# Properties

 Label 3549.2.a.h.1.2 Level $3549$ Weight $2$ Character 3549.1 Self dual yes Analytic conductor $28.339$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3549 = 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3549.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.3389076774$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.169.1 Defining polynomial: $$x^{3} - x^{2} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.273891$$ of defining polynomial Character $$\chi$$ $$=$$ 3549.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.27389 q^{2} -1.00000 q^{3} -0.377203 q^{4} +1.10331 q^{5} +1.27389 q^{6} -1.00000 q^{7} +3.02830 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.27389 q^{2} -1.00000 q^{3} -0.377203 q^{4} +1.10331 q^{5} +1.27389 q^{6} -1.00000 q^{7} +3.02830 q^{8} +1.00000 q^{9} -1.40550 q^{10} -0.348907 q^{11} +0.377203 q^{12} +1.27389 q^{14} -1.10331 q^{15} -3.10331 q^{16} +0.726109 q^{17} -1.27389 q^{18} +2.30219 q^{19} -0.416173 q^{20} +1.00000 q^{21} +0.444469 q^{22} +0.0750160 q^{23} -3.02830 q^{24} -3.78270 q^{25} -1.00000 q^{27} +0.377203 q^{28} +0.480515 q^{29} +1.40550 q^{30} -6.85772 q^{31} -2.10331 q^{32} +0.348907 q^{33} -0.924984 q^{34} -1.10331 q^{35} -0.377203 q^{36} -1.10331 q^{37} -2.93273 q^{38} +3.34116 q^{40} +3.09556 q^{41} -1.27389 q^{42} -4.62280 q^{43} +0.131609 q^{44} +1.10331 q^{45} -0.0955622 q^{46} +4.70769 q^{47} +3.10331 q^{48} +1.00000 q^{49} +4.81875 q^{50} -0.726109 q^{51} +5.54778 q^{53} +1.27389 q^{54} -0.384953 q^{55} -3.02830 q^{56} -2.30219 q^{57} -0.612124 q^{58} -3.92498 q^{59} +0.416173 q^{60} +5.54778 q^{61} +8.73598 q^{62} -1.00000 q^{63} +8.88601 q^{64} -0.444469 q^{66} +8.12386 q^{67} -0.273891 q^{68} -0.0750160 q^{69} +1.40550 q^{70} -9.52936 q^{71} +3.02830 q^{72} -10.4338 q^{73} +1.40550 q^{74} +3.78270 q^{75} -0.868391 q^{76} +0.348907 q^{77} +17.1054 q^{79} -3.42392 q^{80} +1.00000 q^{81} -3.94341 q^{82} -11.9143 q^{83} -0.377203 q^{84} +0.801125 q^{85} +5.88894 q^{86} -0.480515 q^{87} -1.05659 q^{88} -11.9610 q^{89} -1.40550 q^{90} -0.0282963 q^{92} +6.85772 q^{93} -5.99708 q^{94} +2.54003 q^{95} +2.10331 q^{96} +10.1033 q^{97} -1.27389 q^{98} -0.348907 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + 13q^{10} - 8q^{11} - 4q^{12} + 2q^{14} - 6q^{16} + 4q^{17} - 2q^{18} - 7q^{19} - 13q^{20} + 3q^{21} + q^{22} + 9q^{23} + 3q^{24} + 11q^{25} - 3q^{27} - 4q^{28} - 7q^{29} - 13q^{30} - 7q^{31} - 3q^{32} + 8q^{33} + 6q^{34} + 4q^{36} - 4q^{38} + 13q^{40} + 2q^{41} - 2q^{42} - 19q^{43} - 15q^{44} + 7q^{46} - 17q^{47} + 6q^{48} + 3q^{49} - 16q^{50} - 4q^{51} + 13q^{53} + 2q^{54} + 3q^{56} + 7q^{57} + 22q^{58} - 3q^{59} + 13q^{60} + 13q^{61} - 17q^{62} - 3q^{63} + q^{64} - q^{66} + 5q^{67} + q^{68} - 9q^{69} - 13q^{70} + 8q^{71} - 3q^{72} - 2q^{73} - 13q^{74} - 11q^{75} - 18q^{76} + 8q^{77} - q^{79} - 26q^{80} + 3q^{81} - 36q^{82} + 2q^{83} + 4q^{84} + 13q^{85} + 17q^{86} + 7q^{87} + 21q^{88} - 19q^{89} + 13q^{90} + 12q^{92} + 7q^{93} + 7q^{94} + 3q^{96} + 27q^{97} - 2q^{98} - 8q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.27389 −0.900777 −0.450388 0.892833i $$-0.648714\pi$$
−0.450388 + 0.892833i $$0.648714\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −0.377203 −0.188601
$$5$$ 1.10331 0.493416 0.246708 0.969090i $$-0.420651\pi$$
0.246708 + 0.969090i $$0.420651\pi$$
$$6$$ 1.27389 0.520064
$$7$$ −1.00000 −0.377964
$$8$$ 3.02830 1.07066
$$9$$ 1.00000 0.333333
$$10$$ −1.40550 −0.444458
$$11$$ −0.348907 −0.105199 −0.0525996 0.998616i $$-0.516751\pi$$
−0.0525996 + 0.998616i $$0.516751\pi$$
$$12$$ 0.377203 0.108889
$$13$$ 0 0
$$14$$ 1.27389 0.340462
$$15$$ −1.10331 −0.284874
$$16$$ −3.10331 −0.775828
$$17$$ 0.726109 0.176107 0.0880537 0.996116i $$-0.471935\pi$$
0.0880537 + 0.996116i $$0.471935\pi$$
$$18$$ −1.27389 −0.300259
$$19$$ 2.30219 0.528158 0.264079 0.964501i $$-0.414932\pi$$
0.264079 + 0.964501i $$0.414932\pi$$
$$20$$ −0.416173 −0.0930590
$$21$$ 1.00000 0.218218
$$22$$ 0.444469 0.0947611
$$23$$ 0.0750160 0.0156419 0.00782096 0.999969i $$-0.497510\pi$$
0.00782096 + 0.999969i $$0.497510\pi$$
$$24$$ −3.02830 −0.618148
$$25$$ −3.78270 −0.756540
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0.377203 0.0712846
$$29$$ 0.480515 0.0892294 0.0446147 0.999004i $$-0.485794\pi$$
0.0446147 + 0.999004i $$0.485794\pi$$
$$30$$ 1.40550 0.256608
$$31$$ −6.85772 −1.23168 −0.615841 0.787870i $$-0.711184\pi$$
−0.615841 + 0.787870i $$0.711184\pi$$
$$32$$ −2.10331 −0.371817
$$33$$ 0.348907 0.0607368
$$34$$ −0.924984 −0.158633
$$35$$ −1.10331 −0.186494
$$36$$ −0.377203 −0.0628671
$$37$$ −1.10331 −0.181383 −0.0906917 0.995879i $$-0.528908\pi$$
−0.0906917 + 0.995879i $$0.528908\pi$$
$$38$$ −2.93273 −0.475752
$$39$$ 0 0
$$40$$ 3.34116 0.528283
$$41$$ 3.09556 0.483446 0.241723 0.970345i $$-0.422288\pi$$
0.241723 + 0.970345i $$0.422288\pi$$
$$42$$ −1.27389 −0.196566
$$43$$ −4.62280 −0.704970 −0.352485 0.935817i $$-0.614663\pi$$
−0.352485 + 0.935817i $$0.614663\pi$$
$$44$$ 0.131609 0.0198407
$$45$$ 1.10331 0.164472
$$46$$ −0.0955622 −0.0140899
$$47$$ 4.70769 0.686687 0.343343 0.939210i $$-0.388441\pi$$
0.343343 + 0.939210i $$0.388441\pi$$
$$48$$ 3.10331 0.447925
$$49$$ 1.00000 0.142857
$$50$$ 4.81875 0.681474
$$51$$ −0.726109 −0.101676
$$52$$ 0 0
$$53$$ 5.54778 0.762046 0.381023 0.924565i $$-0.375572\pi$$
0.381023 + 0.924565i $$0.375572\pi$$
$$54$$ 1.27389 0.173355
$$55$$ −0.384953 −0.0519070
$$56$$ −3.02830 −0.404673
$$57$$ −2.30219 −0.304932
$$58$$ −0.612124 −0.0803758
$$59$$ −3.92498 −0.510989 −0.255495 0.966810i $$-0.582238\pi$$
−0.255495 + 0.966810i $$0.582238\pi$$
$$60$$ 0.416173 0.0537276
$$61$$ 5.54778 0.710321 0.355160 0.934805i $$-0.384426\pi$$
0.355160 + 0.934805i $$0.384426\pi$$
$$62$$ 8.73598 1.10947
$$63$$ −1.00000 −0.125988
$$64$$ 8.88601 1.11075
$$65$$ 0 0
$$66$$ −0.444469 −0.0547103
$$67$$ 8.12386 0.992487 0.496244 0.868183i $$-0.334712\pi$$
0.496244 + 0.868183i $$0.334712\pi$$
$$68$$ −0.273891 −0.0332141
$$69$$ −0.0750160 −0.00903087
$$70$$ 1.40550 0.167989
$$71$$ −9.52936 −1.13093 −0.565463 0.824773i $$-0.691303\pi$$
−0.565463 + 0.824773i $$0.691303\pi$$
$$72$$ 3.02830 0.356888
$$73$$ −10.4338 −1.22118 −0.610592 0.791946i $$-0.709068\pi$$
−0.610592 + 0.791946i $$0.709068\pi$$
$$74$$ 1.40550 0.163386
$$75$$ 3.78270 0.436789
$$76$$ −0.868391 −0.0996113
$$77$$ 0.348907 0.0397616
$$78$$ 0 0
$$79$$ 17.1054 1.92451 0.962256 0.272146i $$-0.0877335\pi$$
0.962256 + 0.272146i $$0.0877335\pi$$
$$80$$ −3.42392 −0.382806
$$81$$ 1.00000 0.111111
$$82$$ −3.94341 −0.435477
$$83$$ −11.9143 −1.30777 −0.653883 0.756596i $$-0.726861\pi$$
−0.653883 + 0.756596i $$0.726861\pi$$
$$84$$ −0.377203 −0.0411562
$$85$$ 0.801125 0.0868943
$$86$$ 5.88894 0.635020
$$87$$ −0.480515 −0.0515166
$$88$$ −1.05659 −0.112633
$$89$$ −11.9610 −1.26787 −0.633933 0.773388i $$-0.718561\pi$$
−0.633933 + 0.773388i $$0.718561\pi$$
$$90$$ −1.40550 −0.148153
$$91$$ 0 0
$$92$$ −0.0282963 −0.00295009
$$93$$ 6.85772 0.711112
$$94$$ −5.99708 −0.618551
$$95$$ 2.54003 0.260602
$$96$$ 2.10331 0.214668
$$97$$ 10.1033 1.02584 0.512918 0.858438i $$-0.328564\pi$$
0.512918 + 0.858438i $$0.328564\pi$$
$$98$$ −1.27389 −0.128682
$$99$$ −0.348907 −0.0350664
$$100$$ 1.42685 0.142685
$$101$$ 4.07502 0.405479 0.202740 0.979233i $$-0.435015\pi$$
0.202740 + 0.979233i $$0.435015\pi$$
$$102$$ 0.924984 0.0915871
$$103$$ −4.11106 −0.405075 −0.202538 0.979275i $$-0.564919\pi$$
−0.202538 + 0.979275i $$0.564919\pi$$
$$104$$ 0 0
$$105$$ 1.10331 0.107672
$$106$$ −7.06727 −0.686434
$$107$$ −6.17058 −0.596532 −0.298266 0.954483i $$-0.596408\pi$$
−0.298266 + 0.954483i $$0.596408\pi$$
$$108$$ 0.377203 0.0362964
$$109$$ −8.82167 −0.844963 −0.422481 0.906372i $$-0.638841\pi$$
−0.422481 + 0.906372i $$0.638841\pi$$
$$110$$ 0.490388 0.0467567
$$111$$ 1.10331 0.104722
$$112$$ 3.10331 0.293235
$$113$$ 12.0099 1.12979 0.564897 0.825161i $$-0.308916\pi$$
0.564897 + 0.825161i $$0.308916\pi$$
$$114$$ 2.93273 0.274676
$$115$$ 0.0827661 0.00771798
$$116$$ −0.181252 −0.0168288
$$117$$ 0 0
$$118$$ 5.00000 0.460287
$$119$$ −0.726109 −0.0665623
$$120$$ −3.34116 −0.305004
$$121$$ −10.8783 −0.988933
$$122$$ −7.06727 −0.639840
$$123$$ −3.09556 −0.279117
$$124$$ 2.58675 0.232297
$$125$$ −9.69006 −0.866706
$$126$$ 1.27389 0.113487
$$127$$ −5.85772 −0.519788 −0.259894 0.965637i $$-0.583688\pi$$
−0.259894 + 0.965637i $$0.583688\pi$$
$$128$$ −7.11319 −0.628723
$$129$$ 4.62280 0.407015
$$130$$ 0 0
$$131$$ 22.4055 1.95758 0.978789 0.204872i $$-0.0656778\pi$$
0.978789 + 0.204872i $$0.0656778\pi$$
$$132$$ −0.131609 −0.0114551
$$133$$ −2.30219 −0.199625
$$134$$ −10.3489 −0.894009
$$135$$ −1.10331 −0.0949580
$$136$$ 2.19887 0.188552
$$137$$ 20.2087 1.72655 0.863275 0.504734i $$-0.168409\pi$$
0.863275 + 0.504734i $$0.168409\pi$$
$$138$$ 0.0955622 0.00813480
$$139$$ 4.27389 0.362507 0.181253 0.983436i $$-0.441985\pi$$
0.181253 + 0.983436i $$0.441985\pi$$
$$140$$ 0.416173 0.0351730
$$141$$ −4.70769 −0.396459
$$142$$ 12.1394 1.01871
$$143$$ 0 0
$$144$$ −3.10331 −0.258609
$$145$$ 0.530158 0.0440272
$$146$$ 13.2915 1.10001
$$147$$ −1.00000 −0.0824786
$$148$$ 0.416173 0.0342092
$$149$$ −4.89669 −0.401152 −0.200576 0.979678i $$-0.564281\pi$$
−0.200576 + 0.979678i $$0.564281\pi$$
$$150$$ −4.81875 −0.393449
$$151$$ −8.94553 −0.727977 −0.363988 0.931403i $$-0.618585\pi$$
−0.363988 + 0.931403i $$0.618585\pi$$
$$152$$ 6.97170 0.565480
$$153$$ 0.726109 0.0587025
$$154$$ −0.444469 −0.0358163
$$155$$ −7.56620 −0.607732
$$156$$ 0 0
$$157$$ 6.59450 0.526298 0.263149 0.964755i $$-0.415239\pi$$
0.263149 + 0.964755i $$0.415239\pi$$
$$158$$ −21.7905 −1.73356
$$159$$ −5.54778 −0.439968
$$160$$ −2.32061 −0.183460
$$161$$ −0.0750160 −0.00591209
$$162$$ −1.27389 −0.100086
$$163$$ −9.90444 −0.775775 −0.387888 0.921707i $$-0.626795\pi$$
−0.387888 + 0.921707i $$0.626795\pi$$
$$164$$ −1.16765 −0.0911785
$$165$$ 0.384953 0.0299685
$$166$$ 15.1775 1.17800
$$167$$ −16.7565 −1.29666 −0.648330 0.761360i $$-0.724532\pi$$
−0.648330 + 0.761360i $$0.724532\pi$$
$$168$$ 3.02830 0.233638
$$169$$ 0 0
$$170$$ −1.02055 −0.0782723
$$171$$ 2.30219 0.176053
$$172$$ 1.74373 0.132958
$$173$$ 13.7184 1.04299 0.521494 0.853255i $$-0.325375\pi$$
0.521494 + 0.853255i $$0.325375\pi$$
$$174$$ 0.612124 0.0464050
$$175$$ 3.78270 0.285945
$$176$$ 1.08277 0.0816166
$$177$$ 3.92498 0.295020
$$178$$ 15.2370 1.14206
$$179$$ −22.0304 −1.64663 −0.823315 0.567584i $$-0.807878\pi$$
−0.823315 + 0.567584i $$0.807878\pi$$
$$180$$ −0.416173 −0.0310197
$$181$$ 9.43380 0.701208 0.350604 0.936524i $$-0.385976\pi$$
0.350604 + 0.936524i $$0.385976\pi$$
$$182$$ 0 0
$$183$$ −5.54778 −0.410104
$$184$$ 0.227171 0.0167473
$$185$$ −1.21730 −0.0894975
$$186$$ −8.73598 −0.640553
$$187$$ −0.253344 −0.0185264
$$188$$ −1.77575 −0.129510
$$189$$ 1.00000 0.0727393
$$190$$ −3.23572 −0.234744
$$191$$ −3.37720 −0.244366 −0.122183 0.992508i $$-0.538989\pi$$
−0.122183 + 0.992508i $$0.538989\pi$$
$$192$$ −8.88601 −0.641293
$$193$$ 22.3227 1.60683 0.803413 0.595423i $$-0.203015\pi$$
0.803413 + 0.595423i $$0.203015\pi$$
$$194$$ −12.8705 −0.924049
$$195$$ 0 0
$$196$$ −0.377203 −0.0269431
$$197$$ −25.4514 −1.81334 −0.906669 0.421842i $$-0.861384\pi$$
−0.906669 + 0.421842i $$0.861384\pi$$
$$198$$ 0.444469 0.0315870
$$199$$ −22.3404 −1.58367 −0.791833 0.610738i $$-0.790873\pi$$
−0.791833 + 0.610738i $$0.790873\pi$$
$$200$$ −11.4551 −0.810001
$$201$$ −8.12386 −0.573013
$$202$$ −5.19112 −0.365246
$$203$$ −0.480515 −0.0337256
$$204$$ 0.273891 0.0191762
$$205$$ 3.41537 0.238540
$$206$$ 5.23704 0.364882
$$207$$ 0.0750160 0.00521397
$$208$$ 0 0
$$209$$ −0.803248 −0.0555618
$$210$$ −1.40550 −0.0969887
$$211$$ 10.0382 0.691056 0.345528 0.938408i $$-0.387700\pi$$
0.345528 + 0.938408i $$0.387700\pi$$
$$212$$ −2.09264 −0.143723
$$213$$ 9.52936 0.652941
$$214$$ 7.86064 0.537342
$$215$$ −5.10039 −0.347844
$$216$$ −3.02830 −0.206049
$$217$$ 6.85772 0.465532
$$218$$ 11.2378 0.761123
$$219$$ 10.4338 0.705051
$$220$$ 0.145205 0.00978974
$$221$$ 0 0
$$222$$ −1.40550 −0.0943309
$$223$$ −5.13453 −0.343834 −0.171917 0.985111i $$-0.554996\pi$$
−0.171917 + 0.985111i $$0.554996\pi$$
$$224$$ 2.10331 0.140533
$$225$$ −3.78270 −0.252180
$$226$$ −15.2993 −1.01769
$$227$$ −28.9709 −1.92287 −0.961433 0.275039i $$-0.911309\pi$$
−0.961433 + 0.275039i $$0.911309\pi$$
$$228$$ 0.868391 0.0575106
$$229$$ 0.679390 0.0448953 0.0224477 0.999748i $$-0.492854\pi$$
0.0224477 + 0.999748i $$0.492854\pi$$
$$230$$ −0.105435 −0.00695218
$$231$$ −0.348907 −0.0229564
$$232$$ 1.45514 0.0955348
$$233$$ −18.3022 −1.19902 −0.599508 0.800369i $$-0.704637\pi$$
−0.599508 + 0.800369i $$0.704637\pi$$
$$234$$ 0 0
$$235$$ 5.19405 0.338822
$$236$$ 1.48052 0.0963733
$$237$$ −17.1054 −1.11112
$$238$$ 0.924984 0.0599578
$$239$$ −5.45222 −0.352675 −0.176337 0.984330i $$-0.556425\pi$$
−0.176337 + 0.984330i $$0.556425\pi$$
$$240$$ 3.42392 0.221013
$$241$$ −15.4621 −0.996001 −0.498000 0.867177i $$-0.665932\pi$$
−0.498000 + 0.867177i $$0.665932\pi$$
$$242$$ 13.8577 0.890808
$$243$$ −1.00000 −0.0641500
$$244$$ −2.09264 −0.133967
$$245$$ 1.10331 0.0704880
$$246$$ 3.94341 0.251422
$$247$$ 0 0
$$248$$ −20.7672 −1.31872
$$249$$ 11.9143 0.755039
$$250$$ 12.3441 0.780708
$$251$$ −8.29444 −0.523540 −0.261770 0.965130i $$-0.584306\pi$$
−0.261770 + 0.965130i $$0.584306\pi$$
$$252$$ 0.377203 0.0237615
$$253$$ −0.0261736 −0.00164552
$$254$$ 7.46209 0.468213
$$255$$ −0.801125 −0.0501684
$$256$$ −8.71061 −0.544413
$$257$$ −15.0382 −0.938055 −0.469028 0.883184i $$-0.655396\pi$$
−0.469028 + 0.883184i $$0.655396\pi$$
$$258$$ −5.88894 −0.366629
$$259$$ 1.10331 0.0685565
$$260$$ 0 0
$$261$$ 0.480515 0.0297431
$$262$$ −28.5422 −1.76334
$$263$$ −17.2448 −1.06336 −0.531680 0.846945i $$-0.678439\pi$$
−0.531680 + 0.846945i $$0.678439\pi$$
$$264$$ 1.05659 0.0650288
$$265$$ 6.12094 0.376006
$$266$$ 2.93273 0.179817
$$267$$ 11.9610 0.732003
$$268$$ −3.06434 −0.187185
$$269$$ −24.2944 −1.48126 −0.740629 0.671914i $$-0.765472\pi$$
−0.740629 + 0.671914i $$0.765472\pi$$
$$270$$ 1.40550 0.0855360
$$271$$ 18.3326 1.11363 0.556813 0.830638i $$-0.312024\pi$$
0.556813 + 0.830638i $$0.312024\pi$$
$$272$$ −2.25334 −0.136629
$$273$$ 0 0
$$274$$ −25.7437 −1.55524
$$275$$ 1.31981 0.0795875
$$276$$ 0.0282963 0.00170323
$$277$$ 8.45434 0.507972 0.253986 0.967208i $$-0.418258\pi$$
0.253986 + 0.967208i $$0.418258\pi$$
$$278$$ −5.44447 −0.326538
$$279$$ −6.85772 −0.410561
$$280$$ −3.34116 −0.199672
$$281$$ −25.0120 −1.49209 −0.746045 0.665895i $$-0.768050\pi$$
−0.746045 + 0.665895i $$0.768050\pi$$
$$282$$ 5.99708 0.357121
$$283$$ 27.1415 1.61339 0.806697 0.590966i $$-0.201253\pi$$
0.806697 + 0.590966i $$0.201253\pi$$
$$284$$ 3.59450 0.213294
$$285$$ −2.54003 −0.150458
$$286$$ 0 0
$$287$$ −3.09556 −0.182725
$$288$$ −2.10331 −0.123939
$$289$$ −16.4728 −0.968986
$$290$$ −0.675364 −0.0396587
$$291$$ −10.1033 −0.592267
$$292$$ 3.93566 0.230317
$$293$$ −10.5400 −0.615755 −0.307878 0.951426i $$-0.599619\pi$$
−0.307878 + 0.951426i $$0.599619\pi$$
$$294$$ 1.27389 0.0742948
$$295$$ −4.33048 −0.252130
$$296$$ −3.34116 −0.194201
$$297$$ 0.348907 0.0202456
$$298$$ 6.23784 0.361349
$$299$$ 0 0
$$300$$ −1.42685 −0.0823790
$$301$$ 4.62280 0.266454
$$302$$ 11.3956 0.655745
$$303$$ −4.07502 −0.234104
$$304$$ −7.14440 −0.409760
$$305$$ 6.12094 0.350484
$$306$$ −0.924984 −0.0528778
$$307$$ −4.40842 −0.251602 −0.125801 0.992055i $$-0.540150\pi$$
−0.125801 + 0.992055i $$0.540150\pi$$
$$308$$ −0.131609 −0.00749909
$$309$$ 4.11106 0.233870
$$310$$ 9.63852 0.547431
$$311$$ −2.32836 −0.132029 −0.0660146 0.997819i $$-0.521028\pi$$
−0.0660146 + 0.997819i $$0.521028\pi$$
$$312$$ 0 0
$$313$$ 12.6687 0.716078 0.358039 0.933707i $$-0.383445\pi$$
0.358039 + 0.933707i $$0.383445\pi$$
$$314$$ −8.40067 −0.474077
$$315$$ −1.10331 −0.0621646
$$316$$ −6.45222 −0.362966
$$317$$ −12.2915 −0.690360 −0.345180 0.938536i $$-0.612182\pi$$
−0.345180 + 0.938536i $$0.612182\pi$$
$$318$$ 7.06727 0.396313
$$319$$ −0.167655 −0.00938687
$$320$$ 9.80405 0.548063
$$321$$ 6.17058 0.344408
$$322$$ 0.0955622 0.00532547
$$323$$ 1.67164 0.0930125
$$324$$ −0.377203 −0.0209557
$$325$$ 0 0
$$326$$ 12.6172 0.698800
$$327$$ 8.82167 0.487840
$$328$$ 9.37428 0.517608
$$329$$ −4.70769 −0.259543
$$330$$ −0.490388 −0.0269950
$$331$$ 13.2915 0.730568 0.365284 0.930896i $$-0.380972\pi$$
0.365284 + 0.930896i $$0.380972\pi$$
$$332$$ 4.49411 0.246646
$$333$$ −1.10331 −0.0604611
$$334$$ 21.3460 1.16800
$$335$$ 8.96315 0.489709
$$336$$ −3.10331 −0.169300
$$337$$ −3.40550 −0.185509 −0.0927547 0.995689i $$-0.529567\pi$$
−0.0927547 + 0.995689i $$0.529567\pi$$
$$338$$ 0 0
$$339$$ −12.0099 −0.652287
$$340$$ −0.302187 −0.0163884
$$341$$ 2.39270 0.129572
$$342$$ −2.93273 −0.158584
$$343$$ −1.00000 −0.0539949
$$344$$ −13.9992 −0.754786
$$345$$ −0.0827661 −0.00445598
$$346$$ −17.4757 −0.939499
$$347$$ −33.4981 −1.79827 −0.899137 0.437667i $$-0.855805\pi$$
−0.899137 + 0.437667i $$0.855805\pi$$
$$348$$ 0.181252 0.00971611
$$349$$ −28.1805 −1.50846 −0.754232 0.656607i $$-0.771991\pi$$
−0.754232 + 0.656607i $$0.771991\pi$$
$$350$$ −4.81875 −0.257573
$$351$$ 0 0
$$352$$ 0.733860 0.0391148
$$353$$ −2.99225 −0.159261 −0.0796307 0.996824i $$-0.525374\pi$$
−0.0796307 + 0.996824i $$0.525374\pi$$
$$354$$ −5.00000 −0.265747
$$355$$ −10.5139 −0.558018
$$356$$ 4.51173 0.239121
$$357$$ 0.726109 0.0384298
$$358$$ 28.0643 1.48325
$$359$$ 3.80888 0.201025 0.100512 0.994936i $$-0.467952\pi$$
0.100512 + 0.994936i $$0.467952\pi$$
$$360$$ 3.34116 0.176094
$$361$$ −13.6999 −0.721049
$$362$$ −12.0176 −0.631632
$$363$$ 10.8783 0.570961
$$364$$ 0 0
$$365$$ −11.5117 −0.602552
$$366$$ 7.06727 0.369412
$$367$$ −26.4514 −1.38075 −0.690376 0.723450i $$-0.742555\pi$$
−0.690376 + 0.723450i $$0.742555\pi$$
$$368$$ −0.232798 −0.0121354
$$369$$ 3.09556 0.161149
$$370$$ 1.55070 0.0806173
$$371$$ −5.54778 −0.288026
$$372$$ −2.58675 −0.134117
$$373$$ 2.13161 0.110371 0.0551853 0.998476i $$-0.482425\pi$$
0.0551853 + 0.998476i $$0.482425\pi$$
$$374$$ 0.322733 0.0166881
$$375$$ 9.69006 0.500393
$$376$$ 14.2563 0.735211
$$377$$ 0 0
$$378$$ −1.27389 −0.0655219
$$379$$ −13.9434 −0.716225 −0.358112 0.933678i $$-0.616580\pi$$
−0.358112 + 0.933678i $$0.616580\pi$$
$$380$$ −0.958107 −0.0491498
$$381$$ 5.85772 0.300100
$$382$$ 4.30219 0.220119
$$383$$ −18.6044 −0.950639 −0.475320 0.879813i $$-0.657668\pi$$
−0.475320 + 0.879813i $$0.657668\pi$$
$$384$$ 7.11319 0.362993
$$385$$ 0.384953 0.0196190
$$386$$ −28.4367 −1.44739
$$387$$ −4.62280 −0.234990
$$388$$ −3.81100 −0.193474
$$389$$ −1.23009 −0.0623682 −0.0311841 0.999514i $$-0.509928\pi$$
−0.0311841 + 0.999514i $$0.509928\pi$$
$$390$$ 0 0
$$391$$ 0.0544699 0.00275466
$$392$$ 3.02830 0.152952
$$393$$ −22.4055 −1.13021
$$394$$ 32.4223 1.63341
$$395$$ 18.8726 0.949585
$$396$$ 0.131609 0.00661358
$$397$$ 29.6142 1.48630 0.743148 0.669127i $$-0.233332\pi$$
0.743148 + 0.669127i $$0.233332\pi$$
$$398$$ 28.4592 1.42653
$$399$$ 2.30219 0.115253
$$400$$ 11.7389 0.586945
$$401$$ 8.18045 0.408512 0.204256 0.978917i $$-0.434522\pi$$
0.204256 + 0.978917i $$0.434522\pi$$
$$402$$ 10.3489 0.516157
$$403$$ 0 0
$$404$$ −1.53711 −0.0764740
$$405$$ 1.10331 0.0548240
$$406$$ 0.612124 0.0303792
$$407$$ 0.384953 0.0190814
$$408$$ −2.19887 −0.108861
$$409$$ 10.0673 0.497794 0.248897 0.968530i $$-0.419932\pi$$
0.248897 + 0.968530i $$0.419932\pi$$
$$410$$ −4.35081 −0.214871
$$411$$ −20.2087 −0.996824
$$412$$ 1.55070 0.0763977
$$413$$ 3.92498 0.193136
$$414$$ −0.0955622 −0.00469663
$$415$$ −13.1452 −0.645273
$$416$$ 0 0
$$417$$ −4.27389 −0.209293
$$418$$ 1.02325 0.0500488
$$419$$ 9.45997 0.462150 0.231075 0.972936i $$-0.425776\pi$$
0.231075 + 0.972936i $$0.425776\pi$$
$$420$$ −0.416173 −0.0203071
$$421$$ 31.0643 1.51398 0.756992 0.653424i $$-0.226668\pi$$
0.756992 + 0.653424i $$0.226668\pi$$
$$422$$ −12.7875 −0.622487
$$423$$ 4.70769 0.228896
$$424$$ 16.8003 0.815896
$$425$$ −2.74666 −0.133232
$$426$$ −12.1394 −0.588154
$$427$$ −5.54778 −0.268476
$$428$$ 2.32756 0.112507
$$429$$ 0 0
$$430$$ 6.49734 0.313329
$$431$$ −25.2186 −1.21474 −0.607369 0.794420i $$-0.707775\pi$$
−0.607369 + 0.794420i $$0.707775\pi$$
$$432$$ 3.10331 0.149308
$$433$$ 3.76991 0.181170 0.0905851 0.995889i $$-0.471126\pi$$
0.0905851 + 0.995889i $$0.471126\pi$$
$$434$$ −8.73598 −0.419341
$$435$$ −0.530158 −0.0254191
$$436$$ 3.32756 0.159361
$$437$$ 0.172701 0.00826141
$$438$$ −13.2915 −0.635093
$$439$$ −22.6348 −1.08030 −0.540150 0.841569i $$-0.681632\pi$$
−0.540150 + 0.841569i $$0.681632\pi$$
$$440$$ −1.16575 −0.0555750
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 4.64334 0.220612 0.110306 0.993898i $$-0.464817\pi$$
0.110306 + 0.993898i $$0.464817\pi$$
$$444$$ −0.416173 −0.0197507
$$445$$ −13.1968 −0.625586
$$446$$ 6.54083 0.309717
$$447$$ 4.89669 0.231605
$$448$$ −8.88601 −0.419825
$$449$$ 9.42312 0.444705 0.222352 0.974966i $$-0.428626\pi$$
0.222352 + 0.974966i $$0.428626\pi$$
$$450$$ 4.81875 0.227158
$$451$$ −1.08006 −0.0508581
$$452$$ −4.53016 −0.213081
$$453$$ 8.94553 0.420298
$$454$$ 36.9058 1.73207
$$455$$ 0 0
$$456$$ −6.97170 −0.326480
$$457$$ 12.7162 0.594840 0.297420 0.954747i $$-0.403874\pi$$
0.297420 + 0.954747i $$0.403874\pi$$
$$458$$ −0.865468 −0.0404407
$$459$$ −0.726109 −0.0338919
$$460$$ −0.0312196 −0.00145562
$$461$$ −29.7253 −1.38445 −0.692223 0.721684i $$-0.743368\pi$$
−0.692223 + 0.721684i $$0.743368\pi$$
$$462$$ 0.444469 0.0206786
$$463$$ −15.2838 −0.710297 −0.355148 0.934810i $$-0.615570\pi$$
−0.355148 + 0.934810i $$0.615570\pi$$
$$464$$ −1.49119 −0.0692267
$$465$$ 7.56620 0.350874
$$466$$ 23.3150 1.08005
$$467$$ −26.6503 −1.23323 −0.616614 0.787265i $$-0.711496\pi$$
−0.616614 + 0.787265i $$0.711496\pi$$
$$468$$ 0 0
$$469$$ −8.12386 −0.375125
$$470$$ −6.61665 −0.305203
$$471$$ −6.59450 −0.303859
$$472$$ −11.8860 −0.547098
$$473$$ 1.61292 0.0741623
$$474$$ 21.7905 1.00087
$$475$$ −8.70849 −0.399573
$$476$$ 0.273891 0.0125538
$$477$$ 5.54778 0.254015
$$478$$ 6.94553 0.317681
$$479$$ −22.2117 −1.01488 −0.507439 0.861688i $$-0.669408\pi$$
−0.507439 + 0.861688i $$0.669408\pi$$
$$480$$ 2.32061 0.105921
$$481$$ 0 0
$$482$$ 19.6970 0.897174
$$483$$ 0.0750160 0.00341335
$$484$$ 4.10331 0.186514
$$485$$ 11.1471 0.506164
$$486$$ 1.27389 0.0577848
$$487$$ −0.971704 −0.0440321 −0.0220160 0.999758i $$-0.507008\pi$$
−0.0220160 + 0.999758i $$0.507008\pi$$
$$488$$ 16.8003 0.760515
$$489$$ 9.90444 0.447894
$$490$$ −1.40550 −0.0634940
$$491$$ 26.9349 1.21555 0.607777 0.794108i $$-0.292062\pi$$
0.607777 + 0.794108i $$0.292062\pi$$
$$492$$ 1.16765 0.0526419
$$493$$ 0.348907 0.0157140
$$494$$ 0 0
$$495$$ −0.384953 −0.0173023
$$496$$ 21.2816 0.955574
$$497$$ 9.52936 0.427450
$$498$$ −15.1775 −0.680121
$$499$$ −21.1239 −0.945634 −0.472817 0.881161i $$-0.656763\pi$$
−0.472817 + 0.881161i $$0.656763\pi$$
$$500$$ 3.65512 0.163462
$$501$$ 16.7565 0.748626
$$502$$ 10.5662 0.471593
$$503$$ 12.5654 0.560264 0.280132 0.959962i $$-0.409622\pi$$
0.280132 + 0.959962i $$0.409622\pi$$
$$504$$ −3.02830 −0.134891
$$505$$ 4.49602 0.200070
$$506$$ 0.0333423 0.00148225
$$507$$ 0 0
$$508$$ 2.20955 0.0980328
$$509$$ 27.6065 1.22364 0.611818 0.790998i $$-0.290438\pi$$
0.611818 + 0.790998i $$0.290438\pi$$
$$510$$ 1.02055 0.0451905
$$511$$ 10.4338 0.461564
$$512$$ 25.3227 1.11912
$$513$$ −2.30219 −0.101644
$$514$$ 19.1570 0.844978
$$515$$ −4.53579 −0.199871
$$516$$ −1.74373 −0.0767635
$$517$$ −1.64254 −0.0722389
$$518$$ −1.40550 −0.0617541
$$519$$ −13.7184 −0.602169
$$520$$ 0 0
$$521$$ −36.8783 −1.61567 −0.807833 0.589411i $$-0.799360\pi$$
−0.807833 + 0.589411i $$0.799360\pi$$
$$522$$ −0.612124 −0.0267919
$$523$$ 30.8054 1.34702 0.673512 0.739176i $$-0.264785\pi$$
0.673512 + 0.739176i $$0.264785\pi$$
$$524$$ −8.45142 −0.369202
$$525$$ −3.78270 −0.165091
$$526$$ 21.9680 0.957849
$$527$$ −4.97945 −0.216908
$$528$$ −1.08277 −0.0471213
$$529$$ −22.9944 −0.999755
$$530$$ −7.79740 −0.338697
$$531$$ −3.92498 −0.170330
$$532$$ 0.868391 0.0376495
$$533$$ 0 0
$$534$$ −15.2370 −0.659371
$$535$$ −6.80807 −0.294339
$$536$$ 24.6015 1.06262
$$537$$ 22.0304 0.950683
$$538$$ 30.9485 1.33428
$$539$$ −0.348907 −0.0150285
$$540$$ 0.416173 0.0179092
$$541$$ −41.4981 −1.78414 −0.892072 0.451893i $$-0.850749\pi$$
−0.892072 + 0.451893i $$0.850749\pi$$
$$542$$ −23.3537 −1.00313
$$543$$ −9.43380 −0.404843
$$544$$ −1.52723 −0.0654797
$$545$$ −9.73306 −0.416918
$$546$$ 0 0
$$547$$ −41.3716 −1.76892 −0.884460 0.466615i $$-0.845473\pi$$
−0.884460 + 0.466615i $$0.845473\pi$$
$$548$$ −7.62280 −0.325630
$$549$$ 5.54778 0.236774
$$550$$ −1.68129 −0.0716906
$$551$$ 1.10624 0.0471272
$$552$$ −0.227171 −0.00966903
$$553$$ −17.1054 −0.727397
$$554$$ −10.7699 −0.457569
$$555$$ 1.21730 0.0516714
$$556$$ −1.61212 −0.0683693
$$557$$ 30.0149 1.27177 0.635886 0.771783i $$-0.280635\pi$$
0.635886 + 0.771783i $$0.280635\pi$$
$$558$$ 8.73598 0.369824
$$559$$ 0 0
$$560$$ 3.42392 0.144687
$$561$$ 0.253344 0.0106962
$$562$$ 31.8625 1.34404
$$563$$ 25.9349 1.09302 0.546512 0.837451i $$-0.315955\pi$$
0.546512 + 0.837451i $$0.315955\pi$$
$$564$$ 1.77575 0.0747727
$$565$$ 13.2506 0.557459
$$566$$ −34.5753 −1.45331
$$567$$ −1.00000 −0.0419961
$$568$$ −28.8577 −1.21084
$$569$$ −6.56620 −0.275270 −0.137635 0.990483i $$-0.543950\pi$$
−0.137635 + 0.990483i $$0.543950\pi$$
$$570$$ 3.23572 0.135529
$$571$$ −30.0539 −1.25772 −0.628858 0.777520i $$-0.716477\pi$$
−0.628858 + 0.777520i $$0.716477\pi$$
$$572$$ 0 0
$$573$$ 3.37720 0.141085
$$574$$ 3.94341 0.164595
$$575$$ −0.283763 −0.0118337
$$576$$ 8.88601 0.370251
$$577$$ 21.1706 0.881343 0.440671 0.897669i $$-0.354740\pi$$
0.440671 + 0.897669i $$0.354740\pi$$
$$578$$ 20.9845 0.872840
$$579$$ −22.3227 −0.927701
$$580$$ −0.199977 −0.00830360
$$581$$ 11.9143 0.494289
$$582$$ 12.8705 0.533500
$$583$$ −1.93566 −0.0801667
$$584$$ −31.5966 −1.30748
$$585$$ 0 0
$$586$$ 13.4268 0.554658
$$587$$ 0.0312196 0.00128857 0.000644286 1.00000i $$-0.499795\pi$$
0.000644286 1.00000i $$0.499795\pi$$
$$588$$ 0.377203 0.0155556
$$589$$ −15.7877 −0.650523
$$590$$ 5.51656 0.227113
$$591$$ 25.4514 1.04693
$$592$$ 3.42392 0.140722
$$593$$ −10.3150 −0.423586 −0.211793 0.977315i $$-0.567930\pi$$
−0.211793 + 0.977315i $$0.567930\pi$$
$$594$$ −0.444469 −0.0182368
$$595$$ −0.801125 −0.0328429
$$596$$ 1.84704 0.0756579
$$597$$ 22.3404 0.914330
$$598$$ 0 0
$$599$$ 44.0176 1.79851 0.899256 0.437423i $$-0.144109\pi$$
0.899256 + 0.437423i $$0.144109\pi$$
$$600$$ 11.4551 0.467654
$$601$$ 5.04672 0.205860 0.102930 0.994689i $$-0.467178\pi$$
0.102930 + 0.994689i $$0.467178\pi$$
$$602$$ −5.88894 −0.240015
$$603$$ 8.12386 0.330829
$$604$$ 3.37428 0.137297
$$605$$ −12.0021 −0.487956
$$606$$ 5.19112 0.210875
$$607$$ 16.6482 0.675728 0.337864 0.941195i $$-0.390296\pi$$
0.337864 + 0.941195i $$0.390296\pi$$
$$608$$ −4.84222 −0.196378
$$609$$ 0.480515 0.0194715
$$610$$ −7.79740 −0.315708
$$611$$ 0 0
$$612$$ −0.273891 −0.0110714
$$613$$ 38.0120 1.53529 0.767645 0.640875i $$-0.221428\pi$$
0.767645 + 0.640875i $$0.221428\pi$$
$$614$$ 5.61585 0.226637
$$615$$ −3.41537 −0.137721
$$616$$ 1.05659 0.0425713
$$617$$ 15.6666 0.630713 0.315357 0.948973i $$-0.397876\pi$$
0.315357 + 0.948973i $$0.397876\pi$$
$$618$$ −5.23704 −0.210665
$$619$$ 3.26109 0.131074 0.0655372 0.997850i $$-0.479124\pi$$
0.0655372 + 0.997850i $$0.479124\pi$$
$$620$$ 2.85399 0.114619
$$621$$ −0.0750160 −0.00301029
$$622$$ 2.96608 0.118929
$$623$$ 11.9610 0.479209
$$624$$ 0 0
$$625$$ 8.22234 0.328894
$$626$$ −16.1386 −0.645027
$$627$$ 0.803248 0.0320786
$$628$$ −2.48746 −0.0992606
$$629$$ −0.801125 −0.0319430
$$630$$ 1.40550 0.0559964
$$631$$ 1.59955 0.0636770 0.0318385 0.999493i $$-0.489864\pi$$
0.0318385 + 0.999493i $$0.489864\pi$$
$$632$$ 51.8003 2.06051
$$633$$ −10.0382 −0.398981
$$634$$ 15.6580 0.621860
$$635$$ −6.46289 −0.256472
$$636$$ 2.09264 0.0829785
$$637$$ 0 0
$$638$$ 0.213574 0.00845548
$$639$$ −9.52936 −0.376976
$$640$$ −7.84806 −0.310222
$$641$$ 31.3559 1.23848 0.619241 0.785201i $$-0.287440\pi$$
0.619241 + 0.785201i $$0.287440\pi$$
$$642$$ −7.86064 −0.310235
$$643$$ −18.6631 −0.736000 −0.368000 0.929826i $$-0.619957\pi$$
−0.368000 + 0.929826i $$0.619957\pi$$
$$644$$ 0.0282963 0.00111503
$$645$$ 5.10039 0.200828
$$646$$ −2.12949 −0.0837835
$$647$$ 6.53711 0.257000 0.128500 0.991709i $$-0.458984\pi$$
0.128500 + 0.991709i $$0.458984\pi$$
$$648$$ 3.02830 0.118963
$$649$$ 1.36945 0.0537557
$$650$$ 0 0
$$651$$ −6.85772 −0.268775
$$652$$ 3.73598 0.146312
$$653$$ 15.8521 0.620340 0.310170 0.950681i $$-0.399614\pi$$
0.310170 + 0.950681i $$0.399614\pi$$
$$654$$ −11.2378 −0.439434
$$655$$ 24.7203 0.965901
$$656$$ −9.60650 −0.375071
$$657$$ −10.4338 −0.407061
$$658$$ 5.99708 0.233790
$$659$$ −49.7974 −1.93983 −0.969916 0.243441i $$-0.921724\pi$$
−0.969916 + 0.243441i $$0.921724\pi$$
$$660$$ −0.145205 −0.00565211
$$661$$ 3.57103 0.138897 0.0694485 0.997586i $$-0.477876\pi$$
0.0694485 + 0.997586i $$0.477876\pi$$
$$662$$ −16.9319 −0.658078
$$663$$ 0 0
$$664$$ −36.0801 −1.40018
$$665$$ −2.54003 −0.0984982
$$666$$ 1.40550 0.0544620
$$667$$ 0.0360463 0.00139572
$$668$$ 6.32061 0.244552
$$669$$ 5.13453 0.198512
$$670$$ −11.4181 −0.441119
$$671$$ −1.93566 −0.0747252
$$672$$ −2.10331 −0.0811370
$$673$$ 11.7339 0.452307 0.226154 0.974092i $$-0.427385\pi$$
0.226154 + 0.974092i $$0.427385\pi$$
$$674$$ 4.33823 0.167102
$$675$$ 3.78270 0.145596
$$676$$ 0 0
$$677$$ 17.3326 0.666146 0.333073 0.942901i $$-0.391914\pi$$
0.333073 + 0.942901i $$0.391914\pi$$
$$678$$ 15.2993 0.587565
$$679$$ −10.1033 −0.387730
$$680$$ 2.42605 0.0930346
$$681$$ 28.9709 1.11017
$$682$$ −3.04804 −0.116716
$$683$$ 16.9640 0.649108 0.324554 0.945867i $$-0.394786\pi$$
0.324554 + 0.945867i $$0.394786\pi$$
$$684$$ −0.868391 −0.0332038
$$685$$ 22.2966 0.851908
$$686$$ 1.27389 0.0486374
$$687$$ −0.679390 −0.0259203
$$688$$ 14.3460 0.546935
$$689$$ 0 0
$$690$$ 0.105435 0.00401384
$$691$$ 36.6425 1.39395 0.696974 0.717096i $$-0.254529\pi$$
0.696974 + 0.717096i $$0.254529\pi$$
$$692$$ −5.17460 −0.196709
$$693$$ 0.348907 0.0132539
$$694$$ 42.6730 1.61984
$$695$$ 4.71544 0.178867
$$696$$ −1.45514 −0.0551570
$$697$$ 2.24772 0.0851384
$$698$$ 35.8988 1.35879
$$699$$ 18.3022 0.692252
$$700$$ −1.42685 −0.0539297
$$701$$ −14.6092 −0.551782 −0.275891 0.961189i $$-0.588973\pi$$
−0.275891 + 0.961189i $$0.588973\pi$$
$$702$$ 0 0
$$703$$ −2.54003 −0.0957991
$$704$$ −3.10039 −0.116850
$$705$$ −5.19405 −0.195619
$$706$$ 3.81180 0.143459
$$707$$ −4.07502 −0.153257
$$708$$ −1.48052 −0.0556412
$$709$$ −23.5860 −0.885789 −0.442894 0.896574i $$-0.646048\pi$$
−0.442894 + 0.896574i $$0.646048\pi$$
$$710$$ 13.3935 0.502649
$$711$$ 17.1054 0.641504
$$712$$ −36.2215 −1.35746
$$713$$ −0.514439 −0.0192659
$$714$$ −0.924984 −0.0346167
$$715$$ 0 0
$$716$$ 8.30994 0.310557
$$717$$ 5.45222 0.203617
$$718$$ −4.85209 −0.181078
$$719$$ 34.1337 1.27297 0.636487 0.771288i $$-0.280387\pi$$
0.636487 + 0.771288i $$0.280387\pi$$
$$720$$ −3.42392 −0.127602
$$721$$ 4.11106 0.153104
$$722$$ 17.4522 0.649504
$$723$$ 15.4621 0.575041
$$724$$ −3.55845 −0.132249
$$725$$ −1.81765 −0.0675057
$$726$$ −13.8577 −0.514308
$$727$$ −15.3481 −0.569230 −0.284615 0.958642i $$-0.591866\pi$$
−0.284615 + 0.958642i $$0.591866\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 14.6647 0.542765
$$731$$ −3.35666 −0.124150
$$732$$ 2.09264 0.0773462
$$733$$ 23.2894 0.860213 0.430107 0.902778i $$-0.358476\pi$$
0.430107 + 0.902778i $$0.358476\pi$$
$$734$$ 33.6962 1.24375
$$735$$ −1.10331 −0.0406963
$$736$$ −0.157782 −0.00581593
$$737$$ −2.83447 −0.104409
$$738$$ −3.94341 −0.145159
$$739$$ −10.2661 −0.377646 −0.188823 0.982011i $$-0.560467\pi$$
−0.188823 + 0.982011i $$0.560467\pi$$
$$740$$ 0.459168 0.0168794
$$741$$ 0 0
$$742$$ 7.06727 0.259447
$$743$$ −45.3481 −1.66366 −0.831830 0.555030i $$-0.812707\pi$$
−0.831830 + 0.555030i $$0.812707\pi$$
$$744$$ 20.7672 0.761363
$$745$$ −5.40258 −0.197935
$$746$$ −2.71544 −0.0994192
$$747$$ −11.9143 −0.435922
$$748$$ 0.0955622 0.00349410
$$749$$ 6.17058 0.225468
$$750$$ −12.3441 −0.450742
$$751$$ −29.6113 −1.08053 −0.540266 0.841494i $$-0.681676\pi$$
−0.540266 + 0.841494i $$0.681676\pi$$
$$752$$ −14.6094 −0.532751
$$753$$ 8.29444 0.302266
$$754$$ 0 0
$$755$$ −9.86971 −0.359196
$$756$$ −0.377203 −0.0137187
$$757$$ −25.9816 −0.944316 −0.472158 0.881514i $$-0.656525\pi$$
−0.472158 + 0.881514i $$0.656525\pi$$
$$758$$ 17.7624 0.645159
$$759$$ 0.0261736 0.000950041 0
$$760$$ 7.69197 0.279017
$$761$$ −54.0665 −1.95991 −0.979954 0.199224i $$-0.936158\pi$$
−0.979954 + 0.199224i $$0.936158\pi$$
$$762$$ −7.46209 −0.270323
$$763$$ 8.82167 0.319366
$$764$$ 1.27389 0.0460877
$$765$$ 0.801125 0.0289648
$$766$$ 23.6999 0.856313
$$767$$ 0 0
$$768$$ 8.71061 0.314317
$$769$$ 25.7934 0.930133 0.465066 0.885276i $$-0.346030\pi$$
0.465066 + 0.885276i $$0.346030\pi$$
$$770$$ −0.490388 −0.0176724
$$771$$ 15.0382 0.541586
$$772$$ −8.42020 −0.303050
$$773$$ −0.121736 −0.00437853 −0.00218927 0.999998i $$-0.500697\pi$$
−0.00218927 + 0.999998i $$0.500697\pi$$
$$774$$ 5.88894 0.211673
$$775$$ 25.9407 0.931818
$$776$$ 30.5958 1.09833
$$777$$ −1.10331 −0.0395811
$$778$$ 1.56701 0.0561799
$$779$$ 7.12656 0.255336
$$780$$ 0 0
$$781$$ 3.32486 0.118973
$$782$$ −0.0693886 −0.00248133
$$783$$ −0.480515 −0.0171722
$$784$$ −3.10331 −0.110833
$$785$$ 7.27579 0.259684
$$786$$ 28.5422 1.01806
$$787$$ −20.2215 −0.720820 −0.360410 0.932794i $$-0.617363\pi$$
−0.360410 + 0.932794i $$0.617363\pi$$
$$788$$ 9.60035 0.341998
$$789$$ 17.2448 0.613931
$$790$$ −24.0417 −0.855364
$$791$$ −12.0099 −0.427022
$$792$$ −1.05659 −0.0375444
$$793$$ 0 0
$$794$$ −37.7253 −1.33882
$$795$$ −6.12094 −0.217087
$$796$$ 8.42685 0.298682
$$797$$ 32.4124 1.14811 0.574054 0.818818i $$-0.305370\pi$$
0.574054 + 0.818818i $$0.305370\pi$$
$$798$$ −2.93273 −0.103818
$$799$$ 3.41830 0.120931
$$800$$ 7.95620 0.281294
$$801$$ −11.9610 −0.422622
$$802$$ −10.4210 −0.367978
$$803$$ 3.64042 0.128468
$$804$$ 3.06434 0.108071
$$805$$ −0.0827661 −0.00291712
$$806$$ 0 0
$$807$$ 24.2944 0.855205
$$808$$ 12.3404 0.434132
$$809$$ 24.4629 0.860069 0.430035 0.902812i $$-0.358501\pi$$
0.430035 + 0.902812i $$0.358501\pi$$
$$810$$ −1.40550 −0.0493842
$$811$$ −45.2448 −1.58876 −0.794380 0.607421i $$-0.792204\pi$$
−0.794380 + 0.607421i $$0.792204\pi$$
$$812$$ 0.181252 0.00636069
$$813$$ −18.3326 −0.642953
$$814$$ −0.490388 −0.0171881
$$815$$ −10.9277 −0.382780
$$816$$ 2.25334 0.0788828
$$817$$ −10.6425 −0.372335
$$818$$ −12.8246 −0.448401
$$819$$ 0 0
$$820$$ −1.28829 −0.0449890
$$821$$ 36.7331 1.28199 0.640996 0.767544i $$-0.278521\pi$$
0.640996 + 0.767544i $$0.278521\pi$$
$$822$$ 25.7437 0.897916
$$823$$ 42.6476 1.48660 0.743301 0.668957i $$-0.233259\pi$$
0.743301 + 0.668957i $$0.233259\pi$$
$$824$$ −12.4495 −0.433699
$$825$$ −1.31981 −0.0459499
$$826$$ −5.00000 −0.173972
$$827$$ −50.0635 −1.74088 −0.870440 0.492275i $$-0.836166\pi$$
−0.870440 + 0.492275i $$0.836166\pi$$
$$828$$ −0.0282963 −0.000983363 0
$$829$$ 25.8911 0.899234 0.449617 0.893222i $$-0.351561\pi$$
0.449617 + 0.893222i $$0.351561\pi$$
$$830$$ 16.7456 0.581247
$$831$$ −8.45434 −0.293278
$$832$$ 0 0
$$833$$ 0.726109 0.0251582
$$834$$ 5.44447 0.188527
$$835$$ −18.4877 −0.639793
$$836$$ 0.302987 0.0104790
$$837$$ 6.85772 0.237037
$$838$$ −12.0510 −0.416294
$$839$$ −28.9349 −0.998942 −0.499471 0.866331i $$-0.666472\pi$$
−0.499471 + 0.866331i $$0.666472\pi$$
$$840$$ 3.34116 0.115281
$$841$$ −28.7691 −0.992038
$$842$$ −39.5726 −1.36376
$$843$$ 25.0120 0.861459
$$844$$ −3.78643 −0.130334
$$845$$ 0 0
$$846$$ −5.99708 −0.206184
$$847$$ 10.8783 0.373782
$$848$$ −17.2165 −0.591217
$$849$$ −27.1415 −0.931493
$$850$$ 3.49894 0.120013
$$851$$ −0.0827661 −0.00283719
$$852$$ −3.59450 −0.123146
$$853$$ 40.3094 1.38017 0.690083 0.723730i $$-0.257574\pi$$
0.690083 + 0.723730i $$0.257574\pi$$
$$854$$ 7.06727 0.241837
$$855$$ 2.54003 0.0868672
$$856$$ −18.6863 −0.638686
$$857$$ 40.0459 1.36794 0.683971 0.729509i $$-0.260251\pi$$
0.683971 + 0.729509i $$0.260251\pi$$
$$858$$ 0 0
$$859$$ 6.66659 0.227461 0.113731 0.993512i $$-0.463720\pi$$
0.113731 + 0.993512i $$0.463720\pi$$
$$860$$ 1.92388 0.0656038
$$861$$ 3.09556 0.105496
$$862$$ 32.1258 1.09421
$$863$$ −5.85289 −0.199235 −0.0996174 0.995026i $$-0.531762\pi$$
−0.0996174 + 0.995026i $$0.531762\pi$$
$$864$$ 2.10331 0.0715561
$$865$$ 15.1356 0.514627
$$866$$ −4.80245 −0.163194
$$867$$ 16.4728 0.559444
$$868$$ −2.58675 −0.0878000
$$869$$ −5.96820 −0.202457
$$870$$ 0.675364 0.0228970
$$871$$ 0 0
$$872$$ −26.7146 −0.904672
$$873$$ 10.1033 0.341945
$$874$$ −0.220002 −0.00744168
$$875$$ 9.69006 0.327584
$$876$$ −3.93566 −0.132974
$$877$$ 4.36733 0.147474 0.0737371 0.997278i $$-0.476507\pi$$
0.0737371 + 0.997278i $$0.476507\pi$$
$$878$$ 28.8342 0.973109
$$879$$ 10.5400 0.355506
$$880$$ 1.19463 0.0402709
$$881$$ 3.14733 0.106036 0.0530181 0.998594i $$-0.483116\pi$$
0.0530181 + 0.998594i $$0.483116\pi$$
$$882$$ −1.27389 −0.0428941
$$883$$ 14.9554 0.503289 0.251645 0.967820i $$-0.419029\pi$$
0.251645 + 0.967820i $$0.419029\pi$$
$$884$$ 0 0
$$885$$ 4.33048 0.145568
$$886$$ −5.91511 −0.198722
$$887$$ 22.5264 0.756364 0.378182 0.925731i $$-0.376549\pi$$
0.378182 + 0.925731i $$0.376549\pi$$
$$888$$ 3.34116 0.112122
$$889$$ 5.85772 0.196462
$$890$$ 16.8112 0.563513
$$891$$ −0.348907 −0.0116888
$$892$$ 1.93676 0.0648475
$$893$$ 10.8380 0.362679
$$894$$ −6.23784 −0.208625
$$895$$ −24.3064 −0.812474
$$896$$ 7.11319 0.237635
$$897$$ 0 0
$$898$$ −12.0040 −0.400580
$$899$$ −3.29524 −0.109902
$$900$$ 1.42685 0.0475615
$$901$$ 4.02830 0.134202
$$902$$ 1.37588 0.0458118
$$903$$ −4.62280 −0.153837
$$904$$ 36.3695 1.20963
$$905$$ 10.4084 0.345988
$$906$$ −11.3956 −0.378594
$$907$$ −18.6532 −0.619370 −0.309685 0.950839i $$-0.600224\pi$$
−0.309685 + 0.950839i $$0.600224\pi$$
$$908$$ 10.9279 0.362655
$$909$$ 4.07502 0.135160
$$910$$ 0 0
$$911$$ 18.3278 0.607226 0.303613 0.952795i $$-0.401807\pi$$
0.303613 + 0.952795i $$0.401807\pi$$
$$912$$ 7.14440 0.236575
$$913$$ 4.15698 0.137576
$$914$$ −16.1991 −0.535818
$$915$$ −6.12094 −0.202352
$$916$$ −0.256268 −0.00846732
$$917$$ −22.4055 −0.739895
$$918$$ 0.924984 0.0305290
$$919$$ 15.1054 0.498282 0.249141 0.968467i $$-0.419852\pi$$
0.249141 + 0.968467i $$0.419852\pi$$
$$920$$ 0.250640 0.00826337
$$921$$ 4.40842 0.145262
$$922$$ 37.8668 1.24708
$$923$$ 0 0
$$924$$ 0.131609 0.00432960
$$925$$ 4.17350 0.137224
$$926$$ 19.4698 0.639819
$$927$$ −4.11106 −0.135025
$$928$$ −1.01067 −0.0331770
$$929$$ −6.25817 −0.205324 −0.102662 0.994716i $$-0.532736\pi$$
−0.102662 + 0.994716i $$0.532736\pi$$
$$930$$ −9.63852 −0.316059
$$931$$ 2.30219 0.0754511
$$932$$ 6.90364 0.226136
$$933$$ 2.32836 0.0762271
$$934$$ 33.9496 1.11086
$$935$$ −0.279518 −0.00914121
$$936$$ 0 0
$$937$$ −22.6610 −0.740301 −0.370151 0.928972i $$-0.620694\pi$$
−0.370151 + 0.928972i $$0.620694\pi$$
$$938$$ 10.3489 0.337904
$$939$$ −12.6687 −0.413428
$$940$$ −1.95921 −0.0639024
$$941$$ 17.2944 0.563783 0.281891 0.959446i $$-0.409038\pi$$
0.281891 + 0.959446i $$0.409038\pi$$
$$942$$ 8.40067 0.273709
$$943$$ 0.232217 0.00756202
$$944$$ 12.1805 0.396440
$$945$$ 1.10331 0.0358908
$$946$$ −2.05469 −0.0668037
$$947$$ −27.6121 −0.897273 −0.448637 0.893714i $$-0.648090\pi$$
−0.448637 + 0.893714i $$0.648090\pi$$
$$948$$ 6.45222 0.209558
$$949$$ 0 0
$$950$$ 11.0937 0.359926
$$951$$ 12.2915 0.398580
$$952$$ −2.19887 −0.0712659
$$953$$ 4.39080 0.142232 0.0711160 0.997468i $$-0.477344\pi$$
0.0711160 + 0.997468i $$0.477344\pi$$
$$954$$ −7.06727 −0.228811
$$955$$ −3.72611 −0.120574
$$956$$ 2.05659 0.0665150
$$957$$ 0.167655 0.00541951
$$958$$ 28.2952 0.914178
$$959$$ −20.2087 −0.652574
$$960$$ −9.80405 −0.316424
$$961$$ 16.0283 0.517042
$$962$$ 0 0
$$963$$ −6.17058 −0.198844
$$964$$ 5.83235 0.187847
$$965$$ 24.6289 0.792834
$$966$$ −0.0955622 −0.00307466
$$967$$ −18.4875 −0.594517 −0.297258 0.954797i $$-0.596072\pi$$
−0.297258 + 0.954797i $$0.596072\pi$$
$$968$$ −32.9426 −1.05882
$$969$$ −1.67164 −0.0537008
$$970$$ −14.2002 −0.455941
$$971$$ −52.3374 −1.67959 −0.839794 0.542905i $$-0.817324\pi$$
−0.839794 + 0.542905i $$0.817324\pi$$
$$972$$ 0.377203 0.0120988
$$973$$ −4.27389 −0.137015
$$974$$ 1.23784 0.0396631
$$975$$ 0 0
$$976$$ −17.2165 −0.551087
$$977$$ 51.6447 1.65226 0.826130 0.563480i $$-0.190538\pi$$
0.826130 + 0.563480i $$0.190538\pi$$
$$978$$ −12.6172 −0.403453
$$979$$ 4.17328 0.133379
$$980$$ −0.416173 −0.0132941
$$981$$ −8.82167 −0.281654
$$982$$ −34.3121 −1.09494
$$983$$ 37.5419 1.19740 0.598701 0.800973i $$-0.295684\pi$$
0.598701 + 0.800973i $$0.295684\pi$$
$$984$$ −9.37428 −0.298841
$$985$$ −28.0809 −0.894731
$$986$$ −0.444469 −0.0141548
$$987$$ 4.70769 0.149847
$$988$$ 0 0
$$989$$ −0.346784 −0.0110271
$$990$$ 0.490388 0.0155856
$$991$$ −50.5724 −1.60648 −0.803242 0.595653i $$-0.796893\pi$$
−0.803242 + 0.595653i $$0.796893\pi$$
$$992$$ 14.4239 0.457960
$$993$$ −13.2915 −0.421793
$$994$$ −12.1394 −0.385037
$$995$$ −24.6484 −0.781406
$$996$$ −4.49411 −0.142401
$$997$$ −25.9221 −0.820960 −0.410480 0.911870i $$-0.634639\pi$$
−0.410480 + 0.911870i $$0.634639\pi$$
$$998$$ 26.9095 0.851805
$$999$$ 1.10331 0.0349073
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 3549.2.a.h.1.2 3
13.3 even 3 273.2.k.d.22.2 6
13.9 even 3 273.2.k.d.211.2 yes 6
13.12 even 2 3549.2.a.s.1.2 3
39.29 odd 6 819.2.o.d.568.2 6
39.35 odd 6 819.2.o.d.757.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.k.d.22.2 6 13.3 even 3
273.2.k.d.211.2 yes 6 13.9 even 3
819.2.o.d.568.2 6 39.29 odd 6
819.2.o.d.757.2 6 39.35 odd 6
3549.2.a.h.1.2 3 1.1 even 1 trivial
3549.2.a.s.1.2 3 13.12 even 2