# Properties

 Label 1183.2.a.p.1.6 Level $1183$ Weight $2$ Character 1183.1 Self dual yes Analytic conductor $9.446$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Learn more

## 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 1.6 Root $$-1.70320$$ of defining polynomial Character $$\chi$$ $$=$$ 1183.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.70320 q^{2} -0.345949 q^{3} +5.30727 q^{4} +3.25812 q^{5} -0.935168 q^{6} -1.00000 q^{7} +8.94020 q^{8} -2.88032 q^{9} +O(q^{10})$$ $$q+2.70320 q^{2} -0.345949 q^{3} +5.30727 q^{4} +3.25812 q^{5} -0.935168 q^{6} -1.00000 q^{7} +8.94020 q^{8} -2.88032 q^{9} +8.80735 q^{10} -1.84603 q^{11} -1.83605 q^{12} -2.70320 q^{14} -1.12715 q^{15} +13.5526 q^{16} +2.15314 q^{17} -7.78607 q^{18} -2.40096 q^{19} +17.2917 q^{20} +0.345949 q^{21} -4.99017 q^{22} +1.81263 q^{23} -3.09285 q^{24} +5.61537 q^{25} +2.03429 q^{27} -5.30727 q^{28} -2.73406 q^{29} -3.04689 q^{30} +1.74236 q^{31} +18.7549 q^{32} +0.638632 q^{33} +5.82036 q^{34} -3.25812 q^{35} -15.2866 q^{36} -5.93565 q^{37} -6.49025 q^{38} +29.1283 q^{40} -4.22131 q^{41} +0.935168 q^{42} -8.68223 q^{43} -9.79737 q^{44} -9.38444 q^{45} +4.89989 q^{46} +5.87774 q^{47} -4.68850 q^{48} +1.00000 q^{49} +15.1794 q^{50} -0.744877 q^{51} -9.30628 q^{53} +5.49909 q^{54} -6.01459 q^{55} -8.94020 q^{56} +0.830609 q^{57} -7.39071 q^{58} -10.7523 q^{59} -5.98206 q^{60} +10.1101 q^{61} +4.70994 q^{62} +2.88032 q^{63} +23.5929 q^{64} +1.72635 q^{66} -0.826916 q^{67} +11.4273 q^{68} -0.627077 q^{69} -8.80735 q^{70} -2.35425 q^{71} -25.7506 q^{72} +3.19482 q^{73} -16.0452 q^{74} -1.94263 q^{75} -12.7425 q^{76} +1.84603 q^{77} +0.801911 q^{79} +44.1559 q^{80} +7.93720 q^{81} -11.4110 q^{82} +9.97031 q^{83} +1.83605 q^{84} +7.01520 q^{85} -23.4698 q^{86} +0.945847 q^{87} -16.5039 q^{88} +15.1135 q^{89} -25.3680 q^{90} +9.62010 q^{92} -0.602768 q^{93} +15.8887 q^{94} -7.82261 q^{95} -6.48823 q^{96} +9.23171 q^{97} +2.70320 q^{98} +5.31715 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.70320 1.91145 0.955724 0.294264i $$-0.0950745\pi$$
0.955724 + 0.294264i $$0.0950745\pi$$
$$3$$ −0.345949 −0.199734 −0.0998669 0.995001i $$-0.531842\pi$$
−0.0998669 + 0.995001i $$0.531842\pi$$
$$4$$ 5.30727 2.65363
$$5$$ 3.25812 1.45708 0.728539 0.685005i $$-0.240200\pi$$
0.728539 + 0.685005i $$0.240200\pi$$
$$6$$ −0.935168 −0.381781
$$7$$ −1.00000 −0.377964
$$8$$ 8.94020 3.16084
$$9$$ −2.88032 −0.960106
$$10$$ 8.80735 2.78513
$$11$$ −1.84603 −0.556598 −0.278299 0.960494i $$-0.589771\pi$$
−0.278299 + 0.960494i $$0.589771\pi$$
$$12$$ −1.83605 −0.530021
$$13$$ 0 0
$$14$$ −2.70320 −0.722460
$$15$$ −1.12715 −0.291028
$$16$$ 13.5526 3.38814
$$17$$ 2.15314 0.522213 0.261107 0.965310i $$-0.415913\pi$$
0.261107 + 0.965310i $$0.415913\pi$$
$$18$$ −7.78607 −1.83519
$$19$$ −2.40096 −0.550817 −0.275408 0.961327i $$-0.588813\pi$$
−0.275408 + 0.961327i $$0.588813\pi$$
$$20$$ 17.2917 3.86655
$$21$$ 0.345949 0.0754923
$$22$$ −4.99017 −1.06391
$$23$$ 1.81263 0.377959 0.188979 0.981981i $$-0.439482\pi$$
0.188979 + 0.981981i $$0.439482\pi$$
$$24$$ −3.09285 −0.631326
$$25$$ 5.61537 1.12307
$$26$$ 0 0
$$27$$ 2.03429 0.391500
$$28$$ −5.30727 −1.00298
$$29$$ −2.73406 −0.507703 −0.253851 0.967243i $$-0.581697\pi$$
−0.253851 + 0.967243i $$0.581697\pi$$
$$30$$ −3.04689 −0.556284
$$31$$ 1.74236 0.312937 0.156468 0.987683i $$-0.449989\pi$$
0.156468 + 0.987683i $$0.449989\pi$$
$$32$$ 18.7549 3.31542
$$33$$ 0.638632 0.111172
$$34$$ 5.82036 0.998183
$$35$$ −3.25812 −0.550723
$$36$$ −15.2866 −2.54777
$$37$$ −5.93565 −0.975815 −0.487908 0.872895i $$-0.662240\pi$$
−0.487908 + 0.872895i $$0.662240\pi$$
$$38$$ −6.49025 −1.05286
$$39$$ 0 0
$$40$$ 29.1283 4.60558
$$41$$ −4.22131 −0.659259 −0.329629 0.944110i $$-0.606924\pi$$
−0.329629 + 0.944110i $$0.606924\pi$$
$$42$$ 0.935168 0.144300
$$43$$ −8.68223 −1.32403 −0.662014 0.749492i $$-0.730298\pi$$
−0.662014 + 0.749492i $$0.730298\pi$$
$$44$$ −9.79737 −1.47701
$$45$$ −9.38444 −1.39895
$$46$$ 4.89989 0.722449
$$47$$ 5.87774 0.857357 0.428678 0.903457i $$-0.358979\pi$$
0.428678 + 0.903457i $$0.358979\pi$$
$$48$$ −4.68850 −0.676727
$$49$$ 1.00000 0.142857
$$50$$ 15.1794 2.14670
$$51$$ −0.744877 −0.104304
$$52$$ 0 0
$$53$$ −9.30628 −1.27832 −0.639158 0.769076i $$-0.720717\pi$$
−0.639158 + 0.769076i $$0.720717\pi$$
$$54$$ 5.49909 0.748331
$$55$$ −6.01459 −0.811007
$$56$$ −8.94020 −1.19468
$$57$$ 0.830609 0.110017
$$58$$ −7.39071 −0.970447
$$59$$ −10.7523 −1.39982 −0.699912 0.714229i $$-0.746778\pi$$
−0.699912 + 0.714229i $$0.746778\pi$$
$$60$$ −5.98206 −0.772281
$$61$$ 10.1101 1.29446 0.647231 0.762294i $$-0.275927\pi$$
0.647231 + 0.762294i $$0.275927\pi$$
$$62$$ 4.70994 0.598163
$$63$$ 2.88032 0.362886
$$64$$ 23.5929 2.94911
$$65$$ 0 0
$$66$$ 1.72635 0.212499
$$67$$ −0.826916 −0.101024 −0.0505119 0.998723i $$-0.516085\pi$$
−0.0505119 + 0.998723i $$0.516085\pi$$
$$68$$ 11.4273 1.38576
$$69$$ −0.627077 −0.0754912
$$70$$ −8.80735 −1.05268
$$71$$ −2.35425 −0.279398 −0.139699 0.990194i $$-0.544613\pi$$
−0.139699 + 0.990194i $$0.544613\pi$$
$$72$$ −25.7506 −3.03474
$$73$$ 3.19482 0.373925 0.186963 0.982367i $$-0.440136\pi$$
0.186963 + 0.982367i $$0.440136\pi$$
$$74$$ −16.0452 −1.86522
$$75$$ −1.94263 −0.224316
$$76$$ −12.7425 −1.46167
$$77$$ 1.84603 0.210374
$$78$$ 0 0
$$79$$ 0.801911 0.0902220 0.0451110 0.998982i $$-0.485636\pi$$
0.0451110 + 0.998982i $$0.485636\pi$$
$$80$$ 44.1559 4.93678
$$81$$ 7.93720 0.881911
$$82$$ −11.4110 −1.26014
$$83$$ 9.97031 1.09438 0.547192 0.837007i $$-0.315697\pi$$
0.547192 + 0.837007i $$0.315697\pi$$
$$84$$ 1.83605 0.200329
$$85$$ 7.01520 0.760905
$$86$$ −23.4698 −2.53081
$$87$$ 0.945847 0.101405
$$88$$ −16.5039 −1.75932
$$89$$ 15.1135 1.60202 0.801012 0.598648i $$-0.204295\pi$$
0.801012 + 0.598648i $$0.204295\pi$$
$$90$$ −25.3680 −2.67402
$$91$$ 0 0
$$92$$ 9.62010 1.00296
$$93$$ −0.602768 −0.0625041
$$94$$ 15.8887 1.63879
$$95$$ −7.82261 −0.802583
$$96$$ −6.48823 −0.662202
$$97$$ 9.23171 0.937338 0.468669 0.883374i $$-0.344734\pi$$
0.468669 + 0.883374i $$0.344734\pi$$
$$98$$ 2.70320 0.273064
$$99$$ 5.31715 0.534394
$$100$$ 29.8023 2.98023
$$101$$ 14.8234 1.47498 0.737491 0.675357i $$-0.236011\pi$$
0.737491 + 0.675357i $$0.236011\pi$$
$$102$$ −2.01355 −0.199371
$$103$$ −4.28286 −0.422003 −0.211001 0.977486i $$-0.567672\pi$$
−0.211001 + 0.977486i $$0.567672\pi$$
$$104$$ 0 0
$$105$$ 1.12715 0.109998
$$106$$ −25.1567 −2.44343
$$107$$ −19.1258 −1.84896 −0.924479 0.381233i $$-0.875500\pi$$
−0.924479 + 0.381233i $$0.875500\pi$$
$$108$$ 10.7965 1.03890
$$109$$ 4.27153 0.409139 0.204569 0.978852i $$-0.434421\pi$$
0.204569 + 0.978852i $$0.434421\pi$$
$$110$$ −16.2586 −1.55020
$$111$$ 2.05343 0.194903
$$112$$ −13.5526 −1.28060
$$113$$ 2.74976 0.258676 0.129338 0.991601i $$-0.458715\pi$$
0.129338 + 0.991601i $$0.458715\pi$$
$$114$$ 2.24530 0.210291
$$115$$ 5.90576 0.550715
$$116$$ −14.5104 −1.34726
$$117$$ 0 0
$$118$$ −29.0655 −2.67569
$$119$$ −2.15314 −0.197378
$$120$$ −10.0769 −0.919891
$$121$$ −7.59218 −0.690198
$$122$$ 27.3295 2.47430
$$123$$ 1.46036 0.131676
$$124$$ 9.24717 0.830420
$$125$$ 2.00495 0.179329
$$126$$ 7.78607 0.693638
$$127$$ −9.73438 −0.863786 −0.431893 0.901925i $$-0.642154\pi$$
−0.431893 + 0.901925i $$0.642154\pi$$
$$128$$ 26.2666 2.32166
$$129$$ 3.00361 0.264453
$$130$$ 0 0
$$131$$ −18.6615 −1.63046 −0.815230 0.579138i $$-0.803389\pi$$
−0.815230 + 0.579138i $$0.803389\pi$$
$$132$$ 3.38939 0.295009
$$133$$ 2.40096 0.208189
$$134$$ −2.23532 −0.193102
$$135$$ 6.62797 0.570445
$$136$$ 19.2495 1.65063
$$137$$ 8.42156 0.719502 0.359751 0.933048i $$-0.382862\pi$$
0.359751 + 0.933048i $$0.382862\pi$$
$$138$$ −1.69511 −0.144298
$$139$$ −17.6362 −1.49588 −0.747941 0.663765i $$-0.768957\pi$$
−0.747941 + 0.663765i $$0.768957\pi$$
$$140$$ −17.2917 −1.46142
$$141$$ −2.03340 −0.171243
$$142$$ −6.36399 −0.534054
$$143$$ 0 0
$$144$$ −39.0357 −3.25298
$$145$$ −8.90791 −0.739762
$$146$$ 8.63623 0.714739
$$147$$ −0.345949 −0.0285334
$$148$$ −31.5021 −2.58946
$$149$$ −4.02104 −0.329416 −0.164708 0.986342i $$-0.552668\pi$$
−0.164708 + 0.986342i $$0.552668\pi$$
$$150$$ −5.25132 −0.428768
$$151$$ 18.9010 1.53814 0.769069 0.639165i $$-0.220720\pi$$
0.769069 + 0.639165i $$0.220720\pi$$
$$152$$ −21.4650 −1.74104
$$153$$ −6.20173 −0.501380
$$154$$ 4.99017 0.402120
$$155$$ 5.67682 0.455973
$$156$$ 0 0
$$157$$ −11.5735 −0.923670 −0.461835 0.886966i $$-0.652809\pi$$
−0.461835 + 0.886966i $$0.652809\pi$$
$$158$$ 2.16772 0.172455
$$159$$ 3.21950 0.255323
$$160$$ 61.1056 4.83082
$$161$$ −1.81263 −0.142855
$$162$$ 21.4558 1.68573
$$163$$ 4.40542 0.345059 0.172529 0.985004i $$-0.444806\pi$$
0.172529 + 0.985004i $$0.444806\pi$$
$$164$$ −22.4037 −1.74943
$$165$$ 2.08074 0.161985
$$166$$ 26.9517 2.09186
$$167$$ −9.01909 −0.697918 −0.348959 0.937138i $$-0.613465\pi$$
−0.348959 + 0.937138i $$0.613465\pi$$
$$168$$ 3.09285 0.238619
$$169$$ 0 0
$$170$$ 18.9635 1.45443
$$171$$ 6.91552 0.528843
$$172$$ −46.0789 −3.51349
$$173$$ 6.09200 0.463166 0.231583 0.972815i $$-0.425609\pi$$
0.231583 + 0.972815i $$0.425609\pi$$
$$174$$ 2.55681 0.193831
$$175$$ −5.61537 −0.424482
$$176$$ −25.0184 −1.88583
$$177$$ 3.71974 0.279592
$$178$$ 40.8547 3.06219
$$179$$ 3.87964 0.289978 0.144989 0.989433i $$-0.453685\pi$$
0.144989 + 0.989433i $$0.453685\pi$$
$$180$$ −49.8057 −3.71230
$$181$$ 6.58392 0.489379 0.244690 0.969601i $$-0.421314\pi$$
0.244690 + 0.969601i $$0.421314\pi$$
$$182$$ 0 0
$$183$$ −3.49757 −0.258548
$$184$$ 16.2052 1.19467
$$185$$ −19.3391 −1.42184
$$186$$ −1.62940 −0.119473
$$187$$ −3.97476 −0.290663
$$188$$ 31.1948 2.27511
$$189$$ −2.03429 −0.147973
$$190$$ −21.1460 −1.53410
$$191$$ −13.7434 −0.994435 −0.497218 0.867626i $$-0.665645\pi$$
−0.497218 + 0.867626i $$0.665645\pi$$
$$192$$ −8.16195 −0.589038
$$193$$ 22.7530 1.63780 0.818899 0.573937i $$-0.194585\pi$$
0.818899 + 0.573937i $$0.194585\pi$$
$$194$$ 24.9551 1.79167
$$195$$ 0 0
$$196$$ 5.30727 0.379091
$$197$$ 14.5272 1.03502 0.517509 0.855678i $$-0.326859\pi$$
0.517509 + 0.855678i $$0.326859\pi$$
$$198$$ 14.3733 1.02147
$$199$$ 23.8404 1.69000 0.845001 0.534765i $$-0.179600\pi$$
0.845001 + 0.534765i $$0.179600\pi$$
$$200$$ 50.2025 3.54985
$$201$$ 0.286071 0.0201779
$$202$$ 40.0705 2.81935
$$203$$ 2.73406 0.191894
$$204$$ −3.95326 −0.276784
$$205$$ −13.7536 −0.960591
$$206$$ −11.5774 −0.806637
$$207$$ −5.22095 −0.362881
$$208$$ 0 0
$$209$$ 4.43223 0.306584
$$210$$ 3.04689 0.210256
$$211$$ 4.31527 0.297076 0.148538 0.988907i $$-0.452543\pi$$
0.148538 + 0.988907i $$0.452543\pi$$
$$212$$ −49.3909 −3.39218
$$213$$ 0.814450 0.0558052
$$214$$ −51.7007 −3.53419
$$215$$ −28.2878 −1.92921
$$216$$ 18.1870 1.23747
$$217$$ −1.74236 −0.118279
$$218$$ 11.5468 0.782047
$$219$$ −1.10525 −0.0746856
$$220$$ −31.9210 −2.15212
$$221$$ 0 0
$$222$$ 5.55083 0.372548
$$223$$ 23.3947 1.56662 0.783312 0.621629i $$-0.213529\pi$$
0.783312 + 0.621629i $$0.213529\pi$$
$$224$$ −18.7549 −1.25311
$$225$$ −16.1741 −1.07827
$$226$$ 7.43315 0.494446
$$227$$ −26.7229 −1.77366 −0.886829 0.462097i $$-0.847097\pi$$
−0.886829 + 0.462097i $$0.847097\pi$$
$$228$$ 4.40826 0.291944
$$229$$ −3.00670 −0.198688 −0.0993442 0.995053i $$-0.531674\pi$$
−0.0993442 + 0.995053i $$0.531674\pi$$
$$230$$ 15.9644 1.05266
$$231$$ −0.638632 −0.0420189
$$232$$ −24.4431 −1.60477
$$233$$ −11.7148 −0.767462 −0.383731 0.923445i $$-0.625361\pi$$
−0.383731 + 0.923445i $$0.625361\pi$$
$$234$$ 0 0
$$235$$ 19.1504 1.24924
$$236$$ −57.0651 −3.71462
$$237$$ −0.277420 −0.0180204
$$238$$ −5.82036 −0.377278
$$239$$ −1.42797 −0.0923677 −0.0461838 0.998933i $$-0.514706\pi$$
−0.0461838 + 0.998933i $$0.514706\pi$$
$$240$$ −15.2757 −0.986043
$$241$$ 2.67969 0.172614 0.0863069 0.996269i $$-0.472493\pi$$
0.0863069 + 0.996269i $$0.472493\pi$$
$$242$$ −20.5232 −1.31928
$$243$$ −8.84874 −0.567647
$$244$$ 53.6569 3.43503
$$245$$ 3.25812 0.208154
$$246$$ 3.94764 0.251692
$$247$$ 0 0
$$248$$ 15.5770 0.989143
$$249$$ −3.44922 −0.218586
$$250$$ 5.41978 0.342777
$$251$$ 10.9339 0.690143 0.345072 0.938576i $$-0.387855\pi$$
0.345072 + 0.938576i $$0.387855\pi$$
$$252$$ 15.2866 0.962967
$$253$$ −3.34616 −0.210371
$$254$$ −26.3139 −1.65108
$$255$$ −2.42690 −0.151978
$$256$$ 23.8178 1.48861
$$257$$ 4.15138 0.258956 0.129478 0.991582i $$-0.458670\pi$$
0.129478 + 0.991582i $$0.458670\pi$$
$$258$$ 8.11935 0.505488
$$259$$ 5.93565 0.368823
$$260$$ 0 0
$$261$$ 7.87497 0.487448
$$262$$ −50.4456 −3.11654
$$263$$ 4.05360 0.249955 0.124978 0.992160i $$-0.460114\pi$$
0.124978 + 0.992160i $$0.460114\pi$$
$$264$$ 5.70949 0.351395
$$265$$ −30.3210 −1.86260
$$266$$ 6.49025 0.397943
$$267$$ −5.22849 −0.319979
$$268$$ −4.38866 −0.268080
$$269$$ 4.00022 0.243898 0.121949 0.992536i $$-0.461086\pi$$
0.121949 + 0.992536i $$0.461086\pi$$
$$270$$ 17.9167 1.09038
$$271$$ −2.78502 −0.169178 −0.0845888 0.996416i $$-0.526958\pi$$
−0.0845888 + 0.996416i $$0.526958\pi$$
$$272$$ 29.1806 1.76933
$$273$$ 0 0
$$274$$ 22.7651 1.37529
$$275$$ −10.3661 −0.625101
$$276$$ −3.32807 −0.200326
$$277$$ 16.6924 1.00295 0.501474 0.865173i $$-0.332791\pi$$
0.501474 + 0.865173i $$0.332791\pi$$
$$278$$ −47.6741 −2.85930
$$279$$ −5.01855 −0.300453
$$280$$ −29.1283 −1.74075
$$281$$ 13.3731 0.797774 0.398887 0.917000i $$-0.369397\pi$$
0.398887 + 0.917000i $$0.369397\pi$$
$$282$$ −5.49668 −0.327322
$$283$$ −18.8862 −1.12267 −0.561335 0.827589i $$-0.689712\pi$$
−0.561335 + 0.827589i $$0.689712\pi$$
$$284$$ −12.4946 −0.741419
$$285$$ 2.70623 0.160303
$$286$$ 0 0
$$287$$ 4.22131 0.249176
$$288$$ −54.0200 −3.18316
$$289$$ −12.3640 −0.727293
$$290$$ −24.0798 −1.41402
$$291$$ −3.19370 −0.187218
$$292$$ 16.9558 0.992262
$$293$$ 3.41790 0.199676 0.0998380 0.995004i $$-0.468168\pi$$
0.0998380 + 0.995004i $$0.468168\pi$$
$$294$$ −0.935168 −0.0545401
$$295$$ −35.0322 −2.03965
$$296$$ −53.0659 −3.08439
$$297$$ −3.75536 −0.217908
$$298$$ −10.8697 −0.629663
$$299$$ 0 0
$$300$$ −10.3101 −0.595252
$$301$$ 8.68223 0.500435
$$302$$ 51.0930 2.94007
$$303$$ −5.12814 −0.294604
$$304$$ −32.5391 −1.86625
$$305$$ 32.9399 1.88613
$$306$$ −16.7645 −0.958362
$$307$$ −16.3679 −0.934165 −0.467083 0.884214i $$-0.654695\pi$$
−0.467083 + 0.884214i $$0.654695\pi$$
$$308$$ 9.79737 0.558257
$$309$$ 1.48165 0.0842883
$$310$$ 15.3456 0.871569
$$311$$ 23.6979 1.34378 0.671891 0.740650i $$-0.265482\pi$$
0.671891 + 0.740650i $$0.265482\pi$$
$$312$$ 0 0
$$313$$ −5.18025 −0.292805 −0.146403 0.989225i $$-0.546769\pi$$
−0.146403 + 0.989225i $$0.546769\pi$$
$$314$$ −31.2856 −1.76555
$$315$$ 9.38444 0.528753
$$316$$ 4.25596 0.239416
$$317$$ 6.06537 0.340665 0.170332 0.985387i $$-0.445516\pi$$
0.170332 + 0.985387i $$0.445516\pi$$
$$318$$ 8.70294 0.488037
$$319$$ 5.04715 0.282586
$$320$$ 76.8686 4.29709
$$321$$ 6.61655 0.369300
$$322$$ −4.89989 −0.273060
$$323$$ −5.16959 −0.287644
$$324$$ 42.1248 2.34027
$$325$$ 0 0
$$326$$ 11.9087 0.659562
$$327$$ −1.47773 −0.0817188
$$328$$ −37.7394 −2.08381
$$329$$ −5.87774 −0.324050
$$330$$ 5.62465 0.309627
$$331$$ 17.2749 0.949512 0.474756 0.880118i $$-0.342536\pi$$
0.474756 + 0.880118i $$0.342536\pi$$
$$332$$ 52.9151 2.90410
$$333$$ 17.0966 0.936886
$$334$$ −24.3804 −1.33403
$$335$$ −2.69419 −0.147200
$$336$$ 4.68850 0.255779
$$337$$ 8.35464 0.455106 0.227553 0.973766i $$-0.426927\pi$$
0.227553 + 0.973766i $$0.426927\pi$$
$$338$$ 0 0
$$339$$ −0.951279 −0.0516664
$$340$$ 37.2315 2.01916
$$341$$ −3.21644 −0.174180
$$342$$ 18.6940 1.01086
$$343$$ −1.00000 −0.0539949
$$344$$ −77.6208 −4.18504
$$345$$ −2.04309 −0.109997
$$346$$ 16.4679 0.885318
$$347$$ 28.8220 1.54725 0.773623 0.633646i $$-0.218443\pi$$
0.773623 + 0.633646i $$0.218443\pi$$
$$348$$ 5.01986 0.269093
$$349$$ −11.7221 −0.627467 −0.313734 0.949511i $$-0.601580\pi$$
−0.313734 + 0.949511i $$0.601580\pi$$
$$350$$ −15.1794 −0.811376
$$351$$ 0 0
$$352$$ −34.6220 −1.84536
$$353$$ −17.8362 −0.949326 −0.474663 0.880168i $$-0.657430\pi$$
−0.474663 + 0.880168i $$0.657430\pi$$
$$354$$ 10.0552 0.534426
$$355$$ −7.67043 −0.407104
$$356$$ 80.2113 4.25119
$$357$$ 0.744877 0.0394231
$$358$$ 10.4874 0.554278
$$359$$ −5.68162 −0.299864 −0.149932 0.988696i $$-0.547906\pi$$
−0.149932 + 0.988696i $$0.547906\pi$$
$$360$$ −83.8987 −4.42185
$$361$$ −13.2354 −0.696601
$$362$$ 17.7976 0.935423
$$363$$ 2.62651 0.137856
$$364$$ 0 0
$$365$$ 10.4091 0.544838
$$366$$ −9.45462 −0.494201
$$367$$ −19.6316 −1.02476 −0.512381 0.858758i $$-0.671236\pi$$
−0.512381 + 0.858758i $$0.671236\pi$$
$$368$$ 24.5658 1.28058
$$369$$ 12.1587 0.632958
$$370$$ −52.2773 −2.71777
$$371$$ 9.30628 0.483158
$$372$$ −3.19905 −0.165863
$$373$$ 32.0645 1.66024 0.830119 0.557586i $$-0.188272\pi$$
0.830119 + 0.557586i $$0.188272\pi$$
$$374$$ −10.7445 −0.555587
$$375$$ −0.693612 −0.0358180
$$376$$ 52.5482 2.70997
$$377$$ 0 0
$$378$$ −5.49909 −0.282843
$$379$$ 19.0231 0.977150 0.488575 0.872522i $$-0.337517\pi$$
0.488575 + 0.872522i $$0.337517\pi$$
$$380$$ −41.5167 −2.12976
$$381$$ 3.36760 0.172527
$$382$$ −37.1510 −1.90081
$$383$$ 0.699829 0.0357596 0.0178798 0.999840i $$-0.494308\pi$$
0.0178798 + 0.999840i $$0.494308\pi$$
$$384$$ −9.08689 −0.463714
$$385$$ 6.01459 0.306532
$$386$$ 61.5059 3.13057
$$387$$ 25.0076 1.27121
$$388$$ 48.9951 2.48735
$$389$$ −20.0547 −1.01681 −0.508407 0.861117i $$-0.669765\pi$$
−0.508407 + 0.861117i $$0.669765\pi$$
$$390$$ 0 0
$$391$$ 3.90284 0.197375
$$392$$ 8.94020 0.451548
$$393$$ 6.45592 0.325658
$$394$$ 39.2698 1.97838
$$395$$ 2.61272 0.131460
$$396$$ 28.2195 1.41809
$$397$$ 22.2803 1.11822 0.559108 0.829095i $$-0.311144\pi$$
0.559108 + 0.829095i $$0.311144\pi$$
$$398$$ 64.4453 3.23035
$$399$$ −0.830609 −0.0415824
$$400$$ 76.1027 3.80513
$$401$$ −4.80749 −0.240074 −0.120037 0.992769i $$-0.538301\pi$$
−0.120037 + 0.992769i $$0.538301\pi$$
$$402$$ 0.773306 0.0385690
$$403$$ 0 0
$$404$$ 78.6717 3.91406
$$405$$ 25.8604 1.28501
$$406$$ 7.39071 0.366795
$$407$$ 10.9574 0.543137
$$408$$ −6.65935 −0.329687
$$409$$ 36.8035 1.81981 0.909907 0.414811i $$-0.136152\pi$$
0.909907 + 0.414811i $$0.136152\pi$$
$$410$$ −37.1786 −1.83612
$$411$$ −2.91343 −0.143709
$$412$$ −22.7303 −1.11984
$$413$$ 10.7523 0.529084
$$414$$ −14.1132 −0.693628
$$415$$ 32.4845 1.59460
$$416$$ 0 0
$$417$$ 6.10122 0.298778
$$418$$ 11.9812 0.586019
$$419$$ 29.2667 1.42977 0.714887 0.699240i $$-0.246478\pi$$
0.714887 + 0.699240i $$0.246478\pi$$
$$420$$ 5.98206 0.291895
$$421$$ 7.53862 0.367410 0.183705 0.982981i $$-0.441191\pi$$
0.183705 + 0.982981i $$0.441191\pi$$
$$422$$ 11.6650 0.567845
$$423$$ −16.9298 −0.823154
$$424$$ −83.2000 −4.04055
$$425$$ 12.0907 0.586484
$$426$$ 2.20162 0.106669
$$427$$ −10.1101 −0.489261
$$428$$ −101.506 −4.90646
$$429$$ 0 0
$$430$$ −76.4674 −3.68759
$$431$$ 31.2261 1.50411 0.752055 0.659101i $$-0.229063\pi$$
0.752055 + 0.659101i $$0.229063\pi$$
$$432$$ 27.5699 1.32646
$$433$$ 5.88404 0.282769 0.141384 0.989955i $$-0.454845\pi$$
0.141384 + 0.989955i $$0.454845\pi$$
$$434$$ −4.70994 −0.226084
$$435$$ 3.08169 0.147755
$$436$$ 22.6702 1.08570
$$437$$ −4.35204 −0.208186
$$438$$ −2.98770 −0.142758
$$439$$ −9.95642 −0.475194 −0.237597 0.971364i $$-0.576360\pi$$
−0.237597 + 0.971364i $$0.576360\pi$$
$$440$$ −53.7716 −2.56346
$$441$$ −2.88032 −0.137158
$$442$$ 0 0
$$443$$ −35.8813 −1.70477 −0.852385 0.522915i $$-0.824845\pi$$
−0.852385 + 0.522915i $$0.824845\pi$$
$$444$$ 10.8981 0.517202
$$445$$ 49.2416 2.33427
$$446$$ 63.2404 2.99452
$$447$$ 1.39108 0.0657956
$$448$$ −23.5929 −1.11466
$$449$$ −3.99528 −0.188549 −0.0942744 0.995546i $$-0.530053\pi$$
−0.0942744 + 0.995546i $$0.530053\pi$$
$$450$$ −43.7217 −2.06106
$$451$$ 7.79266 0.366942
$$452$$ 14.5937 0.686432
$$453$$ −6.53877 −0.307218
$$454$$ −72.2371 −3.39026
$$455$$ 0 0
$$456$$ 7.42580 0.347745
$$457$$ −41.2222 −1.92829 −0.964147 0.265369i $$-0.914506\pi$$
−0.964147 + 0.265369i $$0.914506\pi$$
$$458$$ −8.12770 −0.379783
$$459$$ 4.38011 0.204446
$$460$$ 31.3435 1.46140
$$461$$ 24.7266 1.15163 0.575816 0.817579i $$-0.304684\pi$$
0.575816 + 0.817579i $$0.304684\pi$$
$$462$$ −1.72635 −0.0803169
$$463$$ −24.4057 −1.13423 −0.567115 0.823639i $$-0.691940\pi$$
−0.567115 + 0.823639i $$0.691940\pi$$
$$464$$ −37.0536 −1.72017
$$465$$ −1.96389 −0.0910733
$$466$$ −31.6674 −1.46696
$$467$$ −4.44860 −0.205857 −0.102928 0.994689i $$-0.532821\pi$$
−0.102928 + 0.994689i $$0.532821\pi$$
$$468$$ 0 0
$$469$$ 0.826916 0.0381834
$$470$$ 51.7673 2.38785
$$471$$ 4.00386 0.184488
$$472$$ −96.1273 −4.42462
$$473$$ 16.0276 0.736951
$$474$$ −0.749922 −0.0344450
$$475$$ −13.4823 −0.618608
$$476$$ −11.4273 −0.523769
$$477$$ 26.8051 1.22732
$$478$$ −3.86008 −0.176556
$$479$$ −31.6766 −1.44734 −0.723671 0.690145i $$-0.757547\pi$$
−0.723671 + 0.690145i $$0.757547\pi$$
$$480$$ −21.1394 −0.964879
$$481$$ 0 0
$$482$$ 7.24372 0.329942
$$483$$ 0.627077 0.0285330
$$484$$ −40.2938 −1.83153
$$485$$ 30.0780 1.36577
$$486$$ −23.9199 −1.08503
$$487$$ 26.9156 1.21966 0.609832 0.792531i $$-0.291237\pi$$
0.609832 + 0.792531i $$0.291237\pi$$
$$488$$ 90.3860 4.09158
$$489$$ −1.52405 −0.0689200
$$490$$ 8.80735 0.397875
$$491$$ 9.72716 0.438980 0.219490 0.975615i $$-0.429561\pi$$
0.219490 + 0.975615i $$0.429561\pi$$
$$492$$ 7.75052 0.349421
$$493$$ −5.88682 −0.265129
$$494$$ 0 0
$$495$$ 17.3239 0.778653
$$496$$ 23.6134 1.06027
$$497$$ 2.35425 0.105602
$$498$$ −9.32392 −0.417815
$$499$$ −7.87525 −0.352545 −0.176272 0.984341i $$-0.556404\pi$$
−0.176272 + 0.984341i $$0.556404\pi$$
$$500$$ 10.6408 0.475872
$$501$$ 3.12015 0.139398
$$502$$ 29.5565 1.31917
$$503$$ −9.75206 −0.434823 −0.217411 0.976080i $$-0.569761\pi$$
−0.217411 + 0.976080i $$0.569761\pi$$
$$504$$ 25.7506 1.14702
$$505$$ 48.2964 2.14916
$$506$$ −9.04533 −0.402114
$$507$$ 0 0
$$508$$ −51.6630 −2.29217
$$509$$ 23.0256 1.02059 0.510295 0.859999i $$-0.329536\pi$$
0.510295 + 0.859999i $$0.329536\pi$$
$$510$$ −6.56039 −0.290499
$$511$$ −3.19482 −0.141331
$$512$$ 11.8512 0.523752
$$513$$ −4.88424 −0.215645
$$514$$ 11.2220 0.494981
$$515$$ −13.9541 −0.614891
$$516$$ 15.9410 0.701762
$$517$$ −10.8505 −0.477203
$$518$$ 16.0452 0.704987
$$519$$ −2.10752 −0.0925100
$$520$$ 0 0
$$521$$ 0.486481 0.0213131 0.0106566 0.999943i $$-0.496608\pi$$
0.0106566 + 0.999943i $$0.496608\pi$$
$$522$$ 21.2876 0.931733
$$523$$ −34.6270 −1.51413 −0.757065 0.653339i $$-0.773368\pi$$
−0.757065 + 0.653339i $$0.773368\pi$$
$$524$$ −99.0414 −4.32664
$$525$$ 1.94263 0.0847834
$$526$$ 10.9577 0.477777
$$527$$ 3.75154 0.163420
$$528$$ 8.65510 0.376665
$$529$$ −19.7144 −0.857147
$$530$$ −81.9636 −3.56027
$$531$$ 30.9699 1.34398
$$532$$ 12.7425 0.552458
$$533$$ 0 0
$$534$$ −14.1336 −0.611622
$$535$$ −62.3141 −2.69408
$$536$$ −7.39279 −0.319320
$$537$$ −1.34216 −0.0579185
$$538$$ 10.8134 0.466198
$$539$$ −1.84603 −0.0795140
$$540$$ 35.1764 1.51375
$$541$$ 22.5384 0.969002 0.484501 0.874791i $$-0.339001\pi$$
0.484501 + 0.874791i $$0.339001\pi$$
$$542$$ −7.52844 −0.323374
$$543$$ −2.27770 −0.0977456
$$544$$ 40.3818 1.73136
$$545$$ 13.9172 0.596146
$$546$$ 0 0
$$547$$ 39.3716 1.68341 0.841704 0.539940i $$-0.181553\pi$$
0.841704 + 0.539940i $$0.181553\pi$$
$$548$$ 44.6955 1.90930
$$549$$ −29.1202 −1.24282
$$550$$ −28.0217 −1.19485
$$551$$ 6.56436 0.279651
$$552$$ −5.60619 −0.238615
$$553$$ −0.801911 −0.0341007
$$554$$ 45.1227 1.91708
$$555$$ 6.69034 0.283989
$$556$$ −93.6000 −3.96952
$$557$$ 0.726975 0.0308029 0.0154015 0.999881i $$-0.495097\pi$$
0.0154015 + 0.999881i $$0.495097\pi$$
$$558$$ −13.5661 −0.574300
$$559$$ 0 0
$$560$$ −44.1559 −1.86593
$$561$$ 1.37506 0.0580552
$$562$$ 36.1502 1.52490
$$563$$ 41.6077 1.75355 0.876777 0.480897i $$-0.159689\pi$$
0.876777 + 0.480897i $$0.159689\pi$$
$$564$$ −10.7918 −0.454417
$$565$$ 8.95907 0.376911
$$566$$ −51.0532 −2.14593
$$567$$ −7.93720 −0.333331
$$568$$ −21.0474 −0.883130
$$569$$ −25.3888 −1.06435 −0.532177 0.846633i $$-0.678626\pi$$
−0.532177 + 0.846633i $$0.678626\pi$$
$$570$$ 7.31546 0.306411
$$571$$ 16.9992 0.711393 0.355697 0.934602i $$-0.384244\pi$$
0.355697 + 0.934602i $$0.384244\pi$$
$$572$$ 0 0
$$573$$ 4.75451 0.198622
$$574$$ 11.4110 0.476288
$$575$$ 10.1786 0.424476
$$576$$ −67.9551 −2.83146
$$577$$ 15.9759 0.665084 0.332542 0.943088i $$-0.392094\pi$$
0.332542 + 0.943088i $$0.392094\pi$$
$$578$$ −33.4223 −1.39018
$$579$$ −7.87139 −0.327124
$$580$$ −47.2767 −1.96306
$$581$$ −9.97031 −0.413638
$$582$$ −8.63320 −0.357858
$$583$$ 17.1796 0.711508
$$584$$ 28.5623 1.18192
$$585$$ 0 0
$$586$$ 9.23926 0.381670
$$587$$ 15.9815 0.659627 0.329814 0.944046i $$-0.393014\pi$$
0.329814 + 0.944046i $$0.393014\pi$$
$$588$$ −1.83605 −0.0757172
$$589$$ −4.18333 −0.172371
$$590$$ −94.6989 −3.89869
$$591$$ −5.02566 −0.206728
$$592$$ −80.4433 −3.30620
$$593$$ −29.0532 −1.19307 −0.596536 0.802586i $$-0.703457\pi$$
−0.596536 + 0.802586i $$0.703457\pi$$
$$594$$ −10.1515 −0.416520
$$595$$ −7.01520 −0.287595
$$596$$ −21.3407 −0.874151
$$597$$ −8.24757 −0.337551
$$598$$ 0 0
$$599$$ 3.45554 0.141190 0.0705948 0.997505i $$-0.477510\pi$$
0.0705948 + 0.997505i $$0.477510\pi$$
$$600$$ −17.3675 −0.709026
$$601$$ 15.5304 0.633497 0.316748 0.948510i $$-0.397409\pi$$
0.316748 + 0.948510i $$0.397409\pi$$
$$602$$ 23.4698 0.956556
$$603$$ 2.38178 0.0969936
$$604$$ 100.313 4.08166
$$605$$ −24.7363 −1.00567
$$606$$ −13.8624 −0.563120
$$607$$ 15.4784 0.628250 0.314125 0.949382i $$-0.398289\pi$$
0.314125 + 0.949382i $$0.398289\pi$$
$$608$$ −45.0296 −1.82619
$$609$$ −0.945847 −0.0383276
$$610$$ 89.0429 3.60524
$$611$$ 0 0
$$612$$ −32.9142 −1.33048
$$613$$ 7.13223 0.288068 0.144034 0.989573i $$-0.453993\pi$$
0.144034 + 0.989573i $$0.453993\pi$$
$$614$$ −44.2456 −1.78561
$$615$$ 4.75803 0.191862
$$616$$ 16.5039 0.664959
$$617$$ −4.96685 −0.199958 −0.0999789 0.994990i $$-0.531878\pi$$
−0.0999789 + 0.994990i $$0.531878\pi$$
$$618$$ 4.00520 0.161113
$$619$$ −42.3570 −1.70247 −0.851235 0.524784i $$-0.824146\pi$$
−0.851235 + 0.524784i $$0.824146\pi$$
$$620$$ 30.1284 1.20999
$$621$$ 3.68741 0.147971
$$622$$ 64.0600 2.56857
$$623$$ −15.1135 −0.605508
$$624$$ 0 0
$$625$$ −21.5445 −0.861779
$$626$$ −14.0032 −0.559682
$$627$$ −1.53333 −0.0612352
$$628$$ −61.4239 −2.45108
$$629$$ −12.7803 −0.509583
$$630$$ 25.3680 1.01068
$$631$$ 6.26775 0.249515 0.124758 0.992187i $$-0.460185\pi$$
0.124758 + 0.992187i $$0.460185\pi$$
$$632$$ 7.16924 0.285177
$$633$$ −1.49286 −0.0593361
$$634$$ 16.3959 0.651163
$$635$$ −31.7158 −1.25860
$$636$$ 17.0868 0.677534
$$637$$ 0 0
$$638$$ 13.6434 0.540149
$$639$$ 6.78098 0.268251
$$640$$ 85.5797 3.38283
$$641$$ −31.5637 −1.24669 −0.623345 0.781947i $$-0.714227\pi$$
−0.623345 + 0.781947i $$0.714227\pi$$
$$642$$ 17.8858 0.705897
$$643$$ 18.2504 0.719725 0.359863 0.933005i $$-0.382824\pi$$
0.359863 + 0.933005i $$0.382824\pi$$
$$644$$ −9.62010 −0.379085
$$645$$ 9.78613 0.385329
$$646$$ −13.9744 −0.549816
$$647$$ 23.0273 0.905298 0.452649 0.891689i $$-0.350479\pi$$
0.452649 + 0.891689i $$0.350479\pi$$
$$648$$ 70.9601 2.78758
$$649$$ 19.8490 0.779140
$$650$$ 0 0
$$651$$ 0.602768 0.0236243
$$652$$ 23.3807 0.915660
$$653$$ 28.8124 1.12752 0.563759 0.825939i $$-0.309355\pi$$
0.563759 + 0.825939i $$0.309355\pi$$
$$654$$ −3.99460 −0.156201
$$655$$ −60.8014 −2.37571
$$656$$ −57.2096 −2.23366
$$657$$ −9.20210 −0.359008
$$658$$ −15.8887 −0.619406
$$659$$ −31.2228 −1.21627 −0.608134 0.793835i $$-0.708082\pi$$
−0.608134 + 0.793835i $$0.708082\pi$$
$$660$$ 11.0431 0.429850
$$661$$ −26.5582 −1.03299 −0.516496 0.856289i $$-0.672764\pi$$
−0.516496 + 0.856289i $$0.672764\pi$$
$$662$$ 46.6973 1.81494
$$663$$ 0 0
$$664$$ 89.1366 3.45917
$$665$$ 7.82261 0.303348
$$666$$ 46.2154 1.79081
$$667$$ −4.95584 −0.191891
$$668$$ −47.8667 −1.85202
$$669$$ −8.09337 −0.312908
$$670$$ −7.28293 −0.281364
$$671$$ −18.6635 −0.720495
$$672$$ 6.48823 0.250289
$$673$$ −19.7386 −0.760867 −0.380434 0.924808i $$-0.624225\pi$$
−0.380434 + 0.924808i $$0.624225\pi$$
$$674$$ 22.5842 0.869912
$$675$$ 11.4233 0.439683
$$676$$ 0 0
$$677$$ −13.1440 −0.505163 −0.252582 0.967576i $$-0.581280\pi$$
−0.252582 + 0.967576i $$0.581280\pi$$
$$678$$ −2.57149 −0.0987576
$$679$$ −9.23171 −0.354280
$$680$$ 62.7172 2.40510
$$681$$ 9.24475 0.354260
$$682$$ −8.69468 −0.332936
$$683$$ 6.76255 0.258762 0.129381 0.991595i $$-0.458701\pi$$
0.129381 + 0.991595i $$0.458701\pi$$
$$684$$ 36.7025 1.40336
$$685$$ 27.4385 1.04837
$$686$$ −2.70320 −0.103209
$$687$$ 1.04017 0.0396848
$$688$$ −117.666 −4.48599
$$689$$ 0 0
$$690$$ −5.52288 −0.210253
$$691$$ −9.17090 −0.348877 −0.174439 0.984668i $$-0.555811\pi$$
−0.174439 + 0.984668i $$0.555811\pi$$
$$692$$ 32.3319 1.22907
$$693$$ −5.31715 −0.201982
$$694$$ 77.9115 2.95748
$$695$$ −57.4609 −2.17962
$$696$$ 8.45605 0.320526
$$697$$ −9.08908 −0.344273
$$698$$ −31.6870 −1.19937
$$699$$ 4.05272 0.153288
$$700$$ −29.8023 −1.12642
$$701$$ −47.4700 −1.79292 −0.896459 0.443127i $$-0.853869\pi$$
−0.896459 + 0.443127i $$0.853869\pi$$
$$702$$ 0 0
$$703$$ 14.2512 0.537495
$$704$$ −43.5532 −1.64147
$$705$$ −6.62507 −0.249515
$$706$$ −48.2148 −1.81459
$$707$$ −14.8234 −0.557491
$$708$$ 19.7416 0.741936
$$709$$ −34.9719 −1.31340 −0.656699 0.754153i $$-0.728048\pi$$
−0.656699 + 0.754153i $$0.728048\pi$$
$$710$$ −20.7347 −0.778158
$$711$$ −2.30976 −0.0866227
$$712$$ 135.117 5.06374
$$713$$ 3.15825 0.118277
$$714$$ 2.01355 0.0753552
$$715$$ 0 0
$$716$$ 20.5903 0.769496
$$717$$ 0.494005 0.0184490
$$718$$ −15.3585 −0.573175
$$719$$ −8.36101 −0.311813 −0.155907 0.987772i $$-0.549830\pi$$
−0.155907 + 0.987772i $$0.549830\pi$$
$$720$$ −127.183 −4.73984
$$721$$ 4.28286 0.159502
$$722$$ −35.7779 −1.33152
$$723$$ −0.927035 −0.0344768
$$724$$ 34.9427 1.29863
$$725$$ −15.3528 −0.570188
$$726$$ 7.09997 0.263505
$$727$$ 27.4014 1.01626 0.508131 0.861280i $$-0.330337\pi$$
0.508131 + 0.861280i $$0.330337\pi$$
$$728$$ 0 0
$$729$$ −20.7504 −0.768532
$$730$$ 28.1379 1.04143
$$731$$ −18.6941 −0.691425
$$732$$ −18.5625 −0.686092
$$733$$ 12.1569 0.449026 0.224513 0.974471i $$-0.427921\pi$$
0.224513 + 0.974471i $$0.427921\pi$$
$$734$$ −53.0681 −1.95878
$$735$$ −1.12715 −0.0415754
$$736$$ 33.9956 1.25309
$$737$$ 1.52651 0.0562297
$$738$$ 32.8674 1.20987
$$739$$ −48.4439 −1.78204 −0.891019 0.453966i $$-0.850009\pi$$
−0.891019 + 0.453966i $$0.850009\pi$$
$$740$$ −102.638 −3.77304
$$741$$ 0 0
$$742$$ 25.1567 0.923531
$$743$$ 16.9906 0.623326 0.311663 0.950193i $$-0.399114\pi$$
0.311663 + 0.950193i $$0.399114\pi$$
$$744$$ −5.38886 −0.197565
$$745$$ −13.1010 −0.479985
$$746$$ 86.6767 3.17346
$$747$$ −28.7177 −1.05073
$$748$$ −21.0951 −0.771313
$$749$$ 19.1258 0.698841
$$750$$ −1.87497 −0.0684642
$$751$$ 43.0323 1.57027 0.785136 0.619323i $$-0.212593\pi$$
0.785136 + 0.619323i $$0.212593\pi$$
$$752$$ 79.6585 2.90485
$$753$$ −3.78258 −0.137845
$$754$$ 0 0
$$755$$ 61.5817 2.24119
$$756$$ −10.7965 −0.392666
$$757$$ 29.1785 1.06051 0.530255 0.847838i $$-0.322096\pi$$
0.530255 + 0.847838i $$0.322096\pi$$
$$758$$ 51.4231 1.86777
$$759$$ 1.15760 0.0420183
$$760$$ −69.9357 −2.53683
$$761$$ 29.4251 1.06666 0.533330 0.845907i $$-0.320941\pi$$
0.533330 + 0.845907i $$0.320941\pi$$
$$762$$ 9.10328 0.329777
$$763$$ −4.27153 −0.154640
$$764$$ −72.9398 −2.63887
$$765$$ −20.2060 −0.730550
$$766$$ 1.89178 0.0683526
$$767$$ 0 0
$$768$$ −8.23976 −0.297327
$$769$$ −17.1864 −0.619759 −0.309879 0.950776i $$-0.600289\pi$$
−0.309879 + 0.950776i $$0.600289\pi$$
$$770$$ 16.2586 0.585920
$$771$$ −1.43617 −0.0517223
$$772$$ 120.756 4.34612
$$773$$ 21.9601 0.789851 0.394926 0.918713i $$-0.370770\pi$$
0.394926 + 0.918713i $$0.370770\pi$$
$$774$$ 67.6004 2.42985
$$775$$ 9.78399 0.351451
$$776$$ 82.5333 2.96277
$$777$$ −2.05343 −0.0736665
$$778$$ −54.2118 −1.94359
$$779$$ 10.1352 0.363131
$$780$$ 0 0
$$781$$ 4.34600 0.155512
$$782$$ 10.5501 0.377272
$$783$$ −5.56188 −0.198765
$$784$$ 13.5526 0.484020
$$785$$ −37.7081 −1.34586
$$786$$ 17.4516 0.622478
$$787$$ 2.33567 0.0832578 0.0416289 0.999133i $$-0.486745\pi$$
0.0416289 + 0.999133i $$0.486745\pi$$
$$788$$ 77.0996 2.74656
$$789$$ −1.40234 −0.0499246
$$790$$ 7.06271 0.251280
$$791$$ −2.74976 −0.0977704
$$792$$ 47.5364 1.68913
$$793$$ 0 0
$$794$$ 60.2280 2.13741
$$795$$ 10.4895 0.372025
$$796$$ 126.527 4.48465
$$797$$ 27.8040 0.984869 0.492434 0.870350i $$-0.336107\pi$$
0.492434 + 0.870350i $$0.336107\pi$$
$$798$$ −2.24530 −0.0794827
$$799$$ 12.6556 0.447723
$$800$$ 105.315 3.72346
$$801$$ −43.5316 −1.53811
$$802$$ −12.9956 −0.458890
$$803$$ −5.89773 −0.208126
$$804$$ 1.51825 0.0535447
$$805$$ −5.90576 −0.208151
$$806$$ 0 0
$$807$$ −1.38387 −0.0487147
$$808$$ 132.524 4.66218
$$809$$ 15.0203 0.528087 0.264043 0.964511i $$-0.414944\pi$$
0.264043 + 0.964511i $$0.414944\pi$$
$$810$$ 69.9056 2.45623
$$811$$ −43.6933 −1.53428 −0.767139 0.641481i $$-0.778320\pi$$
−0.767139 + 0.641481i $$0.778320\pi$$
$$812$$ 14.5104 0.509215
$$813$$ 0.963474 0.0337905
$$814$$ 29.6199 1.03818
$$815$$ 14.3534 0.502778
$$816$$ −10.0950 −0.353395
$$817$$ 20.8456 0.729297
$$818$$ 99.4870 3.47848
$$819$$ 0 0
$$820$$ −72.9939 −2.54906
$$821$$ −18.0701 −0.630651 −0.315326 0.948984i $$-0.602114\pi$$
−0.315326 + 0.948984i $$0.602114\pi$$
$$822$$ −7.87557 −0.274692
$$823$$ 4.45550 0.155309 0.0776544 0.996980i $$-0.475257\pi$$
0.0776544 + 0.996980i $$0.475257\pi$$
$$824$$ −38.2896 −1.33388
$$825$$ 3.58615 0.124854
$$826$$ 29.0655 1.01132
$$827$$ 11.8352 0.411549 0.205774 0.978599i $$-0.434029\pi$$
0.205774 + 0.978599i $$0.434029\pi$$
$$828$$ −27.7090 −0.962953
$$829$$ −3.53894 −0.122913 −0.0614563 0.998110i $$-0.519574\pi$$
−0.0614563 + 0.998110i $$0.519574\pi$$
$$830$$ 87.8120 3.04800
$$831$$ −5.77471 −0.200323
$$832$$ 0 0
$$833$$ 2.15314 0.0746019
$$834$$ 16.4928 0.571099
$$835$$ −29.3853 −1.01692
$$836$$ 23.5230 0.813561
$$837$$ 3.54447 0.122515
$$838$$ 79.1137 2.73294
$$839$$ 33.4853 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$840$$ 10.0769 0.347686
$$841$$ −21.5249 −0.742238
$$842$$ 20.3784 0.702285
$$843$$ −4.62642 −0.159343
$$844$$ 22.9023 0.788330
$$845$$ 0 0
$$846$$ −45.7645 −1.57342
$$847$$ 7.59218 0.260870
$$848$$ −126.124 −4.33112
$$849$$ 6.53368 0.224235
$$850$$ 32.6835 1.12103
$$851$$ −10.7591 −0.368818
$$852$$ 4.32250 0.148087
$$853$$ 22.0871 0.756248 0.378124 0.925755i $$-0.376569\pi$$
0.378124 + 0.925755i $$0.376569\pi$$
$$854$$ −27.3295 −0.935196
$$855$$ 22.5316 0.770565
$$856$$ −170.988 −5.84426
$$857$$ −6.89363 −0.235482 −0.117741 0.993044i $$-0.537565\pi$$
−0.117741 + 0.993044i $$0.537565\pi$$
$$858$$ 0 0
$$859$$ −37.4834 −1.27892 −0.639459 0.768825i $$-0.720842\pi$$
−0.639459 + 0.768825i $$0.720842\pi$$
$$860$$ −150.131 −5.11942
$$861$$ −1.46036 −0.0497689
$$862$$ 84.4103 2.87503
$$863$$ 17.5248 0.596552 0.298276 0.954480i $$-0.403588\pi$$
0.298276 + 0.954480i $$0.403588\pi$$
$$864$$ 38.1528 1.29799
$$865$$ 19.8485 0.674869
$$866$$ 15.9057 0.540498
$$867$$ 4.27731 0.145265
$$868$$ −9.24717 −0.313869
$$869$$ −1.48035 −0.0502174
$$870$$ 8.33040 0.282427
$$871$$ 0 0
$$872$$ 38.1883 1.29322
$$873$$ −26.5903 −0.899944
$$874$$ −11.7644 −0.397937
$$875$$ −2.00495 −0.0677798
$$876$$ −5.86584 −0.198188
$$877$$ −46.7491 −1.57860 −0.789302 0.614005i $$-0.789557\pi$$
−0.789302 + 0.614005i $$0.789557\pi$$
$$878$$ −26.9142 −0.908309
$$879$$ −1.18242 −0.0398821
$$880$$ −81.5131 −2.74781
$$881$$ −2.91875 −0.0983350 −0.0491675 0.998791i $$-0.515657\pi$$
−0.0491675 + 0.998791i $$0.515657\pi$$
$$882$$ −7.78607 −0.262171
$$883$$ −28.5505 −0.960801 −0.480400 0.877049i $$-0.659509\pi$$
−0.480400 + 0.877049i $$0.659509\pi$$
$$884$$ 0 0
$$885$$ 12.1194 0.407388
$$886$$ −96.9941 −3.25858
$$887$$ 0.422914 0.0142001 0.00710004 0.999975i $$-0.497740\pi$$
0.00710004 + 0.999975i $$0.497740\pi$$
$$888$$ 18.3581 0.616058
$$889$$ 9.73438 0.326480
$$890$$ 133.110 4.46184
$$891$$ −14.6523 −0.490870
$$892$$ 124.162 4.15725
$$893$$ −14.1122 −0.472247
$$894$$ 3.76035 0.125765
$$895$$ 12.6404 0.422521
$$896$$ −26.2666 −0.877504
$$897$$ 0 0
$$898$$ −10.8000 −0.360401
$$899$$ −4.76372 −0.158879
$$900$$ −85.8401 −2.86134
$$901$$ −20.0377 −0.667553
$$902$$ 21.0651 0.701391
$$903$$ −3.00361 −0.0999539
$$904$$ 24.5834 0.817633
$$905$$ 21.4512 0.713063
$$906$$ −17.6756 −0.587232
$$907$$ −22.4284 −0.744723 −0.372361 0.928088i $$-0.621452\pi$$
−0.372361 + 0.928088i $$0.621452\pi$$
$$908$$ −141.825 −4.70664
$$909$$ −42.6961 −1.41614
$$910$$ 0 0
$$911$$ −32.5788 −1.07938 −0.539692 0.841863i $$-0.681459\pi$$
−0.539692 + 0.841863i $$0.681459\pi$$
$$912$$ 11.2569 0.372752
$$913$$ −18.4055 −0.609132
$$914$$ −111.432 −3.68583
$$915$$ −11.3955 −0.376724
$$916$$ −15.9574 −0.527247
$$917$$ 18.6615 0.616256
$$918$$ 11.8403 0.390788
$$919$$ −9.87913 −0.325883 −0.162941 0.986636i $$-0.552098\pi$$
−0.162941 + 0.986636i $$0.552098\pi$$
$$920$$ 52.7987 1.74072
$$921$$ 5.66246 0.186584
$$922$$ 66.8408 2.20129
$$923$$ 0 0
$$924$$ −3.38939 −0.111503
$$925$$ −33.3309 −1.09591
$$926$$ −65.9734 −2.16802
$$927$$ 12.3360 0.405168
$$928$$ −51.2769 −1.68325
$$929$$ 29.9136 0.981434 0.490717 0.871319i $$-0.336735\pi$$
0.490717 + 0.871319i $$0.336735\pi$$
$$930$$ −5.30878 −0.174082
$$931$$ −2.40096 −0.0786881
$$932$$ −62.1736 −2.03656
$$933$$ −8.19826 −0.268399
$$934$$ −12.0255 −0.393485
$$935$$ −12.9502 −0.423518
$$936$$ 0 0
$$937$$ −31.8296 −1.03983 −0.519914 0.854219i $$-0.674036\pi$$
−0.519914 + 0.854219i $$0.674036\pi$$
$$938$$ 2.23532 0.0729856
$$939$$ 1.79210 0.0584831
$$940$$ 101.636 3.31501
$$941$$ 42.0885 1.37205 0.686023 0.727580i $$-0.259355\pi$$
0.686023 + 0.727580i $$0.259355\pi$$
$$942$$ 10.8232 0.352640
$$943$$ −7.65167 −0.249173
$$944$$ −145.721 −4.74280
$$945$$ −6.62797 −0.215608
$$946$$ 43.3258 1.40864
$$947$$ 60.3377 1.96071 0.980357 0.197234i $$-0.0631958\pi$$
0.980357 + 0.197234i $$0.0631958\pi$$
$$948$$ −1.47234 −0.0478195
$$949$$ 0 0
$$950$$ −36.4452 −1.18244
$$951$$ −2.09831 −0.0680423
$$952$$ −19.2495 −0.623880
$$953$$ −17.3754 −0.562844 −0.281422 0.959584i $$-0.590806\pi$$
−0.281422 + 0.959584i $$0.590806\pi$$
$$954$$ 72.4593 2.34596
$$955$$ −44.7776 −1.44897
$$956$$ −7.57862 −0.245110
$$957$$ −1.74606 −0.0564421
$$958$$ −85.6281 −2.76652
$$959$$ −8.42156 −0.271946
$$960$$ −26.5926 −0.858274
$$961$$ −27.9642 −0.902070
$$962$$ 0 0
$$963$$ 55.0883 1.77520
$$964$$ 14.2218 0.458054
$$965$$ 74.1322 2.38640
$$966$$ 1.69511 0.0545393
$$967$$ −18.8630 −0.606594 −0.303297 0.952896i $$-0.598087\pi$$
−0.303297 + 0.952896i $$0.598087\pi$$
$$968$$ −67.8756 −2.18160
$$969$$ 1.78842 0.0574522
$$970$$ 81.3068 2.61061
$$971$$ −1.56446 −0.0502060 −0.0251030 0.999685i $$-0.507991\pi$$
−0.0251030 + 0.999685i $$0.507991\pi$$
$$972$$ −46.9626 −1.50633
$$973$$ 17.6362 0.565390
$$974$$ 72.7582 2.33132
$$975$$ 0 0
$$976$$ 137.017 4.38582
$$977$$ 27.8755 0.891817 0.445909 0.895078i $$-0.352881\pi$$
0.445909 + 0.895078i $$0.352881\pi$$
$$978$$ −4.11981 −0.131737
$$979$$ −27.8999 −0.891684
$$980$$ 17.2917 0.552364
$$981$$ −12.3034 −0.392817
$$982$$ 26.2944 0.839088
$$983$$ 33.4239 1.06606 0.533029 0.846097i $$-0.321054\pi$$
0.533029 + 0.846097i $$0.321054\pi$$
$$984$$ 13.0559 0.416207
$$985$$ 47.3313 1.50810
$$986$$ −15.9132 −0.506780
$$987$$ 2.03340 0.0647238
$$988$$ 0 0
$$989$$ −15.7376 −0.500428
$$990$$ 46.8300 1.48835
$$991$$ −18.9110 −0.600726 −0.300363 0.953825i $$-0.597108\pi$$
−0.300363 + 0.953825i $$0.597108\pi$$
$$992$$ 32.6777 1.03752
$$993$$ −5.97622 −0.189650
$$994$$ 6.36399 0.201853
$$995$$ 77.6750 2.46246
$$996$$ −18.3059 −0.580046
$$997$$ 43.5775 1.38011 0.690057 0.723755i $$-0.257585\pi$$
0.690057 + 0.723755i $$0.257585\pi$$
$$998$$ −21.2884 −0.673871
$$999$$ −12.0748 −0.382031
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.1.6 6
7.6 odd 2 8281.2.a.ch.1.6 6
13.5 odd 4 1183.2.c.i.337.1 12
13.6 odd 12 91.2.q.a.36.1 12
13.8 odd 4 1183.2.c.i.337.12 12
13.11 odd 12 91.2.q.a.43.1 yes 12
13.12 even 2 1183.2.a.m.1.1 6
39.11 even 12 819.2.ct.a.316.6 12
39.32 even 12 819.2.ct.a.127.6 12
52.11 even 12 1456.2.cc.c.225.4 12
52.19 even 12 1456.2.cc.c.673.4 12
91.6 even 12 637.2.q.h.491.1 12
91.11 odd 12 637.2.u.h.30.6 12
91.19 even 12 637.2.u.i.361.6 12
91.24 even 12 637.2.u.i.30.6 12
91.32 odd 12 637.2.k.h.569.1 12
91.37 odd 12 637.2.k.h.459.6 12
91.45 even 12 637.2.k.g.569.1 12
91.58 odd 12 637.2.u.h.361.6 12
91.76 even 12 637.2.q.h.589.1 12
91.89 even 12 637.2.k.g.459.6 12
91.90 odd 2 8281.2.a.by.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.q.a.36.1 12 13.6 odd 12
91.2.q.a.43.1 yes 12 13.11 odd 12
637.2.k.g.459.6 12 91.89 even 12
637.2.k.g.569.1 12 91.45 even 12
637.2.k.h.459.6 12 91.37 odd 12
637.2.k.h.569.1 12 91.32 odd 12
637.2.q.h.491.1 12 91.6 even 12
637.2.q.h.589.1 12 91.76 even 12
637.2.u.h.30.6 12 91.11 odd 12
637.2.u.h.361.6 12 91.58 odd 12
637.2.u.i.30.6 12 91.24 even 12
637.2.u.i.361.6 12 91.19 even 12
819.2.ct.a.127.6 12 39.32 even 12
819.2.ct.a.316.6 12 39.11 even 12
1183.2.a.m.1.1 6 13.12 even 2
1183.2.a.p.1.6 6 1.1 even 1 trivial
1183.2.c.i.337.1 12 13.5 odd 4
1183.2.c.i.337.12 12 13.8 odd 4
1456.2.cc.c.225.4 12 52.11 even 12
1456.2.cc.c.673.4 12 52.19 even 12
8281.2.a.by.1.1 6 91.90 odd 2
8281.2.a.ch.1.6 6 7.6 odd 2