0.230109
$$246$$ 57.6497 3.67561
$$247$$ 0 0
$$248$$ −8.03060 −0.509944
$$249$$ 10.7828 0.683331
$$250$$ 22.5861 1.42847
$$251$$ −12.9827 −0.819459 −0.409730 0.912207i $$-0.634377\pi$$
−0.409730 + 0.912207i $$0.634377\pi$$
$$252$$ −5.18087 −0.326364
$$253$$ 3.42012 0.215021
$$254$$ −25.9669 −1.62931
$$255$$ −40.4053 −2.53028
$$256$$ 6.60537 0.412836
$$257$$ −4.58521 −0.286018 −0.143009 0.989721i $$-0.545678\pi$$
−0.143009 + 0.989721i $$0.545678\pi$$
$$258$$ −17.3772 −1.08186
$$259$$ −9.63812 −0.598884
$$260$$ 0 0
$$261$$ 2.71144 0.167834
$$262$$ −17.3026 −1.06896
$$263$$ −2.66499 −0.164330 −0.0821652 0.996619i $$-0.526184\pi$$
−0.0821652 + 0.996619i $$0.526184\pi$$
$$264$$ −1.90087 −0.116990
$$265$$ 17.7219 1.08865
$$266$$ 5.01776 0.307659
$$267$$ −8.18453 −0.500885
$$268$$ 20.6409 1.26084
$$269$$ 11.9256 0.727119 0.363559 0.931571i $$-0.381561\pi$$
0.363559 + 0.931571i $$0.381561\pi$$
$$270$$ −15.2060 −0.925409
$$271$$ 13.0283 0.791414 0.395707 0.918377i $$-0.370500\pi$$
0.395707 + 0.918377i $$0.370500\pi$$
$$272$$ 14.3790 0.871853
$$273$$ 0 0
$$274$$ 15.7267 0.950082
$$275$$ −7.06698 −0.426155
$$276$$ −21.3760 −1.28668
$$277$$ 21.3649 1.28369 0.641846 0.766833i $$-0.278169\pi$$
0.641846 + 0.766833i $$0.278169\pi$$
$$278$$ 35.1858 2.11030
$$279$$ −17.9128 −1.07241
$$280$$ −3.41525 −0.204100
$$281$$ −17.2678 −1.03011 −0.515054 0.857158i $$-0.672228\pi$$
−0.515054 + 0.857158i $$0.672228\pi$$
$$282$$ −14.2276 −0.847242
$$283$$ −21.2402 −1.26260 −0.631299 0.775539i $$-0.717478\pi$$
−0.631299 + 0.775539i $$0.717478\pi$$
$$284$$ 15.7967 0.937360
$$285$$ −19.3774 −1.14782
$$286$$ 0 0
$$287$$ −12.0841 −0.713304
$$288$$ −16.9444 −0.998456
$$289$$ 7.60320 0.447247
$$290$$ 9.73985 0.571944
$$291$$ 1.04782 0.0614243
$$292$$ 17.5106 1.02473
$$293$$ −0.420060 −0.0245402 −0.0122701 0.999925i $$-0.503906\pi$$
−0.0122701 + 0.999925i $$0.503906\pi$$
$$294$$ 4.77070 0.278233
$$295$$ 26.3919 1.53660
$$296$$ 9.13898 0.531192
$$297$$ 1.77404 0.102940
$$298$$ 5.31910 0.308127
$$299$$ 0 0
$$300$$ 44.1690 2.55010
$$301$$ 3.64250 0.209950
$$302$$ 33.5335 1.92963
$$303$$ 13.1696 0.756577
$$304$$ 6.89581 0.395502
$$305$$ −5.54409 −0.317453
$$306$$ −22.1296 −1.26507
$$307$$ 14.0807 0.803628 0.401814 0.915721i $$-0.368380\pi$$
0.401814 + 0.915721i $$0.368380\pi$$
$$308$$ 2.17122 0.123717
$$309$$ 18.6335 1.06002
$$310$$ −64.3453 −3.65456
$$311$$ 10.3848 0.588867 0.294434 0.955672i $$-0.404869\pi$$
0.294434 + 0.955672i $$0.404869\pi$$
$$312$$ 0 0
$$313$$ 6.84759 0.387048 0.193524 0.981096i $$-0.438008\pi$$
0.193524 + 0.981096i $$0.438008\pi$$
$$314$$ 27.3864 1.54550
$$315$$ −7.61796 −0.429223
$$316$$ 1.85564 0.104388
$$317$$ 0.701249 0.0393861 0.0196930 0.999806i $$-0.493731\pi$$
0.0196930 + 0.999806i $$0.493731\pi$$
$$318$$ 23.4734 1.31632
$$319$$ −1.13632 −0.0636217
$$320$$ −39.9840 −2.23518
$$321$$ −8.66799 −0.483800
$$322$$ 8.13910 0.453574
$$323$$ 11.7991 0.656520
$$324$$ −26.6305 −1.47947
$$325$$ 0 0
$$326$$ −4.89268 −0.270980
$$327$$ 23.5619 1.30298
$$328$$ 11.4583 0.632680
$$329$$ 2.98229 0.164419
$$330$$ −15.2307 −0.838424
$$331$$ −4.19865 −0.230778 −0.115389 0.993320i $$-0.536812\pi$$
−0.115389 + 0.993320i $$0.536812\pi$$
$$332$$ 11.6785 0.640940
$$333$$ 20.3851 1.11710
$$334$$ 29.0923 1.59186
$$335$$ 30.3504 1.65822
$$336$$ 6.55628 0.357674
$$337$$ 20.4278 1.11278 0.556388 0.830923i $$-0.312187\pi$$
0.556388 + 0.830923i $$0.312187\pi$$
$$338$$ 0 0
$$339$$ −11.1049 −0.603138
$$340$$ −43.7617 −2.37331
$$341$$ 7.50697 0.406525
$$342$$ −10.6128 −0.573876
$$343$$ −1.00000 −0.0539949
$$344$$ −3.45386 −0.186220
$$345$$ −31.4313 −1.69220
$$346$$ −7.77165 −0.417807
$$347$$ 7.97000 0.427852 0.213926 0.976850i $$-0.431375\pi$$
0.213926 + 0.976850i $$0.431375\pi$$
$$348$$ 7.10207 0.380711
$$349$$ −21.5972 −1.15607 −0.578037 0.816011i $$-0.696181\pi$$
−0.578037 + 0.816011i $$0.696181\pi$$
$$350$$ −16.8178 −0.898947
$$351$$ 0 0
$$352$$ 7.10110 0.378490
$$353$$ 21.6176 1.15059 0.575295 0.817946i $$-0.304887\pi$$
0.575295 + 0.817946i $$0.304887\pi$$
$$354$$ 34.9571 1.85795
$$355$$ 23.2275 1.23279
$$356$$ −8.86441 −0.469813
$$357$$ 11.2181 0.593727
$$358$$ −12.4301 −0.656953
$$359$$ −13.6834 −0.722180 −0.361090 0.932531i $$-0.617595\pi$$
−0.361090 + 0.932531i $$0.617595\pi$$
$$360$$ 7.22344 0.380708
$$361$$ −13.3414 −0.702180
$$362$$ 4.46226 0.234531
$$363$$ −23.1012 −1.21250
$$364$$ 0 0
$$365$$ 25.7477 1.34769
$$366$$ −7.34335 −0.383843
$$367$$ −11.4128 −0.595741 −0.297871 0.954606i $$-0.596276\pi$$
−0.297871 + 0.954606i $$0.596276\pi$$
$$368$$ 11.1854 0.583080
$$369$$ 25.5586 1.33053
$$370$$ 73.2261 3.80685
$$371$$ −4.92032 −0.255450
$$372$$ −46.9190 −2.43264
$$373$$ −31.2808 −1.61966 −0.809830 0.586664i $$-0.800441\pi$$
−0.809830 + 0.586664i $$0.800441\pi$$
$$374$$ 9.27416 0.479555
$$375$$ 24.2164 1.25053
$$376$$ −2.82784 −0.145835
$$377$$ 0 0
$$378$$ 4.22181 0.217146
$$379$$ 27.4151 1.40822 0.704108 0.710093i $$-0.251347\pi$$
0.704108 + 0.710093i $$0.251347\pi$$
$$380$$ −20.9871 −1.07661
$$381$$ −27.8412 −1.42635
$$382$$ −23.9767 −1.22675
$$383$$ −16.1006 −0.822705 −0.411352 0.911476i $$-0.634943\pi$$
−0.411352 + 0.911476i $$0.634943\pi$$
$$384$$ −16.7228 −0.853383
$$385$$ 3.19256 0.162708
$$386$$ −29.7199 −1.51271
$$387$$ −7.70408 −0.391620
$$388$$ 1.13486 0.0576139
$$389$$ 21.1380 1.07174 0.535870 0.844301i $$-0.319984\pi$$
0.535870 + 0.844301i $$0.319984\pi$$
$$390$$ 0 0
$$391$$ 19.1388 0.967893
$$392$$ 0.948212 0.0478919
$$393$$ −18.5515 −0.935800
$$394$$ −48.4255 −2.43964
$$395$$ 2.72853 0.137287
$$396$$ −4.59224 −0.230769
$$397$$ 13.0984 0.657390 0.328695 0.944436i $$-0.393391\pi$$
0.328695 + 0.944436i $$0.393391\pi$$
$$398$$ −6.64426 −0.333046
$$399$$ 5.37995 0.269335
$$400$$ −23.1123 −1.15562
$$401$$ −19.4447 −0.971022 −0.485511 0.874230i $$-0.661367\pi$$
−0.485511 + 0.874230i $$0.661367\pi$$
$$402$$ 40.2002 2.00501
$$403$$ 0 0
$$404$$ 14.2636 0.709642
$$405$$ −39.1575 −1.94575
$$406$$ −2.70418 −0.134206
$$407$$ −8.54308 −0.423465
$$408$$ −10.6372 −0.526618
$$409$$ −24.0559 −1.18949 −0.594743 0.803916i $$-0.702746\pi$$
−0.594743 + 0.803916i $$0.702746\pi$$
$$410$$ 91.8098 4.53416
$$411$$ 16.8618 0.831733
$$412$$ 20.1813 0.994261
$$413$$ −7.32746 −0.360561
$$414$$ −17.2146 −0.846053
$$415$$ 17.1721 0.842944
$$416$$ 0 0
$$417$$ 37.7256 1.84743
$$418$$ 4.44767 0.217542
$$419$$ 39.0238 1.90644 0.953218 0.302283i $$-0.0977487\pi$$
0.953218 + 0.302283i $$0.0977487\pi$$
$$420$$ −19.9537 −0.973640
$$421$$ −22.0284 −1.07360 −0.536799 0.843710i $$-0.680367\pi$$
−0.536799 + 0.843710i $$0.680367\pi$$
$$422$$ −31.3536 −1.52627
$$423$$ −6.30771 −0.306691
$$424$$ 4.66551 0.226577
$$425$$ −39.5465 −1.91828
$$426$$ 30.7657 1.49060
$$427$$ 1.53926 0.0744902
$$428$$ −9.38803 −0.453787
$$429$$ 0 0
$$430$$ −27.6741 −1.33456
$$431$$ −35.8797 −1.72826 −0.864131 0.503267i $$-0.832131\pi$$
−0.864131 + 0.503267i $$0.832131\pi$$
$$432$$ 5.80195 0.279147
$$433$$ 12.2136 0.586946 0.293473 0.955967i $$-0.405189\pi$$
0.293473 + 0.955967i $$0.405189\pi$$
$$434$$ 17.8648 0.857540
$$435$$ 10.4429 0.500699
$$436$$ 25.5192 1.22215
$$437$$ 9.17853 0.439069
$$438$$ 34.1037 1.62954
$$439$$ −15.7553 −0.751960 −0.375980 0.926628i $$-0.622694\pi$$
−0.375980 + 0.926628i $$0.622694\pi$$
$$440$$ −3.02722 −0.144317
$$441$$ 2.11505 0.100717
$$442$$ 0 0
$$443$$ −15.0706 −0.716028 −0.358014 0.933716i $$-0.616546\pi$$
−0.358014 + 0.933716i $$0.616546\pi$$
$$444$$ 53.3947 2.53400
$$445$$ −13.0342 −0.617883
$$446$$ 9.24099 0.437574
$$447$$ 5.70305 0.269745
$$448$$ 11.1012 0.524482
$$449$$ 30.7826 1.45272 0.726360 0.687315i $$-0.241211\pi$$
0.726360 + 0.687315i $$0.241211\pi$$
$$450$$ 35.5705 1.67681
$$451$$ −10.7112 −0.504370
$$452$$ −12.0274 −0.565722
$$453$$ 35.9540 1.68926
$$454$$ −28.6166 −1.34304
$$455$$ 0 0
$$456$$ −5.10133 −0.238892
$$457$$ −7.75597 −0.362809 −0.181405 0.983409i $$-0.558064\pi$$
−0.181405 + 0.983409i $$0.558064\pi$$
$$458$$ −34.8429 −1.62810
$$459$$ 9.92745 0.463374
$$460$$ −34.0422 −1.58723
$$461$$ −1.47222 −0.0685681 −0.0342840 0.999412i $$-0.510915\pi$$
−0.0342840 + 0.999412i $$0.510915\pi$$
$$462$$ 4.22867 0.196735
$$463$$ 14.0366 0.652335 0.326168 0.945312i $$-0.394243\pi$$
0.326168 + 0.945312i $$0.394243\pi$$
$$464$$ −3.71630 −0.172525
$$465$$ −68.9898 −3.19933
$$466$$ −34.8104 −1.61256
$$467$$ 31.3806 1.45212 0.726060 0.687631i $$-0.241349\pi$$
0.726060 + 0.687631i $$0.241349\pi$$
$$468$$ 0 0
$$469$$ −8.42649 −0.389099
$$470$$ −22.6581 −1.04514
$$471$$ 29.3632 1.35298
$$472$$ 6.94798 0.319807
$$473$$ 3.22865 0.148454
$$474$$ 3.61404 0.165999
$$475$$ −18.9655 −0.870199
$$476$$ 12.1500 0.556895
$$477$$ 10.4067 0.476492
$$478$$ −64.1698 −2.93506
$$479$$ −41.1951 −1.88225 −0.941125 0.338059i $$-0.890230\pi$$
−0.941125 + 0.338059i $$0.890230\pi$$
$$480$$ −65.2598 −2.97869
$$481$$ 0 0
$$482$$ −62.2572 −2.83574
$$483$$ 8.72660 0.397074
$$484$$ −25.0202 −1.13728
$$485$$ 1.66870 0.0757719
$$486$$ −39.2002 −1.77816
$$487$$ −28.3265 −1.28360 −0.641798 0.766874i $$-0.721811\pi$$
−0.641798 + 0.766874i $$0.721811\pi$$
$$488$$ −1.45955 −0.0660706
$$489$$ −5.24584 −0.237225
$$490$$ 7.59755 0.343222
$$491$$ 34.7863 1.56988 0.784941 0.619571i $$-0.212693\pi$$
0.784941 + 0.619571i $$0.212693\pi$$
$$492$$ 66.9455 3.01814
$$493$$ −6.35879 −0.286386
$$494$$ 0 0
$$495$$ −6.75244 −0.303499
$$496$$ 24.5513 1.10239
$$497$$ −6.44888 −0.289272
$$498$$ 22.7451 1.01923
$$499$$ −0.0694885 −0.00311073 −0.00155537 0.999999i $$-0.500495\pi$$
−0.00155537 + 0.999999i $$0.500495\pi$$
$$500$$ 26.2281 1.17295
$$501$$ 31.1923 1.39357
$$502$$ −27.3855 −1.22228
$$503$$ 25.7372 1.14756 0.573782 0.819008i $$-0.305476\pi$$
0.573782 + 0.819008i $$0.305476\pi$$
$$504$$ −2.00552 −0.0893329
$$505$$ 20.9733 0.933299
$$506$$ 7.21437 0.320718
$$507$$ 0 0
$$508$$ −30.1540 −1.33787
$$509$$ −7.04000 −0.312042 −0.156021 0.987754i $$-0.549867\pi$$
−0.156021 + 0.987754i $$0.549867\pi$$
$$510$$ −85.2304 −3.77407
$$511$$ −7.14859 −0.316235
$$512$$ 28.7215 1.26932
$$513$$ 4.76097 0.210202
$$514$$ −9.67199 −0.426613
$$515$$ 29.6746 1.30762
$$516$$ −20.1793 −0.888342
$$517$$ 2.64346 0.116259
$$518$$ −20.3305 −0.893273
$$519$$ −8.33263 −0.365762
$$520$$ 0 0
$$521$$ 16.3253 0.715225 0.357613 0.933870i $$-0.383591\pi$$
0.357613 + 0.933870i $$0.383591\pi$$
$$522$$ 5.71948 0.250335
$$523$$ −7.08946 −0.310000 −0.155000 0.987914i $$-0.549538\pi$$
−0.155000 + 0.987914i $$0.549538\pi$$
$$524$$ −20.0926 −0.877748
$$525$$ −18.0317 −0.786968
$$526$$ −5.62150 −0.245109
$$527$$ 42.0086 1.82993
$$528$$ 5.81138 0.252908
$$529$$ −8.11189 −0.352691
$$530$$ 37.3824 1.62379
$$531$$ 15.4980 0.672555
$$532$$ 5.82686 0.252626
$$533$$ 0 0
$$534$$ −17.2644 −0.747102
$$535$$ −13.8042 −0.596807
$$536$$ 7.99010 0.345120
$$537$$ −13.3274 −0.575118
$$538$$ 25.1558 1.08454
$$539$$ −0.886384 −0.0381793
$$540$$ −17.6579 −0.759877
$$541$$ 25.5162 1.09703 0.548515 0.836141i $$-0.315194\pi$$
0.548515 + 0.836141i $$0.315194\pi$$
$$542$$ 27.4818 1.18044
$$543$$ 4.78436 0.205316
$$544$$ 39.7374 1.70373
$$545$$ 37.5235 1.60733
$$546$$ 0 0
$$547$$ −13.3073 −0.568978 −0.284489 0.958679i $$-0.591824\pi$$
−0.284489 + 0.958679i $$0.591824\pi$$
$$548$$ 18.2625 0.780136
$$549$$ −3.25562 −0.138947
$$550$$ −14.9070 −0.635637
$$551$$ −3.04952 −0.129914
$$552$$ −8.27466 −0.352193
$$553$$ −0.757551 −0.0322143
$$554$$ 45.0669 1.91471
$$555$$ 78.5118 3.33264
$$556$$ 40.8594 1.73282
$$557$$ 17.0071 0.720612 0.360306 0.932834i $$-0.382672\pi$$
0.360306 + 0.932834i $$0.382672\pi$$
$$558$$ −37.7851 −1.59957
$$559$$ 0 0
$$560$$ 10.4412 0.441220
$$561$$ 9.94358 0.419818
$$562$$ −36.4244 −1.53647
$$563$$ −24.9193 −1.05022 −0.525111 0.851034i $$-0.675976\pi$$
−0.525111 + 0.851034i $$0.675976\pi$$
$$564$$ −16.5218 −0.695691
$$565$$ −17.6851 −0.744019
$$566$$ −44.8038 −1.88325
$$567$$ 10.8717 0.456569
$$568$$ 6.11491 0.256576
$$569$$ 5.88129 0.246557 0.123278 0.992372i $$-0.460659\pi$$
0.123278 + 0.992372i $$0.460659\pi$$
$$570$$ −40.8745 −1.71204
$$571$$ 8.92622 0.373551 0.186775 0.982403i $$-0.440196\pi$$
0.186775 + 0.982403i $$0.440196\pi$$
$$572$$ 0 0
$$573$$ −25.7074 −1.07394
$$574$$ −25.4901 −1.06394
$$575$$ −30.7632 −1.28291
$$576$$ −23.4796 −0.978317
$$577$$ 36.1933 1.50675 0.753374 0.657592i $$-0.228425\pi$$
0.753374 + 0.657592i $$0.228425\pi$$
$$578$$ 16.0381 0.667097
$$579$$ −31.8652 −1.32427
$$580$$ 11.3104 0.469638
$$581$$ −4.76766 −0.197796
$$582$$ 2.21026 0.0916183
$$583$$ −4.36130 −0.180626
$$584$$ 6.77838 0.280491
$$585$$ 0 0
$$586$$ −0.886069 −0.0366032
$$587$$ 36.4895 1.50608 0.753041 0.657974i $$-0.228586\pi$$
0.753041 + 0.657974i $$0.228586\pi$$
$$588$$ 5.53995 0.228464
$$589$$ 20.1463 0.830116
$$590$$ 55.6708 2.29193
$$591$$ −51.9210 −2.13574
$$592$$ −27.9399 −1.14832
$$593$$ 34.9930 1.43699 0.718495 0.695533i $$-0.244832\pi$$
0.718495 + 0.695533i $$0.244832\pi$$
$$594$$ 3.74215 0.153542
$$595$$ 17.8654 0.732410
$$596$$ 6.17679 0.253011
$$597$$ −7.12385 −0.291560
$$598$$ 0 0
$$599$$ −32.5052 −1.32812 −0.664062 0.747677i $$-0.731169\pi$$
−0.664062 + 0.747677i $$0.731169\pi$$
$$600$$ 17.0979 0.698018
$$601$$ 20.0780 0.819000 0.409500 0.912310i $$-0.365703\pi$$
0.409500 + 0.912310i $$0.365703\pi$$
$$602$$ 7.68344 0.313154
$$603$$ 17.8225 0.725788
$$604$$ 38.9406 1.58447
$$605$$ −36.7897 −1.49572
$$606$$ 27.7799 1.12848
$$607$$ 9.71601 0.394361 0.197180 0.980367i $$-0.436822\pi$$
0.197180 + 0.980367i $$0.436822\pi$$
$$608$$ 19.0571 0.772868
$$609$$ −2.89937 −0.117488
$$610$$ −11.6946 −0.473502
$$611$$ 0 0
$$612$$ −25.6979 −1.03878
$$613$$ −11.8816 −0.479893 −0.239947 0.970786i $$-0.577130\pi$$
−0.239947 + 0.970786i $$0.577130\pi$$
$$614$$ 29.7017 1.19866
$$615$$ 98.4368 3.96936
$$616$$ 0.840480 0.0338639
$$617$$ 19.9884 0.804705 0.402352 0.915485i $$-0.368193\pi$$
0.402352 + 0.915485i $$0.368193\pi$$
$$618$$ 39.3052 1.58109
$$619$$ 41.7176 1.67677 0.838386 0.545078i $$-0.183500\pi$$
0.838386 + 0.545078i $$0.183500\pi$$
$$620$$ −74.7207 −3.00086
$$621$$ 7.72257 0.309896
$$622$$ 21.9056 0.878333
$$623$$ 3.61884 0.144986
$$624$$ 0 0
$$625$$ −1.29828 −0.0519312
$$626$$ 14.4442 0.577307
$$627$$ 4.76871 0.190444
$$628$$ 31.8023 1.26905
$$629$$ −47.8066 −1.90618
$$630$$ −16.0692 −0.640213
$$631$$ 17.6415 0.702296 0.351148 0.936320i $$-0.385791\pi$$
0.351148 + 0.936320i $$0.385791\pi$$
$$632$$ 0.718319 0.0285732
$$633$$ −33.6168 −1.33615
$$634$$ 1.47921 0.0587468
$$635$$ −44.3385 −1.75952
$$636$$ 27.2584 1.08086
$$637$$ 0 0
$$638$$ −2.39694 −0.0948958
$$639$$ 13.6397 0.539580
$$640$$ −26.6319 −1.05272
$$641$$ −10.9202 −0.431324 −0.215662 0.976468i $$-0.569191\pi$$
−0.215662 + 0.976468i $$0.569191\pi$$
$$642$$ −18.2842 −0.721618
$$643$$ −17.6351 −0.695462 −0.347731 0.937594i $$-0.613048\pi$$
−0.347731 + 0.937594i $$0.613048\pi$$
$$644$$ 9.45150 0.372441
$$645$$ −29.6716 −1.16832
$$646$$ 24.8889 0.979241
$$647$$ 16.6726 0.655469 0.327735 0.944770i $$-0.393715\pi$$
0.327735 + 0.944770i $$0.393715\pi$$
$$648$$ −10.3087 −0.404963
$$649$$ −6.49495 −0.254949
$$650$$ 0 0
$$651$$ 19.1544 0.750719
$$652$$ −5.68161 −0.222509
$$653$$ −6.77329 −0.265059 −0.132530 0.991179i $$-0.542310\pi$$
−0.132530 + 0.991179i $$0.542310\pi$$
$$654$$ 49.7013 1.94347
$$655$$ −29.5441 −1.15439
$$656$$ −35.0306 −1.36772
$$657$$ 15.1197 0.589874
$$658$$ 6.29081 0.245241
$$659$$ 33.5361 1.30638 0.653190 0.757194i $$-0.273430\pi$$
0.653190 + 0.757194i $$0.273430\pi$$
$$660$$ −17.6866 −0.688451
$$661$$ 25.1661 0.978848 0.489424 0.872046i $$-0.337207\pi$$
0.489424 + 0.872046i $$0.337207\pi$$
$$662$$ −8.85657 −0.344221
$$663$$ 0 0
$$664$$ 4.52075 0.175439
$$665$$ 8.56783 0.332246
$$666$$ 43.0002 1.66622
$$667$$ −4.94651 −0.191529
$$668$$ 33.7833 1.30712
$$669$$ 9.90802 0.383066
$$670$$ 64.0207 2.47334
$$671$$ 1.36438 0.0526713
$$672$$ 18.1188 0.698947
$$673$$ −1.85468 −0.0714927 −0.0357464 0.999361i $$-0.511381\pi$$
−0.0357464 + 0.999361i $$0.511381\pi$$
$$674$$ 43.0902 1.65977
$$675$$ −15.9571 −0.614189
$$676$$ 0 0
$$677$$ −14.7209 −0.565770 −0.282885 0.959154i $$-0.591291\pi$$
−0.282885 + 0.959154i $$0.591291\pi$$
$$678$$ −23.4246 −0.899618
$$679$$ −0.463300 −0.0177798
$$680$$ −16.9402 −0.649627
$$681$$ −30.6822 −1.17575
$$682$$ 15.8351 0.606358
$$683$$ 7.94353 0.303951 0.151975 0.988384i $$-0.451437\pi$$
0.151975 + 0.988384i $$0.451437\pi$$
$$684$$ −12.3241 −0.471224
$$685$$ 26.8533 1.02601
$$686$$ −2.10939 −0.0805368
$$687$$ −37.3579 −1.42529
$$688$$ 10.5592 0.402566
$$689$$ 0 0
$$690$$ −66.3008 −2.52403
$$691$$ 10.2307 0.389193 0.194597 0.980883i $$-0.437660\pi$$
0.194597 + 0.980883i $$0.437660\pi$$
$$692$$ −9.02481 −0.343072
$$693$$ 1.87475 0.0712159
$$694$$ 16.8118 0.638168
$$695$$ 60.0797 2.27895
$$696$$ 2.74922 0.104209
$$697$$ −59.9392 −2.27036
$$698$$ −45.5570 −1.72436
$$699$$ −37.3231 −1.41169
$$700$$ −19.5296 −0.738148
$$701$$ 16.5978 0.626891 0.313445 0.949606i $$-0.398517\pi$$
0.313445 + 0.949606i $$0.398517\pi$$
$$702$$ 0 0
$$703$$ −22.9269 −0.864706
$$704$$ 9.83992 0.370856
$$705$$ −24.2936 −0.914951
$$706$$ 45.5999 1.71618
$$707$$ −5.82303 −0.218998
$$708$$ 40.5938 1.52561
$$709$$ 47.8659 1.79764 0.898820 0.438318i $$-0.144426\pi$$
0.898820 + 0.438318i $$0.144426\pi$$
$$710$$ 48.9957 1.83878
$$711$$ 1.60226 0.0600895
$$712$$ −3.43142 −0.128598
$$713$$ 32.6785 1.22382
$$714$$ 23.6634 0.885581
$$715$$ 0 0
$$716$$ −14.4344 −0.539441
$$717$$ −68.8017 −2.56944
$$718$$ −28.8635 −1.07718
$$719$$ −38.0922 −1.42060 −0.710300 0.703899i $$-0.751441\pi$$
−0.710300 + 0.703899i $$0.751441\pi$$
$$720$$ −22.0836 −0.823009
$$721$$ −8.23888 −0.306832
$$722$$ −28.1423 −1.04735
$$723$$ −66.7511 −2.48250
$$724$$ 5.18179 0.192580
$$725$$ 10.2209 0.379596
$$726$$ −48.7294 −1.80852
$$727$$ −15.4059 −0.571374 −0.285687 0.958323i $$-0.592222\pi$$
−0.285687 + 0.958323i $$0.592222\pi$$
$$728$$ 0 0
$$729$$ −9.41460 −0.348689
$$730$$ 54.3118 2.01017
$$731$$ 18.0674 0.668246
$$732$$ −8.52744 −0.315183
$$733$$ −11.6298 −0.429557 −0.214778 0.976663i $$-0.568903\pi$$
−0.214778 + 0.976663i $$0.568903\pi$$
$$734$$ −24.0740 −0.888586
$$735$$ 8.14596 0.300468
$$736$$ 30.9117 1.13942
$$737$$ −7.46911 −0.275128
$$738$$ 53.9130 1.98456
$$739$$ −2.68901 −0.0989168 −0.0494584 0.998776i $$-0.515750\pi$$
−0.0494584 + 0.998776i $$0.515750\pi$$
$$740$$ 85.0336 3.12590
$$741$$ 0 0
$$742$$ −10.3789 −0.381020
$$743$$ −2.46720 −0.0905126 −0.0452563 0.998975i $$-0.514410\pi$$
−0.0452563 + 0.998975i $$0.514410\pi$$
$$744$$ −18.1624 −0.665866
$$745$$ 9.08236 0.332752
$$746$$ −65.9835 −2.41583
$$747$$ 10.0839 0.368949
$$748$$ 10.7696 0.393775
$$749$$ 3.83260 0.140040
$$750$$ 51.0819 1.86525
$$751$$ −37.9185 −1.38366 −0.691832 0.722058i $$-0.743196\pi$$
−0.691832 + 0.722058i $$0.743196\pi$$
$$752$$ 8.64534 0.315263
$$753$$ −29.3623 −1.07002
$$754$$ 0 0
$$755$$ 57.2583 2.08384
$$756$$ 4.90256 0.178304
$$757$$ 34.6451 1.25920 0.629598 0.776921i $$-0.283219\pi$$
0.629598 + 0.776921i $$0.283219\pi$$
$$758$$ 57.8290 2.10044
$$759$$ 7.73512 0.280767
$$760$$ −8.12411 −0.294692
$$761$$ 22.8595 0.828655 0.414328 0.910128i $$-0.364017\pi$$
0.414328 + 0.910128i $$0.364017\pi$$
$$762$$ −58.7280 −2.12749
$$763$$ −10.4180 −0.377158
$$764$$ −27.8428 −1.00732
$$765$$ −37.7863 −1.36617
$$766$$ −33.9625 −1.22712
$$767$$ 0 0
$$768$$ 14.9390 0.539065
$$769$$ −51.8275 −1.86895 −0.934473 0.356034i $$-0.884129\pi$$
−0.934473 + 0.356034i $$0.884129\pi$$
$$770$$ 6.73435 0.242689
$$771$$ −10.3701 −0.373471
$$772$$ −34.5122 −1.24212
$$773$$ 5.69966 0.205003 0.102501 0.994733i $$-0.467315\pi$$
0.102501 + 0.994733i $$0.467315\pi$$
$$774$$ −16.2509 −0.584126
$$775$$ −67.5234 −2.42551
$$776$$ 0.439306 0.0157702
$$777$$ −21.7980 −0.782001
$$778$$ 44.5883 1.59857
$$779$$ −28.7454 −1.02991
$$780$$ 0 0
$$781$$ −5.71619 −0.204541
$$782$$ 40.3712 1.44367
$$783$$ −2.56579 −0.0916938
$$784$$ −2.89889 −0.103532
$$785$$ 46.7622 1.66902
$$786$$ −39.1324 −1.39580
$$787$$ −16.8141 −0.599358 −0.299679 0.954040i $$-0.596880\pi$$
−0.299679 + 0.954040i $$0.596880\pi$$
$$788$$ −56.2340 −2.00325
$$789$$ −6.02727 −0.214577
$$790$$ 5.75553 0.204773
$$791$$ 4.91011 0.174583
$$792$$ −1.77766 −0.0631664
$$793$$ 0 0
$$794$$ 27.6296 0.980539
$$795$$ 40.0807 1.42152
$$796$$ −7.71562 −0.273473
$$797$$ 42.1301 1.49233 0.746163 0.665764i $$-0.231894\pi$$
0.746163 + 0.665764i $$0.231894\pi$$
$$798$$ 11.3484 0.401729
$$799$$ 14.7926 0.523326
$$800$$ −63.8727 −2.25824
$$801$$ −7.65403 −0.270442
$$802$$ −41.0164 −1.44834
$$803$$ −6.33640 −0.223607
$$804$$ 46.6824 1.64636
$$805$$ 13.8975 0.489823
$$806$$ 0 0
$$807$$ 26.9716 0.949445
$$808$$ 5.52147 0.194245
$$809$$ −30.1686 −1.06067 −0.530336 0.847787i $$-0.677934\pi$$
−0.530336 + 0.847787i $$0.677934\pi$$
$$810$$ −82.5984 −2.90221
$$811$$ 23.7929 0.835480 0.417740 0.908567i $$-0.362822\pi$$
0.417740 + 0.908567i $$0.362822\pi$$
$$812$$ −3.14022 −0.110200
$$813$$ 29.4655 1.03340
$$814$$ −18.0207 −0.631624
$$815$$ −8.35425 −0.292637
$$816$$ 32.5202 1.13843
$$817$$ 8.66468 0.303139
$$818$$ −50.7432 −1.77419
$$819$$ 0 0
$$820$$ 106.614 3.72312
$$821$$ −35.8847 −1.25239 −0.626193 0.779668i $$-0.715388\pi$$
−0.626193 + 0.779668i $$0.715388\pi$$
$$822$$ 35.5682 1.24058
$$823$$ −12.2346 −0.426470 −0.213235 0.977001i $$-0.568400\pi$$
−0.213235 + 0.977001i $$0.568400\pi$$
$$824$$ 7.81220 0.272151
$$825$$ −15.9830 −0.556457
$$826$$ −15.4565 −0.537799
$$827$$ 27.3474 0.950962 0.475481 0.879726i $$-0.342274\pi$$
0.475481 + 0.879726i $$0.342274\pi$$
$$828$$ −19.9904 −0.694715
$$829$$ 23.5738 0.818751 0.409376 0.912366i $$-0.365747\pi$$
0.409376 + 0.912366i $$0.365747\pi$$
$$830$$ 36.2226 1.25730
$$831$$ 48.3199 1.67620
$$832$$ 0 0
$$833$$ −4.96016 −0.171859
$$834$$ 79.5779 2.75556
$$835$$ 49.6751 1.71908
$$836$$ 5.16483 0.178630
$$837$$ 16.9506 0.585898
$$838$$ 82.3163 2.84357
$$839$$ 10.5883 0.365549 0.182775 0.983155i $$-0.441492\pi$$
0.182775 + 0.983155i $$0.441492\pi$$
$$840$$ −7.72409 −0.266507
$$841$$ −27.3565 −0.943329
$$842$$ −46.4664 −1.60134
$$843$$ −39.0536 −1.34508
$$844$$ −36.4092 −1.25326
$$845$$ 0 0
$$846$$ −13.3054 −0.457449
$$847$$ 10.2143 0.350968
$$848$$ −14.2635 −0.489810
$$849$$ −48.0379 −1.64866
$$850$$ −83.4188 −2.86124
$$851$$ −37.1888 −1.27482
$$852$$ 35.7265 1.22397
$$853$$ 21.3925 0.732464 0.366232 0.930524i $$-0.380648\pi$$
0.366232 + 0.930524i $$0.380648\pi$$
$$854$$ 3.24690 0.111107
$$855$$ −18.1214 −0.619739
$$856$$ −3.63412 −0.124212
$$857$$ 7.22129 0.246675 0.123337 0.992365i $$-0.460640\pi$$
0.123337 + 0.992365i $$0.460640\pi$$
$$858$$ 0 0
$$859$$ 57.1073 1.94848 0.974238 0.225524i $$-0.0724095\pi$$
0.974238 + 0.225524i $$0.0724095\pi$$
$$860$$ −32.1364 −1.09584
$$861$$ −27.3301 −0.931406
$$862$$ −75.6841 −2.57781
$$863$$ 51.3361 1.74750 0.873751 0.486374i $$-0.161681\pi$$
0.873751 + 0.486374i $$0.161681\pi$$
$$864$$ 16.0341 0.545493
$$865$$ −13.2701 −0.451197
$$866$$ 25.7631 0.875467
$$867$$ 17.1958 0.583999
$$868$$ 20.7455 0.704148
$$869$$ −0.671481 −0.0227784
$$870$$ 22.0281 0.746823
$$871$$ 0 0
$$872$$ 9.87851 0.334528
$$873$$ 0.979903 0.0331647
$$874$$ 19.3611 0.654899
$$875$$ −10.7074 −0.361977
$$876$$ 39.6029 1.33806
$$877$$ 21.4277 0.723563 0.361781 0.932263i $$-0.382169\pi$$
0.361781 + 0.932263i $$0.382169\pi$$
$$878$$ −33.2341 −1.12160
$$879$$ −0.950027 −0.0320436
$$880$$ 9.25489 0.311982
$$881$$ 29.0619 0.979120 0.489560 0.871970i $$-0.337157\pi$$
0.489560 + 0.871970i $$0.337157\pi$$
$$882$$ 4.46147 0.150225
$$883$$ −4.83594 −0.162742 −0.0813711 0.996684i $$-0.525930\pi$$
−0.0813711 + 0.996684i $$0.525930\pi$$
$$884$$ 0 0
$$885$$ 59.6892 2.00643
$$886$$ −31.7899 −1.06800
$$887$$ −24.9898 −0.839075 −0.419538 0.907738i $$-0.637808\pi$$
−0.419538 + 0.907738i $$0.637808\pi$$
$$888$$ 20.6692 0.693612
$$889$$ 12.3102 0.412869
$$890$$ −27.4943 −0.921611
$$891$$ 9.63651 0.322835
$$892$$ 10.7311 0.359303
$$893$$ 7.09420 0.237398
$$894$$ 12.0299 0.402341
$$895$$ −21.2244 −0.709455
$$896$$ 7.39409 0.247019
$$897$$ 0 0
$$898$$ 64.9324 2.16682
$$899$$ −10.8573 −0.362111
$$900$$ 41.3061 1.37687
$$901$$ −24.4056 −0.813068
$$902$$ −22.5940 −0.752300
$$903$$ 8.23805 0.274145
$$904$$ −4.65582 −0.154850
$$905$$ 7.61931 0.253274
$$906$$ 75.8409 2.51964
$$907$$ 15.0412 0.499435 0.249717 0.968319i $$-0.419662\pi$$
0.249717 + 0.968319i $$0.419662\pi$$
$$908$$ −33.2310 −1.10281
$$909$$ 12.3160 0.408497
$$910$$ 0 0
$$911$$ −9.22150 −0.305522 −0.152761 0.988263i $$-0.548816\pi$$
−0.152761 + 0.988263i $$0.548816\pi$$
$$912$$ 15.5959 0.516432
$$913$$ −4.22598 −0.139860
$$914$$ −16.3604 −0.541153
$$915$$ −12.5388 −0.414519
$$916$$ −40.4612 −1.33687
$$917$$ 8.20265 0.270875
$$918$$ 20.9409 0.691151
$$919$$ −45.0803 −1.48706 −0.743531 0.668701i $$-0.766851\pi$$
−0.743531 + 0.668701i $$0.766851\pi$$
$$920$$ −13.1778 −0.434459
$$921$$ 31.8456 1.04935
$$922$$ −3.10548 −0.102274
$$923$$ 0 0
$$924$$ 4.91053 0.161544
$$925$$ 76.8430 2.52658
$$926$$ 29.6086 0.972999
$$927$$ 17.4257 0.572334
$$928$$ −10.2703 −0.337139
$$929$$ 46.9169 1.53929 0.769647 0.638469i $$-0.220432\pi$$
0.769647 + 0.638469i $$0.220432\pi$$
$$930$$ −145.526 −4.77200
$$931$$ −2.37878 −0.0779612
$$932$$ −40.4235 −1.32412
$$933$$ 23.4867 0.768921
$$934$$ 66.1939 2.16593
$$935$$ 15.8356 0.517880
$$936$$ 0 0
$$937$$ −28.3912 −0.927501 −0.463750 0.885966i $$-0.653497\pi$$
−0.463750 + 0.885966i $$0.653497\pi$$
$$938$$ −17.7747 −0.580366
$$939$$ 15.4868 0.505394
$$940$$ −26.3117 −0.858192
$$941$$ 17.8718 0.582603 0.291302 0.956631i $$-0.405912\pi$$
0.291302 + 0.956631i $$0.405912\pi$$
$$942$$ 61.9384 2.01806
$$943$$ −46.6268 −1.51838
$$944$$ −21.2415 −0.691353
$$945$$ 7.20874 0.234500
$$946$$ 6.81048 0.221428
$$947$$ −9.52234 −0.309435 −0.154717 0.987959i $$-0.549447\pi$$
−0.154717 + 0.987959i $$0.549447\pi$$
$$948$$ 4.19680 0.136306
$$949$$ 0 0
$$950$$ −40.0057 −1.29796
$$951$$ 1.58598 0.0514289
$$952$$ 4.70328 0.152434
$$953$$ −13.4180 −0.434652 −0.217326 0.976099i $$-0.569733\pi$$
−0.217326 + 0.976099i $$0.569733\pi$$
$$954$$ 21.9519 0.710718
$$955$$ −40.9402 −1.32479
$$956$$ −74.5169 −2.41005
$$957$$ −2.56996 −0.0830749
$$958$$ −86.8964 −2.80749
$$959$$ −7.45555 −0.240752
$$960$$ −90.4298 −2.91861
$$961$$ 40.7275 1.31379
$$962$$ 0 0
$$963$$ −8.10615 −0.261217
$$964$$ −72.2960 −2.32850
$$965$$ −50.7468 −1.63360
$$966$$ 18.4078 0.592261
$$967$$ −12.9316 −0.415851 −0.207926 0.978145i $$-0.566671\pi$$
−0.207926 + 0.978145i $$0.566671\pi$$
$$968$$ −9.68534 −0.311299
$$969$$ 26.6854 0.857260
$$970$$ 3.51994 0.113019
$$971$$ 47.5213 1.52503 0.762516 0.646969i $$-0.223964\pi$$
0.762516 + 0.646969i $$0.223964\pi$$
$$972$$ −45.5211 −1.46009
$$973$$ −16.6806 −0.534754
$$974$$ −59.7515 −1.91456
$$975$$ 0 0
$$976$$ 4.46216 0.142830
$$977$$ 36.4942 1.16755 0.583776 0.811915i $$-0.301575\pi$$
0.583776 + 0.811915i $$0.301575\pi$$
$$978$$ −11.0655 −0.353836
$$979$$ 3.20768 0.102518
$$980$$ 8.82263 0.281829
$$981$$ 22.0347 0.703514
$$982$$ 73.3777 2.34158
$$983$$ −44.1843 −1.40926 −0.704629 0.709576i $$-0.748887\pi$$
−0.704629 + 0.709576i $$0.748887\pi$$
$$984$$ 25.9147 0.826130
$$985$$ −82.6865 −2.63461
$$986$$ −13.4132 −0.427162
$$987$$ 6.74490 0.214692
$$988$$ 0 0
$$989$$ 14.0546 0.446911
$$990$$ −14.2435 −0.452689
$$991$$ −50.7097 −1.61085 −0.805424 0.592699i $$-0.798062\pi$$
−0.805424 + 0.592699i $$0.798062\pi$$
$$992$$ 67.8495 2.15422
$$993$$ −9.49586 −0.301342
$$994$$ −13.6032 −0.431467
$$995$$ −11.3451 −0.359663
$$996$$ 26.4126 0.836916
$$997$$ −50.2768 −1.59228 −0.796141 0.605111i $$-0.793129\pi$$
−0.796141 + 0.605111i $$0.793129\pi$$
$$998$$ −0.146578 −0.00463985
$$999$$ −19.2901 −0.610312
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 1183.2.a.p.1.5 6
7.6 odd 2 8281.2.a.ch.1.5 6
13.2 odd 12 91.2.q.a.43.6 yes 12
13.5 odd 4 1183.2.c.i.337.2 12
13.7 odd 12 91.2.q.a.36.6 12
13.8 odd 4 1183.2.c.i.337.11 12
13.12 even 2 1183.2.a.m.1.2 6
39.2 even 12 819.2.ct.a.316.1 12
39.20 even 12 819.2.ct.a.127.1 12
52.7 even 12 1456.2.cc.c.673.5 12
52.15 even 12 1456.2.cc.c.225.5 12
91.2 odd 12 637.2.k.h.459.1 12
91.20 even 12 637.2.q.h.491.6 12
91.33 even 12 637.2.u.i.361.1 12
91.41 even 12 637.2.q.h.589.6 12
91.46 odd 12 637.2.k.h.569.6 12
91.54 even 12 637.2.k.g.459.1 12
91.59 even 12 637.2.k.g.569.6 12
91.67 odd 12 637.2.u.h.30.1 12
91.72 odd 12 637.2.u.h.361.1 12
91.80 even 12 637.2.u.i.30.1 12
91.90 odd 2 8281.2.a.by.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.q.a.36.6 12 13.7 odd 12
91.2.q.a.43.6 yes 12 13.2 odd 12
637.2.k.g.459.1 12 91.54 even 12
637.2.k.g.569.6 12 91.59 even 12
637.2.k.h.459.1 12 91.2 odd 12
637.2.k.h.569.6 12 91.46 odd 12
637.2.q.h.491.6 12 91.20 even 12
637.2.q.h.589.6 12 91.41 even 12
637.2.u.h.30.1 12 91.67 odd 12
637.2.u.h.361.1 12 91.72 odd 12
637.2.u.i.30.1 12 91.80 even 12
637.2.u.i.361.1 12 91.33 even 12
819.2.ct.a.127.1 12 39.20 even 12
819.2.ct.a.316.1 12 39.2 even 12
1183.2.a.m.1.2 6 13.12 even 2
1183.2.a.p.1.5 6 1.1 even 1 trivial
1183.2.c.i.337.2 12 13.5 odd 4
1183.2.c.i.337.11 12 13.8 odd 4
1456.2.cc.c.225.5 12 52.15 even 12
1456.2.cc.c.673.5 12 52.7 even 12
8281.2.a.by.1.2 6 91.90 odd 2
8281.2.a.ch.1.5 6 7.6 odd 2