# Properties

 Label 320.4.a.q.1.2 Level $320$ Weight $4$ Character 320.1 Self dual yes Analytic conductor $18.881$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 320.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.8806112018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 320.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.47214 q^{3} +5.00000 q^{5} +31.3050 q^{7} -7.00000 q^{9} +O(q^{10})$$ $$q+4.47214 q^{3} +5.00000 q^{5} +31.3050 q^{7} -7.00000 q^{9} -8.94427 q^{11} +62.0000 q^{13} +22.3607 q^{15} -46.0000 q^{17} +107.331 q^{19} +140.000 q^{21} -192.302 q^{23} +25.0000 q^{25} -152.053 q^{27} +90.0000 q^{29} +152.053 q^{31} -40.0000 q^{33} +156.525 q^{35} +214.000 q^{37} +277.272 q^{39} -10.0000 q^{41} -67.0820 q^{43} -35.0000 q^{45} -398.020 q^{47} +637.000 q^{49} -205.718 q^{51} +678.000 q^{53} -44.7214 q^{55} +480.000 q^{57} -411.437 q^{59} -250.000 q^{61} -219.135 q^{63} +310.000 q^{65} +49.1935 q^{67} -860.000 q^{69} +366.715 q^{71} +522.000 q^{73} +111.803 q^{75} -280.000 q^{77} -876.539 q^{79} -491.000 q^{81} +380.132 q^{83} -230.000 q^{85} +402.492 q^{87} +970.000 q^{89} +1940.91 q^{91} +680.000 q^{93} +536.656 q^{95} -934.000 q^{97} +62.6099 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5} - 14 q^{9} + O(q^{10})$$ $$2 q + 10 q^{5} - 14 q^{9} + 124 q^{13} - 92 q^{17} + 280 q^{21} + 50 q^{25} + 180 q^{29} - 80 q^{33} + 428 q^{37} - 20 q^{41} - 70 q^{45} + 1274 q^{49} + 1356 q^{53} + 960 q^{57} - 500 q^{61} + 620 q^{65} - 1720 q^{69} + 1044 q^{73} - 560 q^{77} - 982 q^{81} - 460 q^{85} + 1940 q^{89} + 1360 q^{93} - 1868 q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 4.47214 0.860663 0.430331 0.902671i $$-0.358397\pi$$
0.430331 + 0.902671i $$0.358397\pi$$
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 31.3050 1.69031 0.845154 0.534522i $$-0.179509\pi$$
0.845154 + 0.534522i $$0.179509\pi$$
$$8$$ 0 0
$$9$$ −7.00000 −0.259259
$$10$$ 0 0
$$11$$ −8.94427 −0.245164 −0.122582 0.992458i $$-0.539117\pi$$
−0.122582 + 0.992458i $$0.539117\pi$$
$$12$$ 0 0
$$13$$ 62.0000 1.32275 0.661373 0.750057i $$-0.269974\pi$$
0.661373 + 0.750057i $$0.269974\pi$$
$$14$$ 0 0
$$15$$ 22.3607 0.384900
$$16$$ 0 0
$$17$$ −46.0000 −0.656273 −0.328136 0.944630i $$-0.606421\pi$$
−0.328136 + 0.944630i $$0.606421\pi$$
$$18$$ 0 0
$$19$$ 107.331 1.29597 0.647986 0.761652i $$-0.275611\pi$$
0.647986 + 0.761652i $$0.275611\pi$$
$$20$$ 0 0
$$21$$ 140.000 1.45479
$$22$$ 0 0
$$23$$ −192.302 −1.74338 −0.871689 0.490059i $$-0.836975\pi$$
−0.871689 + 0.490059i $$0.836975\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −152.053 −1.08380
$$28$$ 0 0
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ 152.053 0.880950 0.440475 0.897765i $$-0.354810\pi$$
0.440475 + 0.897765i $$0.354810\pi$$
$$32$$ 0 0
$$33$$ −40.0000 −0.211003
$$34$$ 0 0
$$35$$ 156.525 0.755929
$$36$$ 0 0
$$37$$ 214.000 0.950848 0.475424 0.879757i $$-0.342295\pi$$
0.475424 + 0.879757i $$0.342295\pi$$
$$38$$ 0 0
$$39$$ 277.272 1.13844
$$40$$ 0 0
$$41$$ −10.0000 −0.0380912 −0.0190456 0.999819i $$-0.506063\pi$$
−0.0190456 + 0.999819i $$0.506063\pi$$
$$42$$ 0 0
$$43$$ −67.0820 −0.237905 −0.118953 0.992900i $$-0.537954\pi$$
−0.118953 + 0.992900i $$0.537954\pi$$
$$44$$ 0 0
$$45$$ −35.0000 −0.115944
$$46$$ 0 0
$$47$$ −398.020 −1.23526 −0.617630 0.786469i $$-0.711907\pi$$
−0.617630 + 0.786469i $$0.711907\pi$$
$$48$$ 0 0
$$49$$ 637.000 1.85714
$$50$$ 0 0
$$51$$ −205.718 −0.564830
$$52$$ 0 0
$$53$$ 678.000 1.75718 0.878589 0.477578i $$-0.158485\pi$$
0.878589 + 0.477578i $$0.158485\pi$$
$$54$$ 0 0
$$55$$ −44.7214 −0.109640
$$56$$ 0 0
$$57$$ 480.000 1.11540
$$58$$ 0 0
$$59$$ −411.437 −0.907872 −0.453936 0.891034i $$-0.649981\pi$$
−0.453936 + 0.891034i $$0.649981\pi$$
$$60$$ 0 0
$$61$$ −250.000 −0.524741 −0.262371 0.964967i $$-0.584504\pi$$
−0.262371 + 0.964967i $$0.584504\pi$$
$$62$$ 0 0
$$63$$ −219.135 −0.438228
$$64$$ 0 0
$$65$$ 310.000 0.591550
$$66$$ 0 0
$$67$$ 49.1935 0.0897006 0.0448503 0.998994i $$-0.485719\pi$$
0.0448503 + 0.998994i $$0.485719\pi$$
$$68$$ 0 0
$$69$$ −860.000 −1.50046
$$70$$ 0 0
$$71$$ 366.715 0.612973 0.306486 0.951875i $$-0.400847\pi$$
0.306486 + 0.951875i $$0.400847\pi$$
$$72$$ 0 0
$$73$$ 522.000 0.836924 0.418462 0.908234i $$-0.362569\pi$$
0.418462 + 0.908234i $$0.362569\pi$$
$$74$$ 0 0
$$75$$ 111.803 0.172133
$$76$$ 0 0
$$77$$ −280.000 −0.414402
$$78$$ 0 0
$$79$$ −876.539 −1.24833 −0.624166 0.781291i $$-0.714561\pi$$
−0.624166 + 0.781291i $$0.714561\pi$$
$$80$$ 0 0
$$81$$ −491.000 −0.673525
$$82$$ 0 0
$$83$$ 380.132 0.502709 0.251355 0.967895i $$-0.419124\pi$$
0.251355 + 0.967895i $$0.419124\pi$$
$$84$$ 0 0
$$85$$ −230.000 −0.293494
$$86$$ 0 0
$$87$$ 402.492 0.495997
$$88$$ 0 0
$$89$$ 970.000 1.15528 0.577639 0.816292i $$-0.303974\pi$$
0.577639 + 0.816292i $$0.303974\pi$$
$$90$$ 0 0
$$91$$ 1940.91 2.23585
$$92$$ 0 0
$$93$$ 680.000 0.758201
$$94$$ 0 0
$$95$$ 536.656 0.579577
$$96$$ 0 0
$$97$$ −934.000 −0.977663 −0.488832 0.872378i $$-0.662577\pi$$
−0.488832 + 0.872378i $$0.662577\pi$$
$$98$$ 0 0
$$99$$ 62.6099 0.0635609
$$100$$ 0 0
$$101$$ 602.000 0.593082 0.296541 0.955020i $$-0.404167\pi$$
0.296541 + 0.955020i $$0.404167\pi$$
$$102$$ 0 0
$$103$$ 1829.10 1.74978 0.874888 0.484325i $$-0.160935\pi$$
0.874888 + 0.484325i $$0.160935\pi$$
$$104$$ 0 0
$$105$$ 700.000 0.650600
$$106$$ 0 0
$$107$$ −1525.00 −1.37782 −0.688912 0.724845i $$-0.741911\pi$$
−0.688912 + 0.724845i $$0.741911\pi$$
$$108$$ 0 0
$$109$$ −2154.00 −1.89281 −0.946403 0.322989i $$-0.895312\pi$$
−0.946403 + 0.322989i $$0.895312\pi$$
$$110$$ 0 0
$$111$$ 957.037 0.818360
$$112$$ 0 0
$$113$$ −2182.00 −1.81651 −0.908254 0.418420i $$-0.862584\pi$$
−0.908254 + 0.418420i $$0.862584\pi$$
$$114$$ 0 0
$$115$$ −961.509 −0.779663
$$116$$ 0 0
$$117$$ −434.000 −0.342934
$$118$$ 0 0
$$119$$ −1440.03 −1.10930
$$120$$ 0 0
$$121$$ −1251.00 −0.939895
$$122$$ 0 0
$$123$$ −44.7214 −0.0327837
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1310.34 −0.915539 −0.457770 0.889071i $$-0.651352\pi$$
−0.457770 + 0.889071i $$0.651352\pi$$
$$128$$ 0 0
$$129$$ −300.000 −0.204756
$$130$$ 0 0
$$131$$ −205.718 −0.137204 −0.0686019 0.997644i $$-0.521854\pi$$
−0.0686019 + 0.997644i $$0.521854\pi$$
$$132$$ 0 0
$$133$$ 3360.00 2.19059
$$134$$ 0 0
$$135$$ −760.263 −0.484689
$$136$$ 0 0
$$137$$ −2094.00 −1.30586 −0.652929 0.757419i $$-0.726460\pi$$
−0.652929 + 0.757419i $$0.726460\pi$$
$$138$$ 0 0
$$139$$ −1377.42 −0.840511 −0.420256 0.907406i $$-0.638060\pi$$
−0.420256 + 0.907406i $$0.638060\pi$$
$$140$$ 0 0
$$141$$ −1780.00 −1.06314
$$142$$ 0 0
$$143$$ −554.545 −0.324289
$$144$$ 0 0
$$145$$ 450.000 0.257727
$$146$$ 0 0
$$147$$ 2848.75 1.59837
$$148$$ 0 0
$$149$$ 334.000 0.183640 0.0918200 0.995776i $$-0.470732\pi$$
0.0918200 + 0.995776i $$0.470732\pi$$
$$150$$ 0 0
$$151$$ −3139.44 −1.69195 −0.845973 0.533225i $$-0.820980\pi$$
−0.845973 + 0.533225i $$0.820980\pi$$
$$152$$ 0 0
$$153$$ 322.000 0.170145
$$154$$ 0 0
$$155$$ 760.263 0.393973
$$156$$ 0 0
$$157$$ −834.000 −0.423952 −0.211976 0.977275i $$-0.567990\pi$$
−0.211976 + 0.977275i $$0.567990\pi$$
$$158$$ 0 0
$$159$$ 3032.11 1.51234
$$160$$ 0 0
$$161$$ −6020.00 −2.94685
$$162$$ 0 0
$$163$$ −3090.25 −1.48495 −0.742475 0.669874i $$-0.766348\pi$$
−0.742475 + 0.669874i $$0.766348\pi$$
$$164$$ 0 0
$$165$$ −200.000 −0.0943635
$$166$$ 0 0
$$167$$ −4.47214 −0.00207224 −0.00103612 0.999999i $$-0.500330\pi$$
−0.00103612 + 0.999999i $$0.500330\pi$$
$$168$$ 0 0
$$169$$ 1647.00 0.749659
$$170$$ 0 0
$$171$$ −751.319 −0.335993
$$172$$ 0 0
$$173$$ 1838.00 0.807749 0.403874 0.914814i $$-0.367663\pi$$
0.403874 + 0.914814i $$0.367663\pi$$
$$174$$ 0 0
$$175$$ 782.624 0.338062
$$176$$ 0 0
$$177$$ −1840.00 −0.781372
$$178$$ 0 0
$$179$$ 1842.52 0.769365 0.384683 0.923049i $$-0.374311\pi$$
0.384683 + 0.923049i $$0.374311\pi$$
$$180$$ 0 0
$$181$$ −1862.00 −0.764648 −0.382324 0.924028i $$-0.624876\pi$$
−0.382324 + 0.924028i $$0.624876\pi$$
$$182$$ 0 0
$$183$$ −1118.03 −0.451625
$$184$$ 0 0
$$185$$ 1070.00 0.425232
$$186$$ 0 0
$$187$$ 411.437 0.160894
$$188$$ 0 0
$$189$$ −4760.00 −1.83195
$$190$$ 0 0
$$191$$ 2066.13 0.782721 0.391360 0.920237i $$-0.372005\pi$$
0.391360 + 0.920237i $$0.372005\pi$$
$$192$$ 0 0
$$193$$ 3378.00 1.25986 0.629932 0.776650i $$-0.283083\pi$$
0.629932 + 0.776650i $$0.283083\pi$$
$$194$$ 0 0
$$195$$ 1386.36 0.509125
$$196$$ 0 0
$$197$$ −66.0000 −0.0238696 −0.0119348 0.999929i $$-0.503799\pi$$
−0.0119348 + 0.999929i $$0.503799\pi$$
$$198$$ 0 0
$$199$$ 1216.42 0.433316 0.216658 0.976248i $$-0.430484\pi$$
0.216658 + 0.976248i $$0.430484\pi$$
$$200$$ 0 0
$$201$$ 220.000 0.0772020
$$202$$ 0 0
$$203$$ 2817.45 0.974118
$$204$$ 0 0
$$205$$ −50.0000 −0.0170349
$$206$$ 0 0
$$207$$ 1346.11 0.451987
$$208$$ 0 0
$$209$$ −960.000 −0.317725
$$210$$ 0 0
$$211$$ −5286.06 −1.72468 −0.862341 0.506329i $$-0.831002\pi$$
−0.862341 + 0.506329i $$0.831002\pi$$
$$212$$ 0 0
$$213$$ 1640.00 0.527563
$$214$$ 0 0
$$215$$ −335.410 −0.106394
$$216$$ 0 0
$$217$$ 4760.00 1.48908
$$218$$ 0 0
$$219$$ 2334.45 0.720310
$$220$$ 0 0
$$221$$ −2852.00 −0.868083
$$222$$ 0 0
$$223$$ −2965.03 −0.890371 −0.445186 0.895438i $$-0.646862\pi$$
−0.445186 + 0.895438i $$0.646862\pi$$
$$224$$ 0 0
$$225$$ −175.000 −0.0518519
$$226$$ 0 0
$$227$$ −4369.28 −1.27753 −0.638765 0.769402i $$-0.720554\pi$$
−0.638765 + 0.769402i $$0.720554\pi$$
$$228$$ 0 0
$$229$$ 3250.00 0.937843 0.468921 0.883240i $$-0.344643\pi$$
0.468921 + 0.883240i $$0.344643\pi$$
$$230$$ 0 0
$$231$$ −1252.20 −0.356661
$$232$$ 0 0
$$233$$ 3298.00 0.927293 0.463646 0.886020i $$-0.346541\pi$$
0.463646 + 0.886020i $$0.346541\pi$$
$$234$$ 0 0
$$235$$ −1990.10 −0.552425
$$236$$ 0 0
$$237$$ −3920.00 −1.07439
$$238$$ 0 0
$$239$$ 554.545 0.150086 0.0750429 0.997180i $$-0.476091\pi$$
0.0750429 + 0.997180i $$0.476091\pi$$
$$240$$ 0 0
$$241$$ 5150.00 1.37652 0.688259 0.725465i $$-0.258375\pi$$
0.688259 + 0.725465i $$0.258375\pi$$
$$242$$ 0 0
$$243$$ 1909.60 0.504119
$$244$$ 0 0
$$245$$ 3185.00 0.830540
$$246$$ 0 0
$$247$$ 6654.54 1.71424
$$248$$ 0 0
$$249$$ 1700.00 0.432663
$$250$$ 0 0
$$251$$ 1386.36 0.348631 0.174316 0.984690i $$-0.444229\pi$$
0.174316 + 0.984690i $$0.444229\pi$$
$$252$$ 0 0
$$253$$ 1720.00 0.427413
$$254$$ 0 0
$$255$$ −1028.59 −0.252600
$$256$$ 0 0
$$257$$ −4166.00 −1.01116 −0.505580 0.862780i $$-0.668721\pi$$
−0.505580 + 0.862780i $$0.668721\pi$$
$$258$$ 0 0
$$259$$ 6699.26 1.60723
$$260$$ 0 0
$$261$$ −630.000 −0.149410
$$262$$ 0 0
$$263$$ −961.509 −0.225434 −0.112717 0.993627i $$-0.535955\pi$$
−0.112717 + 0.993627i $$0.535955\pi$$
$$264$$ 0 0
$$265$$ 3390.00 0.785834
$$266$$ 0 0
$$267$$ 4337.97 0.994305
$$268$$ 0 0
$$269$$ 1494.00 0.338627 0.169314 0.985562i $$-0.445845\pi$$
0.169314 + 0.985562i $$0.445845\pi$$
$$270$$ 0 0
$$271$$ −5017.74 −1.12474 −0.562372 0.826884i $$-0.690111\pi$$
−0.562372 + 0.826884i $$0.690111\pi$$
$$272$$ 0 0
$$273$$ 8680.00 1.92431
$$274$$ 0 0
$$275$$ −223.607 −0.0490327
$$276$$ 0 0
$$277$$ 1006.00 0.218212 0.109106 0.994030i $$-0.465201\pi$$
0.109106 + 0.994030i $$0.465201\pi$$
$$278$$ 0 0
$$279$$ −1064.37 −0.228395
$$280$$ 0 0
$$281$$ −3210.00 −0.681468 −0.340734 0.940160i $$-0.610676\pi$$
−0.340734 + 0.940160i $$0.610676\pi$$
$$282$$ 0 0
$$283$$ 3635.85 0.763705 0.381853 0.924223i $$-0.375286\pi$$
0.381853 + 0.924223i $$0.375286\pi$$
$$284$$ 0 0
$$285$$ 2400.00 0.498820
$$286$$ 0 0
$$287$$ −313.050 −0.0643858
$$288$$ 0 0
$$289$$ −2797.00 −0.569306
$$290$$ 0 0
$$291$$ −4176.97 −0.841439
$$292$$ 0 0
$$293$$ 3622.00 0.722183 0.361091 0.932530i $$-0.382404\pi$$
0.361091 + 0.932530i $$0.382404\pi$$
$$294$$ 0 0
$$295$$ −2057.18 −0.406013
$$296$$ 0 0
$$297$$ 1360.00 0.265708
$$298$$ 0 0
$$299$$ −11922.7 −2.30605
$$300$$ 0 0
$$301$$ −2100.00 −0.402133
$$302$$ 0 0
$$303$$ 2692.23 0.510443
$$304$$ 0 0
$$305$$ −1250.00 −0.234671
$$306$$ 0 0
$$307$$ 2088.49 0.388261 0.194131 0.980976i $$-0.437811\pi$$
0.194131 + 0.980976i $$0.437811\pi$$
$$308$$ 0 0
$$309$$ 8180.00 1.50597
$$310$$ 0 0
$$311$$ 8899.55 1.62266 0.811330 0.584589i $$-0.198744\pi$$
0.811330 + 0.584589i $$0.198744\pi$$
$$312$$ 0 0
$$313$$ 8778.00 1.58518 0.792591 0.609754i $$-0.208732\pi$$
0.792591 + 0.609754i $$0.208732\pi$$
$$314$$ 0 0
$$315$$ −1095.67 −0.195982
$$316$$ 0 0
$$317$$ 5046.00 0.894043 0.447021 0.894523i $$-0.352485\pi$$
0.447021 + 0.894523i $$0.352485\pi$$
$$318$$ 0 0
$$319$$ −804.984 −0.141287
$$320$$ 0 0
$$321$$ −6820.00 −1.18584
$$322$$ 0 0
$$323$$ −4937.24 −0.850512
$$324$$ 0 0
$$325$$ 1550.00 0.264549
$$326$$ 0 0
$$327$$ −9632.98 −1.62907
$$328$$ 0 0
$$329$$ −12460.0 −2.08797
$$330$$ 0 0
$$331$$ −313.050 −0.0519842 −0.0259921 0.999662i $$-0.508274\pi$$
−0.0259921 + 0.999662i $$0.508274\pi$$
$$332$$ 0 0
$$333$$ −1498.00 −0.246516
$$334$$ 0 0
$$335$$ 245.967 0.0401153
$$336$$ 0 0
$$337$$ −2574.00 −0.416067 −0.208034 0.978122i $$-0.566706\pi$$
−0.208034 + 0.978122i $$0.566706\pi$$
$$338$$ 0 0
$$339$$ −9758.20 −1.56340
$$340$$ 0 0
$$341$$ −1360.00 −0.215977
$$342$$ 0 0
$$343$$ 9203.66 1.44884
$$344$$ 0 0
$$345$$ −4300.00 −0.671027
$$346$$ 0 0
$$347$$ 2643.03 0.408892 0.204446 0.978878i $$-0.434461\pi$$
0.204446 + 0.978878i $$0.434461\pi$$
$$348$$ 0 0
$$349$$ 10170.0 1.55985 0.779925 0.625873i $$-0.215257\pi$$
0.779925 + 0.625873i $$0.215257\pi$$
$$350$$ 0 0
$$351$$ −9427.26 −1.43359
$$352$$ 0 0
$$353$$ −318.000 −0.0479474 −0.0239737 0.999713i $$-0.507632\pi$$
−0.0239737 + 0.999713i $$0.507632\pi$$
$$354$$ 0 0
$$355$$ 1833.58 0.274130
$$356$$ 0 0
$$357$$ −6440.00 −0.954737
$$358$$ 0 0
$$359$$ −12378.9 −1.81987 −0.909933 0.414755i $$-0.863867\pi$$
−0.909933 + 0.414755i $$0.863867\pi$$
$$360$$ 0 0
$$361$$ 4661.00 0.679545
$$362$$ 0 0
$$363$$ −5594.64 −0.808933
$$364$$ 0 0
$$365$$ 2610.00 0.374284
$$366$$ 0 0
$$367$$ 3072.36 0.436991 0.218496 0.975838i $$-0.429885\pi$$
0.218496 + 0.975838i $$0.429885\pi$$
$$368$$ 0 0
$$369$$ 70.0000 0.00987549
$$370$$ 0 0
$$371$$ 21224.8 2.97017
$$372$$ 0 0
$$373$$ 3278.00 0.455036 0.227518 0.973774i $$-0.426939\pi$$
0.227518 + 0.973774i $$0.426939\pi$$
$$374$$ 0 0
$$375$$ 559.017 0.0769800
$$376$$ 0 0
$$377$$ 5580.00 0.762293
$$378$$ 0 0
$$379$$ 5116.12 0.693397 0.346699 0.937977i $$-0.387303\pi$$
0.346699 + 0.937977i $$0.387303\pi$$
$$380$$ 0 0
$$381$$ −5860.00 −0.787971
$$382$$ 0 0
$$383$$ 1149.34 0.153338 0.0766690 0.997057i $$-0.475572\pi$$
0.0766690 + 0.997057i $$0.475572\pi$$
$$384$$ 0 0
$$385$$ −1400.00 −0.185326
$$386$$ 0 0
$$387$$ 469.574 0.0616791
$$388$$ 0 0
$$389$$ −834.000 −0.108703 −0.0543515 0.998522i $$-0.517309\pi$$
−0.0543515 + 0.998522i $$0.517309\pi$$
$$390$$ 0 0
$$391$$ 8845.88 1.14413
$$392$$ 0 0
$$393$$ −920.000 −0.118086
$$394$$ 0 0
$$395$$ −4382.69 −0.558271
$$396$$ 0 0
$$397$$ 8734.00 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 0 0
$$399$$ 15026.4 1.88536
$$400$$ 0 0
$$401$$ 242.000 0.0301369 0.0150685 0.999886i $$-0.495203\pi$$
0.0150685 + 0.999886i $$0.495203\pi$$
$$402$$ 0 0
$$403$$ 9427.26 1.16527
$$404$$ 0 0
$$405$$ −2455.00 −0.301210
$$406$$ 0 0
$$407$$ −1914.07 −0.233113
$$408$$ 0 0
$$409$$ −6514.00 −0.787522 −0.393761 0.919213i $$-0.628826\pi$$
−0.393761 + 0.919213i $$0.628826\pi$$
$$410$$ 0 0
$$411$$ −9364.65 −1.12390
$$412$$ 0 0
$$413$$ −12880.0 −1.53458
$$414$$ 0 0
$$415$$ 1900.66 0.224818
$$416$$ 0 0
$$417$$ −6160.00 −0.723397
$$418$$ 0 0
$$419$$ 16081.8 1.87505 0.937527 0.347913i $$-0.113110\pi$$
0.937527 + 0.347913i $$0.113110\pi$$
$$420$$ 0 0
$$421$$ −7250.00 −0.839295 −0.419648 0.907687i $$-0.637846\pi$$
−0.419648 + 0.907687i $$0.637846\pi$$
$$422$$ 0 0
$$423$$ 2786.14 0.320252
$$424$$ 0 0
$$425$$ −1150.00 −0.131255
$$426$$ 0 0
$$427$$ −7826.24 −0.886975
$$428$$ 0 0
$$429$$ −2480.00 −0.279104
$$430$$ 0 0
$$431$$ −4981.96 −0.556781 −0.278390 0.960468i $$-0.589801\pi$$
−0.278390 + 0.960468i $$0.589801\pi$$
$$432$$ 0 0
$$433$$ 11482.0 1.27434 0.637171 0.770723i $$-0.280105\pi$$
0.637171 + 0.770723i $$0.280105\pi$$
$$434$$ 0 0
$$435$$ 2012.46 0.221816
$$436$$ 0 0
$$437$$ −20640.0 −2.25937
$$438$$ 0 0
$$439$$ −3792.37 −0.412301 −0.206150 0.978520i $$-0.566094\pi$$
−0.206150 + 0.978520i $$0.566094\pi$$
$$440$$ 0 0
$$441$$ −4459.00 −0.481481
$$442$$ 0 0
$$443$$ −746.847 −0.0800988 −0.0400494 0.999198i $$-0.512752\pi$$
−0.0400494 + 0.999198i $$0.512752\pi$$
$$444$$ 0 0
$$445$$ 4850.00 0.516656
$$446$$ 0 0
$$447$$ 1493.69 0.158052
$$448$$ 0 0
$$449$$ −1306.00 −0.137269 −0.0686347 0.997642i $$-0.521864\pi$$
−0.0686347 + 0.997642i $$0.521864\pi$$
$$450$$ 0 0
$$451$$ 89.4427 0.00933857
$$452$$ 0 0
$$453$$ −14040.0 −1.45620
$$454$$ 0 0
$$455$$ 9704.54 0.999902
$$456$$ 0 0
$$457$$ −9526.00 −0.975071 −0.487536 0.873103i $$-0.662104\pi$$
−0.487536 + 0.873103i $$0.662104\pi$$
$$458$$ 0 0
$$459$$ 6994.42 0.711267
$$460$$ 0 0
$$461$$ −1518.00 −0.153363 −0.0766815 0.997056i $$-0.524432\pi$$
−0.0766815 + 0.997056i $$0.524432\pi$$
$$462$$ 0 0
$$463$$ −17293.7 −1.73587 −0.867936 0.496676i $$-0.834554\pi$$
−0.867936 + 0.496676i $$0.834554\pi$$
$$464$$ 0 0
$$465$$ 3400.00 0.339078
$$466$$ 0 0
$$467$$ −16980.7 −1.68260 −0.841299 0.540570i $$-0.818208\pi$$
−0.841299 + 0.540570i $$0.818208\pi$$
$$468$$ 0 0
$$469$$ 1540.00 0.151622
$$470$$ 0 0
$$471$$ −3729.76 −0.364880
$$472$$ 0 0
$$473$$ 600.000 0.0583256
$$474$$ 0 0
$$475$$ 2683.28 0.259195
$$476$$ 0 0
$$477$$ −4746.00 −0.455565
$$478$$ 0 0
$$479$$ −3810.26 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$480$$ 0 0
$$481$$ 13268.0 1.25773
$$482$$ 0 0
$$483$$ −26922.3 −2.53624
$$484$$ 0 0
$$485$$ −4670.00 −0.437224
$$486$$ 0 0
$$487$$ −1310.34 −0.121924 −0.0609620 0.998140i $$-0.519417\pi$$
−0.0609620 + 0.998140i $$0.519417\pi$$
$$488$$ 0 0
$$489$$ −13820.0 −1.27804
$$490$$ 0 0
$$491$$ 2960.55 0.272114 0.136057 0.990701i $$-0.456557\pi$$
0.136057 + 0.990701i $$0.456557\pi$$
$$492$$ 0 0
$$493$$ −4140.00 −0.378207
$$494$$ 0 0
$$495$$ 313.050 0.0284253
$$496$$ 0 0
$$497$$ 11480.0 1.03611
$$498$$ 0 0
$$499$$ −19319.6 −1.73320 −0.866598 0.499006i $$-0.833699\pi$$
−0.866598 + 0.499006i $$0.833699\pi$$
$$500$$ 0 0
$$501$$ −20.0000 −0.00178350
$$502$$ 0 0
$$503$$ −3072.36 −0.272345 −0.136173 0.990685i $$-0.543480\pi$$
−0.136173 + 0.990685i $$0.543480\pi$$
$$504$$ 0 0
$$505$$ 3010.00 0.265234
$$506$$ 0 0
$$507$$ 7365.61 0.645203
$$508$$ 0 0
$$509$$ −18550.0 −1.61535 −0.807676 0.589626i $$-0.799275\pi$$
−0.807676 + 0.589626i $$0.799275\pi$$
$$510$$ 0 0
$$511$$ 16341.2 1.41466
$$512$$ 0 0
$$513$$ −16320.0 −1.40457
$$514$$ 0 0
$$515$$ 9145.52 0.782524
$$516$$ 0 0
$$517$$ 3560.00 0.302841
$$518$$ 0 0
$$519$$ 8219.79 0.695200
$$520$$ 0 0
$$521$$ −2102.00 −0.176757 −0.0883784 0.996087i $$-0.528168\pi$$
−0.0883784 + 0.996087i $$0.528168\pi$$
$$522$$ 0 0
$$523$$ 17696.2 1.47955 0.739773 0.672856i $$-0.234933\pi$$
0.739773 + 0.672856i $$0.234933\pi$$
$$524$$ 0 0
$$525$$ 3500.00 0.290957
$$526$$ 0 0
$$527$$ −6994.42 −0.578144
$$528$$ 0 0
$$529$$ 24813.0 2.03937
$$530$$ 0 0
$$531$$ 2880.06 0.235374
$$532$$ 0 0
$$533$$ −620.000 −0.0503850
$$534$$ 0 0
$$535$$ −7624.99 −0.616182
$$536$$ 0 0
$$537$$ 8240.00 0.662164
$$538$$ 0 0
$$539$$ −5697.50 −0.455304
$$540$$ 0 0
$$541$$ 9922.00 0.788503 0.394251 0.919003i $$-0.371004\pi$$
0.394251 + 0.919003i $$0.371004\pi$$
$$542$$ 0 0
$$543$$ −8327.12 −0.658105
$$544$$ 0 0
$$545$$ −10770.0 −0.846488
$$546$$ 0 0
$$547$$ 3716.34 0.290493 0.145246 0.989396i $$-0.453603\pi$$
0.145246 + 0.989396i $$0.453603\pi$$
$$548$$ 0 0
$$549$$ 1750.00 0.136044
$$550$$ 0 0
$$551$$ 9659.81 0.746864
$$552$$ 0 0
$$553$$ −27440.0 −2.11007
$$554$$ 0 0
$$555$$ 4785.19 0.365982
$$556$$ 0 0
$$557$$ 15094.0 1.14821 0.574105 0.818781i $$-0.305350\pi$$
0.574105 + 0.818781i $$0.305350\pi$$
$$558$$ 0 0
$$559$$ −4159.09 −0.314688
$$560$$ 0 0
$$561$$ 1840.00 0.138476
$$562$$ 0 0
$$563$$ 5657.25 0.423490 0.211745 0.977325i $$-0.432085\pi$$
0.211745 + 0.977325i $$0.432085\pi$$
$$564$$ 0 0
$$565$$ −10910.0 −0.812367
$$566$$ 0 0
$$567$$ −15370.7 −1.13847
$$568$$ 0 0
$$569$$ −5906.00 −0.435136 −0.217568 0.976045i $$-0.569812\pi$$
−0.217568 + 0.976045i $$0.569812\pi$$
$$570$$ 0 0
$$571$$ −4892.52 −0.358573 −0.179287 0.983797i $$-0.557379\pi$$
−0.179287 + 0.983797i $$0.557379\pi$$
$$572$$ 0 0
$$573$$ 9240.00 0.673659
$$574$$ 0 0
$$575$$ −4807.55 −0.348676
$$576$$ 0 0
$$577$$ −13286.0 −0.958585 −0.479292 0.877655i $$-0.659107\pi$$
−0.479292 + 0.877655i $$0.659107\pi$$
$$578$$ 0 0
$$579$$ 15106.9 1.08432
$$580$$ 0 0
$$581$$ 11900.0 0.849734
$$582$$ 0 0
$$583$$ −6064.22 −0.430796
$$584$$ 0 0
$$585$$ −2170.00 −0.153365
$$586$$ 0 0
$$587$$ 9029.24 0.634884 0.317442 0.948278i $$-0.397176\pi$$
0.317442 + 0.948278i $$0.397176\pi$$
$$588$$ 0 0
$$589$$ 16320.0 1.14169
$$590$$ 0 0
$$591$$ −295.161 −0.0205437
$$592$$ 0 0
$$593$$ 11442.0 0.792355 0.396178 0.918174i $$-0.370336\pi$$
0.396178 + 0.918174i $$0.370336\pi$$
$$594$$ 0 0
$$595$$ −7200.14 −0.496096
$$596$$ 0 0
$$597$$ 5440.00 0.372939
$$598$$ 0 0
$$599$$ 14149.8 0.965187 0.482593 0.875845i $$-0.339695\pi$$
0.482593 + 0.875845i $$0.339695\pi$$
$$600$$ 0 0
$$601$$ 3110.00 0.211081 0.105540 0.994415i $$-0.466343\pi$$
0.105540 + 0.994415i $$0.466343\pi$$
$$602$$ 0 0
$$603$$ −344.354 −0.0232557
$$604$$ 0 0
$$605$$ −6255.00 −0.420334
$$606$$ 0 0
$$607$$ 11193.8 0.748502 0.374251 0.927327i $$-0.377900\pi$$
0.374251 + 0.927327i $$0.377900\pi$$
$$608$$ 0 0
$$609$$ 12600.0 0.838387
$$610$$ 0 0
$$611$$ −24677.2 −1.63394
$$612$$ 0 0
$$613$$ 5342.00 0.351976 0.175988 0.984392i $$-0.443688\pi$$
0.175988 + 0.984392i $$0.443688\pi$$
$$614$$ 0 0
$$615$$ −223.607 −0.0146613
$$616$$ 0 0
$$617$$ 19714.0 1.28631 0.643157 0.765734i $$-0.277624\pi$$
0.643157 + 0.765734i $$0.277624\pi$$
$$618$$ 0 0
$$619$$ −13166.0 −0.854903 −0.427451 0.904038i $$-0.640589\pi$$
−0.427451 + 0.904038i $$0.640589\pi$$
$$620$$ 0 0
$$621$$ 29240.0 1.88947
$$622$$ 0 0
$$623$$ 30365.8 1.95278
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −4293.25 −0.273454
$$628$$ 0 0
$$629$$ −9844.00 −0.624016
$$630$$ 0 0
$$631$$ −12262.6 −0.773639 −0.386820 0.922155i $$-0.626426\pi$$
−0.386820 + 0.922155i $$0.626426\pi$$
$$632$$ 0 0
$$633$$ −23640.0 −1.48437
$$634$$ 0 0
$$635$$ −6551.68 −0.409442
$$636$$ 0 0
$$637$$ 39494.0 2.45653
$$638$$ 0 0
$$639$$ −2567.01 −0.158919
$$640$$ 0 0
$$641$$ −2690.00 −0.165754 −0.0828772 0.996560i $$-0.526411\pi$$
−0.0828772 + 0.996560i $$0.526411\pi$$
$$642$$ 0 0
$$643$$ 12240.2 0.750712 0.375356 0.926881i $$-0.377520\pi$$
0.375356 + 0.926881i $$0.377520\pi$$
$$644$$ 0 0
$$645$$ −1500.00 −0.0915697
$$646$$ 0 0
$$647$$ 17973.5 1.09214 0.546068 0.837741i $$-0.316124\pi$$
0.546068 + 0.837741i $$0.316124\pi$$
$$648$$ 0 0
$$649$$ 3680.00 0.222577
$$650$$ 0 0
$$651$$ 21287.4 1.28159
$$652$$ 0 0
$$653$$ 3478.00 0.208430 0.104215 0.994555i $$-0.466767\pi$$
0.104215 + 0.994555i $$0.466767\pi$$
$$654$$ 0 0
$$655$$ −1028.59 −0.0613594
$$656$$ 0 0
$$657$$ −3654.00 −0.216980
$$658$$ 0 0
$$659$$ −10572.1 −0.624934 −0.312467 0.949929i $$-0.601155\pi$$
−0.312467 + 0.949929i $$0.601155\pi$$
$$660$$ 0 0
$$661$$ 110.000 0.00647277 0.00323639 0.999995i $$-0.498970\pi$$
0.00323639 + 0.999995i $$0.498970\pi$$
$$662$$ 0 0
$$663$$ −12754.5 −0.747127
$$664$$ 0 0
$$665$$ 16800.0 0.979663
$$666$$ 0 0
$$667$$ −17307.2 −1.00470
$$668$$ 0 0
$$669$$ −13260.0 −0.766310
$$670$$ 0 0
$$671$$ 2236.07 0.128647
$$672$$ 0 0
$$673$$ −14278.0 −0.817796 −0.408898 0.912580i $$-0.634087\pi$$
−0.408898 + 0.912580i $$0.634087\pi$$
$$674$$ 0 0
$$675$$ −3801.32 −0.216760
$$676$$ 0 0
$$677$$ −18386.0 −1.04377 −0.521884 0.853016i $$-0.674771\pi$$
−0.521884 + 0.853016i $$0.674771\pi$$
$$678$$ 0 0
$$679$$ −29238.8 −1.65255
$$680$$ 0 0
$$681$$ −19540.0 −1.09952
$$682$$ 0 0
$$683$$ 15317.1 0.858113 0.429057 0.903278i $$-0.358846\pi$$
0.429057 + 0.903278i $$0.358846\pi$$
$$684$$ 0 0
$$685$$ −10470.0 −0.583997
$$686$$ 0 0
$$687$$ 14534.4 0.807167
$$688$$ 0 0
$$689$$ 42036.0 2.32430
$$690$$ 0 0
$$691$$ 9507.76 0.523433 0.261717 0.965145i $$-0.415711\pi$$
0.261717 + 0.965145i $$0.415711\pi$$
$$692$$ 0 0
$$693$$ 1960.00 0.107438
$$694$$ 0 0
$$695$$ −6887.09 −0.375888
$$696$$ 0 0
$$697$$ 460.000 0.0249982
$$698$$ 0 0
$$699$$ 14749.1 0.798086
$$700$$ 0 0
$$701$$ 15830.0 0.852911 0.426456 0.904508i $$-0.359762\pi$$
0.426456 + 0.904508i $$0.359762\pi$$
$$702$$ 0 0
$$703$$ 22968.9 1.23227
$$704$$ 0 0
$$705$$ −8900.00 −0.475452
$$706$$ 0 0
$$707$$ 18845.6 1.00249
$$708$$ 0 0
$$709$$ 20050.0 1.06205 0.531025 0.847356i $$-0.321807\pi$$
0.531025 + 0.847356i $$0.321807\pi$$
$$710$$ 0 0
$$711$$ 6135.77 0.323642
$$712$$ 0 0
$$713$$ −29240.0 −1.53583
$$714$$ 0 0
$$715$$ −2772.72 −0.145027
$$716$$ 0 0
$$717$$ 2480.00 0.129173
$$718$$ 0 0
$$719$$ −21126.4 −1.09580 −0.547900 0.836544i $$-0.684573\pi$$
−0.547900 + 0.836544i $$0.684573\pi$$
$$720$$ 0 0
$$721$$ 57260.0 2.95766
$$722$$ 0 0
$$723$$ 23031.5 1.18472
$$724$$ 0 0
$$725$$ 2250.00 0.115259
$$726$$ 0 0
$$727$$ 11336.9 0.578351 0.289175 0.957276i $$-0.406619\pi$$
0.289175 + 0.957276i $$0.406619\pi$$
$$728$$ 0 0
$$729$$ 21797.0 1.10740
$$730$$ 0 0
$$731$$ 3085.77 0.156131
$$732$$ 0 0
$$733$$ 17198.0 0.866607 0.433303 0.901248i $$-0.357348\pi$$
0.433303 + 0.901248i $$0.357348\pi$$
$$734$$ 0 0
$$735$$ 14243.8 0.714815
$$736$$ 0 0
$$737$$ −440.000 −0.0219913
$$738$$ 0 0
$$739$$ 4597.36 0.228845 0.114423 0.993432i $$-0.463498\pi$$
0.114423 + 0.993432i $$0.463498\pi$$
$$740$$ 0 0
$$741$$ 29760.0 1.47539
$$742$$ 0 0
$$743$$ 2419.43 0.119462 0.0597309 0.998215i $$-0.480976\pi$$
0.0597309 + 0.998215i $$0.480976\pi$$
$$744$$ 0 0
$$745$$ 1670.00 0.0821263
$$746$$ 0 0
$$747$$ −2660.92 −0.130332
$$748$$ 0 0
$$749$$ −47740.0 −2.32895
$$750$$ 0 0
$$751$$ −7432.69 −0.361149 −0.180574 0.983561i $$-0.557796\pi$$
−0.180574 + 0.983561i $$0.557796\pi$$
$$752$$ 0 0
$$753$$ 6200.00 0.300054
$$754$$ 0 0
$$755$$ −15697.2 −0.756662
$$756$$ 0 0
$$757$$ −11474.0 −0.550898 −0.275449 0.961316i $$-0.588826\pi$$
−0.275449 + 0.961316i $$0.588826\pi$$
$$758$$ 0 0
$$759$$ 7692.07 0.367858
$$760$$ 0 0
$$761$$ 31802.0 1.51488 0.757439 0.652906i $$-0.226450\pi$$
0.757439 + 0.652906i $$0.226450\pi$$
$$762$$ 0 0
$$763$$ −67430.9 −3.19942
$$764$$ 0 0
$$765$$ 1610.00 0.0760911
$$766$$ 0 0
$$767$$ −25509.1 −1.20089
$$768$$ 0 0
$$769$$ −5310.00 −0.249003 −0.124502 0.992219i $$-0.539733\pi$$
−0.124502 + 0.992219i $$0.539733\pi$$
$$770$$ 0 0
$$771$$ −18630.9 −0.870267
$$772$$ 0 0
$$773$$ −37938.0 −1.76525 −0.882623 0.470082i $$-0.844224\pi$$
−0.882623 + 0.470082i $$0.844224\pi$$
$$774$$ 0 0
$$775$$ 3801.32 0.176190
$$776$$ 0 0
$$777$$ 29960.0 1.38328
$$778$$ 0 0
$$779$$ −1073.31 −0.0493651
$$780$$ 0 0
$$781$$ −3280.00 −0.150279
$$782$$ 0 0
$$783$$ −13684.7 −0.624588
$$784$$ 0 0
$$785$$ −4170.00 −0.189597
$$786$$ 0 0
$$787$$ 37633.0 1.70454 0.852270 0.523103i $$-0.175226\pi$$
0.852270 + 0.523103i $$0.175226\pi$$
$$788$$ 0 0
$$789$$ −4300.00 −0.194023
$$790$$ 0 0
$$791$$ −68307.4 −3.07046
$$792$$ 0 0
$$793$$ −15500.0 −0.694100
$$794$$ 0 0
$$795$$ 15160.5 0.676338
$$796$$ 0 0
$$797$$ 17526.0 0.778924 0.389462 0.921042i $$-0.372661\pi$$
0.389462 + 0.921042i $$0.372661\pi$$
$$798$$ 0 0
$$799$$ 18308.9 0.810667
$$800$$ 0 0
$$801$$ −6790.00 −0.299517
$$802$$ 0 0
$$803$$ −4668.91 −0.205183
$$804$$ 0 0
$$805$$ −30100.0 −1.31787
$$806$$ 0 0
$$807$$ 6681.37 0.291444
$$808$$ 0 0
$$809$$ 8970.00 0.389825 0.194912 0.980821i $$-0.437558\pi$$
0.194912 + 0.980821i $$0.437558\pi$$
$$810$$ 0 0
$$811$$ −3550.88 −0.153746 −0.0768731 0.997041i $$-0.524494\pi$$
−0.0768731 + 0.997041i $$0.524494\pi$$
$$812$$ 0 0
$$813$$ −22440.0 −0.968026
$$814$$ 0 0
$$815$$ −15451.2 −0.664090
$$816$$ 0 0
$$817$$ −7200.00 −0.308318
$$818$$ 0 0
$$819$$ −13586.3 −0.579665
$$820$$ 0 0
$$821$$ 15550.0 0.661022 0.330511 0.943802i $$-0.392779\pi$$
0.330511 + 0.943802i $$0.392779\pi$$
$$822$$ 0 0
$$823$$ 26712.1 1.13138 0.565689 0.824619i $$-0.308610\pi$$
0.565689 + 0.824619i $$0.308610\pi$$
$$824$$ 0 0
$$825$$ −1000.00 −0.0422006
$$826$$ 0 0
$$827$$ −863.122 −0.0362923 −0.0181461 0.999835i $$-0.505776\pi$$
−0.0181461 + 0.999835i $$0.505776\pi$$
$$828$$ 0 0
$$829$$ −19066.0 −0.798781 −0.399391 0.916781i $$-0.630778\pi$$
−0.399391 + 0.916781i $$0.630778\pi$$
$$830$$ 0 0
$$831$$ 4498.97 0.187807
$$832$$ 0 0
$$833$$ −29302.0 −1.21879
$$834$$ 0 0
$$835$$ −22.3607 −0.000926734 0
$$836$$ 0 0
$$837$$ −23120.0 −0.954772
$$838$$ 0 0
$$839$$ 47744.5 1.96463 0.982315 0.187238i $$-0.0599534\pi$$
0.982315 + 0.187238i $$0.0599534\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ 0 0
$$843$$ −14355.6 −0.586514
$$844$$ 0 0
$$845$$ 8235.00 0.335258
$$846$$ 0 0
$$847$$ −39162.5 −1.58871
$$848$$ 0 0
$$849$$ 16260.0 0.657293
$$850$$ 0 0
$$851$$ −41152.6 −1.65769
$$852$$ 0 0
$$853$$ 14462.0 0.580503 0.290252 0.956950i $$-0.406261\pi$$
0.290252 + 0.956950i $$0.406261\pi$$
$$854$$ 0 0
$$855$$ −3756.59 −0.150261
$$856$$ 0 0
$$857$$ 29346.0 1.16971 0.584854 0.811138i $$-0.301152\pi$$
0.584854 + 0.811138i $$0.301152\pi$$
$$858$$ 0 0
$$859$$ 22807.9 0.905932 0.452966 0.891528i $$-0.350366\pi$$
0.452966 + 0.891528i $$0.350366\pi$$
$$860$$ 0 0
$$861$$ −1400.00 −0.0554145
$$862$$ 0 0
$$863$$ −24753.3 −0.976375 −0.488187 0.872739i $$-0.662342\pi$$
−0.488187 + 0.872739i $$0.662342\pi$$
$$864$$ 0 0
$$865$$ 9190.00 0.361236
$$866$$ 0 0
$$867$$ −12508.6 −0.489981
$$868$$ 0 0
$$869$$ 7840.00 0.306046
$$870$$ 0 0
$$871$$ 3050.00 0.118651
$$872$$ 0 0
$$873$$ 6538.00 0.253468
$$874$$ 0 0
$$875$$ 3913.12 0.151186
$$876$$ 0 0
$$877$$ 32126.0 1.23696 0.618482 0.785799i $$-0.287748\pi$$
0.618482 + 0.785799i $$0.287748\pi$$
$$878$$ 0 0
$$879$$ 16198.1 0.621556
$$880$$ 0 0
$$881$$ −33570.0 −1.28377 −0.641885 0.766801i $$-0.721848\pi$$
−0.641885 + 0.766801i $$0.721848\pi$$
$$882$$ 0 0
$$883$$ −6435.40 −0.245265 −0.122632 0.992452i $$-0.539134\pi$$
−0.122632 + 0.992452i $$0.539134\pi$$
$$884$$ 0 0
$$885$$ −9200.00 −0.349440
$$886$$ 0 0
$$887$$ 46827.7 1.77263 0.886314 0.463084i $$-0.153257\pi$$
0.886314 + 0.463084i $$0.153257\pi$$
$$888$$ 0 0
$$889$$ −41020.0 −1.54754
$$890$$ 0 0
$$891$$ 4391.64 0.165124
$$892$$ 0 0
$$893$$ −42720.0 −1.60086
$$894$$ 0 0
$$895$$ 9212.60 0.344071
$$896$$ 0 0
$$897$$ −53320.0 −1.98473
$$898$$ 0 0
$$899$$ 13684.7 0.507688
$$900$$ 0 0
$$901$$ −31188.0 −1.15319
$$902$$ 0 0
$$903$$ −9391.49 −0.346101
$$904$$ 0 0
$$905$$ −9310.00 −0.341961
$$906$$ 0 0
$$907$$ 11980.9 0.438608 0.219304 0.975657i $$-0.429621\pi$$
0.219304 + 0.975657i $$0.429621\pi$$
$$908$$ 0 0
$$909$$ −4214.00 −0.153762
$$910$$ 0 0
$$911$$ 24194.3 0.879903 0.439951 0.898022i $$-0.354996\pi$$
0.439951 + 0.898022i $$0.354996\pi$$
$$912$$ 0 0
$$913$$ −3400.00 −0.123246
$$914$$ 0 0
$$915$$ −5590.17 −0.201973
$$916$$ 0 0
$$917$$ −6440.00 −0.231917
$$918$$ 0 0
$$919$$ 37512.3 1.34648 0.673240 0.739424i $$-0.264902\pi$$
0.673240 + 0.739424i $$0.264902\pi$$
$$920$$ 0 0
$$921$$ 9340.00 0.334162
$$922$$ 0 0
$$923$$ 22736.3 0.810808
$$924$$ 0 0
$$925$$ 5350.00 0.190170
$$926$$ 0 0
$$927$$ −12803.7 −0.453646
$$928$$ 0 0
$$929$$ −21994.0 −0.776749 −0.388374 0.921502i $$-0.626963\pi$$
−0.388374 + 0.921502i $$0.626963\pi$$
$$930$$ 0 0
$$931$$ 68370.0 2.40681
$$932$$ 0 0
$$933$$ 39800.0 1.39656
$$934$$ 0 0
$$935$$ 2057.18 0.0719541
$$936$$ 0 0
$$937$$ −16286.0 −0.567813 −0.283906 0.958852i $$-0.591630\pi$$
−0.283906 + 0.958852i $$0.591630\pi$$
$$938$$ 0 0
$$939$$ 39256.4 1.36431
$$940$$ 0 0
$$941$$ −24302.0 −0.841894 −0.420947 0.907085i $$-0.638302\pi$$
−0.420947 + 0.907085i $$0.638302\pi$$
$$942$$ 0 0
$$943$$ 1923.02 0.0664073
$$944$$ 0 0
$$945$$ −23800.0 −0.819274
$$946$$ 0 0
$$947$$ 19869.7 0.681815 0.340907 0.940097i $$-0.389266\pi$$
0.340907 + 0.940097i $$0.389266\pi$$
$$948$$ 0 0
$$949$$ 32364.0 1.10704
$$950$$ 0 0
$$951$$ 22566.4 0.769470
$$952$$ 0 0
$$953$$ −22422.0 −0.762140 −0.381070 0.924546i $$-0.624444\pi$$
−0.381070 + 0.924546i $$0.624444\pi$$
$$954$$ 0 0
$$955$$ 10330.6 0.350043
$$956$$ 0 0
$$957$$ −3600.00 −0.121600
$$958$$ 0 0
$$959$$ −65552.6 −2.20730
$$960$$ 0 0
$$961$$ −6671.00 −0.223927
$$962$$ 0 0
$$963$$ 10675.0 0.357214
$$964$$ 0 0
$$965$$ 16890.0 0.563428
$$966$$ 0 0
$$967$$ −43777.7 −1.45584 −0.727920 0.685662i $$-0.759513\pi$$
−0.727920 + 0.685662i $$0.759513\pi$$
$$968$$ 0 0
$$969$$ −22080.0 −0.732004
$$970$$ 0 0
$$971$$ −25714.8 −0.849873 −0.424936 0.905223i $$-0.639704\pi$$
−0.424936 + 0.905223i $$0.639704\pi$$
$$972$$ 0 0
$$973$$ −43120.0 −1.42072
$$974$$ 0 0
$$975$$ 6931.81 0.227688
$$976$$ 0 0
$$977$$ 28986.0 0.949175 0.474588 0.880208i $$-0.342597\pi$$
0.474588 + 0.880208i $$0.342597\pi$$
$$978$$ 0 0
$$979$$ −8675.94 −0.283232
$$980$$ 0 0
$$981$$ 15078.0 0.490727
$$982$$ 0 0
$$983$$ −32123.4 −1.04229 −0.521147 0.853467i $$-0.674496\pi$$
−0.521147 + 0.853467i $$0.674496\pi$$
$$984$$ 0 0
$$985$$ −330.000 −0.0106748
$$986$$ 0 0
$$987$$ −55722.8 −1.79704
$$988$$ 0 0
$$989$$ 12900.0 0.414758
$$990$$ 0 0
$$991$$ 11994.3 0.384471 0.192235 0.981349i $$-0.438426\pi$$
0.192235 + 0.981349i $$0.438426\pi$$
$$992$$ 0 0
$$993$$ −1400.00 −0.0447408
$$994$$ 0 0
$$995$$ 6082.10 0.193785
$$996$$ 0 0
$$997$$ 406.000 0.0128968 0.00644842 0.999979i $$-0.497947\pi$$
0.00644842 + 0.999979i $$0.497947\pi$$
$$998$$ 0 0
$$999$$ −32539.3 −1.03053
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 320.4.a.q.1.2 2
4.3 odd 2 inner 320.4.a.q.1.1 2
5.4 even 2 1600.4.a.cg.1.1 2
8.3 odd 2 160.4.a.d.1.2 yes 2
8.5 even 2 160.4.a.d.1.1 2
16.3 odd 4 1280.4.d.v.641.1 4
16.5 even 4 1280.4.d.v.641.2 4
16.11 odd 4 1280.4.d.v.641.4 4
16.13 even 4 1280.4.d.v.641.3 4
20.19 odd 2 1600.4.a.cg.1.2 2
24.5 odd 2 1440.4.a.bb.1.2 2
24.11 even 2 1440.4.a.bb.1.1 2
40.3 even 4 800.4.c.l.449.3 4
40.13 odd 4 800.4.c.l.449.2 4
40.19 odd 2 800.4.a.o.1.1 2
40.27 even 4 800.4.c.l.449.1 4
40.29 even 2 800.4.a.o.1.2 2
40.37 odd 4 800.4.c.l.449.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.d.1.1 2 8.5 even 2
160.4.a.d.1.2 yes 2 8.3 odd 2
320.4.a.q.1.1 2 4.3 odd 2 inner
320.4.a.q.1.2 2 1.1 even 1 trivial
800.4.a.o.1.1 2 40.19 odd 2
800.4.a.o.1.2 2 40.29 even 2
800.4.c.l.449.1 4 40.27 even 4
800.4.c.l.449.2 4 40.13 odd 4
800.4.c.l.449.3 4 40.3 even 4
800.4.c.l.449.4 4 40.37 odd 4
1280.4.d.v.641.1 4 16.3 odd 4
1280.4.d.v.641.2 4 16.5 even 4
1280.4.d.v.641.3 4 16.13 even 4
1280.4.d.v.641.4 4 16.11 odd 4
1440.4.a.bb.1.1 2 24.11 even 2
1440.4.a.bb.1.2 2 24.5 odd 2
1600.4.a.cg.1.1 2 5.4 even 2
1600.4.a.cg.1.2 2 20.19 odd 2