# Properties

 Label 1280.4.d.v.641.3 Level $1280$ Weight $4$ Character 1280.641 Analytic conductor $75.522$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$75.5224448073$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 641.3 Root $$-0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 1280.641 Dual form 1280.4.d.v.641.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.47214i q^{3} -5.00000i q^{5} -31.3050 q^{7} +7.00000 q^{9} +O(q^{10})$$ $$q+4.47214i q^{3} -5.00000i q^{5} -31.3050 q^{7} +7.00000 q^{9} +8.94427i q^{11} +62.0000i q^{13} +22.3607 q^{15} -46.0000 q^{17} +107.331i q^{19} -140.000i q^{21} +192.302 q^{23} -25.0000 q^{25} +152.053i q^{27} +90.0000i q^{29} +152.053 q^{31} -40.0000 q^{33} +156.525i q^{35} -214.000i q^{37} -277.272 q^{39} +10.0000 q^{41} +67.0820i q^{43} -35.0000i q^{45} -398.020 q^{47} +637.000 q^{49} -205.718i q^{51} -678.000i q^{53} +44.7214 q^{55} -480.000 q^{57} +411.437i q^{59} -250.000i q^{61} -219.135 q^{63} +310.000 q^{65} +49.1935i q^{67} +860.000i q^{69} -366.715 q^{71} -522.000 q^{73} -111.803i q^{75} -280.000i q^{77} -876.539 q^{79} -491.000 q^{81} +380.132i q^{83} +230.000i q^{85} -402.492 q^{87} -970.000 q^{89} -1940.91i q^{91} +680.000i q^{93} +536.656 q^{95} -934.000 q^{97} +62.6099i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 28 q^{9} + O(q^{10})$$ $$4 q + 28 q^{9} - 184 q^{17} - 100 q^{25} - 160 q^{33} + 40 q^{41} + 2548 q^{49} - 1920 q^{57} + 1240 q^{65} - 2088 q^{73} - 1964 q^{81} - 3880 q^{89} - 3736 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 4.47214i 0.860663i 0.902671 + 0.430331i $$0.141603\pi$$
−0.902671 + 0.430331i $$0.858397\pi$$
$$4$$ 0 0
$$5$$ − 5.00000i − 0.447214i
$$6$$ 0 0
$$7$$ −31.3050 −1.69031 −0.845154 0.534522i $$-0.820491\pi$$
−0.845154 + 0.534522i $$0.820491\pi$$
$$8$$ 0 0
$$9$$ 7.00000 0.259259
$$10$$ 0 0
$$11$$ 8.94427i 0.245164i 0.992458 + 0.122582i $$0.0391174\pi$$
−0.992458 + 0.122582i $$0.960883\pi$$
$$12$$ 0 0
$$13$$ 62.0000i 1.32275i 0.750057 + 0.661373i $$0.230026\pi$$
−0.750057 + 0.661373i $$0.769974\pi$$
$$14$$ 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
$$19$$ 107.331i 1.29597i 0.761652 + 0.647986i $$0.224389\pi$$
−0.761652 + 0.647986i $$0.775611\pi$$
$$20$$ 0 0
$$21$$ − 140.000i − 1.45479i
$$22$$ 0 0
$$23$$ 192.302 1.74338 0.871689 0.490059i $$-0.163025\pi$$
0.871689 + 0.490059i $$0.163025\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ 152.053i 1.08380i
$$28$$ 0 0
$$29$$ 90.0000i 0.576296i 0.957586 + 0.288148i $$0.0930395\pi$$
−0.957586 + 0.288148i $$0.906961\pi$$
$$30$$ 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
$$35$$ 156.525i 0.755929i
$$36$$ 0 0
$$37$$ − 214.000i − 0.950848i −0.879757 0.475424i $$-0.842295\pi$$
0.879757 0.475424i $$-0.157705\pi$$
$$38$$ 0 0
$$39$$ −277.272 −1.13844
$$40$$ 0 0
$$41$$ 10.0000 0.0380912 0.0190456 0.999819i $$-0.493937\pi$$
0.0190456 + 0.999819i $$0.493937\pi$$
$$42$$ 0 0
$$43$$ 67.0820i 0.237905i 0.992900 + 0.118953i $$0.0379536\pi$$
−0.992900 + 0.118953i $$0.962046\pi$$
$$44$$ 0 0
$$45$$ − 35.0000i − 0.115944i
$$46$$ 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
$$51$$ − 205.718i − 0.564830i
$$52$$ 0 0
$$53$$ − 678.000i − 1.75718i −0.477578 0.878589i $$-0.658485\pi$$
0.477578 0.878589i $$-0.341515\pi$$
$$54$$ 0 0
$$55$$ 44.7214 0.109640
$$56$$ 0 0
$$57$$ −480.000 −1.11540
$$58$$ 0 0
$$59$$ 411.437i 0.907872i 0.891034 + 0.453936i $$0.149981\pi$$
−0.891034 + 0.453936i $$0.850019\pi$$
$$60$$ 0 0
$$61$$ − 250.000i − 0.524741i −0.964967 0.262371i $$-0.915496\pi$$
0.964967 0.262371i $$-0.0845043\pi$$
$$62$$ 0 0
$$63$$ −219.135 −0.438228
$$64$$ 0 0
$$65$$ 310.000 0.591550
$$66$$ 0 0
$$67$$ 49.1935i 0.0897006i 0.998994 + 0.0448503i $$0.0142811\pi$$
−0.998994 + 0.0448503i $$0.985719\pi$$
$$68$$ 0 0
$$69$$ 860.000i 1.50046i
$$70$$ 0 0
$$71$$ −366.715 −0.612973 −0.306486 0.951875i $$-0.599153\pi$$
−0.306486 + 0.951875i $$0.599153\pi$$
$$72$$ 0 0
$$73$$ −522.000 −0.836924 −0.418462 0.908234i $$-0.637431\pi$$
−0.418462 + 0.908234i $$0.637431\pi$$
$$74$$ 0 0
$$75$$ − 111.803i − 0.172133i
$$76$$ 0 0
$$77$$ − 280.000i − 0.414402i
$$78$$ 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
$$83$$ 380.132i 0.502709i 0.967895 + 0.251355i $$0.0808760\pi$$
−0.967895 + 0.251355i $$0.919124\pi$$
$$84$$ 0 0
$$85$$ 230.000i 0.293494i
$$86$$ 0 0
$$87$$ −402.492 −0.495997
$$88$$ 0 0
$$89$$ −970.000 −1.15528 −0.577639 0.816292i $$-0.696026\pi$$
−0.577639 + 0.816292i $$0.696026\pi$$
$$90$$ 0 0
$$91$$ − 1940.91i − 2.23585i
$$92$$ 0 0
$$93$$ 680.000i 0.758201i
$$94$$ 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
$$99$$ 62.6099i 0.0635609i
$$100$$ 0 0
$$101$$ − 602.000i − 0.593082i −0.955020 0.296541i $$-0.904167\pi$$
0.955020 0.296541i $$-0.0958331\pi$$
$$102$$ 0 0
$$103$$ −1829.10 −1.74978 −0.874888 0.484325i $$-0.839065\pi$$
−0.874888 + 0.484325i $$0.839065\pi$$
$$104$$ 0 0
$$105$$ −700.000 −0.650600
$$106$$ 0 0
$$107$$ 1525.00i 1.37782i 0.724845 + 0.688912i $$0.241911\pi$$
−0.724845 + 0.688912i $$0.758089\pi$$
$$108$$ 0 0
$$109$$ − 2154.00i − 1.89281i −0.322989 0.946403i $$-0.604688\pi$$
0.322989 0.946403i $$-0.395312\pi$$
$$110$$ 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
$$115$$ − 961.509i − 0.779663i
$$116$$ 0 0
$$117$$ 434.000i 0.342934i
$$118$$ 0 0
$$119$$ 1440.03 1.10930
$$120$$ 0 0
$$121$$ 1251.00 0.939895
$$122$$ 0 0
$$123$$ 44.7214i 0.0327837i
$$124$$ 0 0
$$125$$ 125.000i 0.0894427i
$$126$$ 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
$$131$$ − 205.718i − 0.137204i −0.997644 0.0686019i $$-0.978146\pi$$
0.997644 0.0686019i $$-0.0218538\pi$$
$$132$$ 0 0
$$133$$ − 3360.00i − 2.19059i
$$134$$ 0 0
$$135$$ 760.263 0.484689
$$136$$ 0 0
$$137$$ 2094.00 1.30586 0.652929 0.757419i $$-0.273540\pi$$
0.652929 + 0.757419i $$0.273540\pi$$
$$138$$ 0 0
$$139$$ 1377.42i 0.840511i 0.907406 + 0.420256i $$0.138060\pi$$
−0.907406 + 0.420256i $$0.861940\pi$$
$$140$$ 0 0
$$141$$ − 1780.00i − 1.06314i
$$142$$ 0 0
$$143$$ −554.545 −0.324289
$$144$$ 0 0
$$145$$ 450.000 0.257727
$$146$$ 0 0
$$147$$ 2848.75i 1.59837i
$$148$$ 0 0
$$149$$ − 334.000i − 0.183640i −0.995776 0.0918200i $$-0.970732\pi$$
0.995776 0.0918200i $$-0.0292684\pi$$
$$150$$ 0 0
$$151$$ 3139.44 1.69195 0.845973 0.533225i $$-0.179020\pi$$
0.845973 + 0.533225i $$0.179020\pi$$
$$152$$ 0 0
$$153$$ −322.000 −0.170145
$$154$$ 0 0
$$155$$ − 760.263i − 0.393973i
$$156$$ 0 0
$$157$$ − 834.000i − 0.423952i −0.977275 0.211976i $$-0.932010\pi$$
0.977275 0.211976i $$-0.0679898\pi$$
$$158$$ 0 0
$$159$$ 3032.11 1.51234
$$160$$ 0 0
$$161$$ −6020.00 −2.94685
$$162$$ 0 0
$$163$$ − 3090.25i − 1.48495i −0.669874 0.742475i $$-0.733652\pi$$
0.669874 0.742475i $$-0.266348\pi$$
$$164$$ 0 0
$$165$$ 200.000i 0.0943635i
$$166$$ 0 0
$$167$$ 4.47214 0.00207224 0.00103612 0.999999i $$-0.499670\pi$$
0.00103612 + 0.999999i $$0.499670\pi$$
$$168$$ 0 0
$$169$$ −1647.00 −0.749659
$$170$$ 0 0
$$171$$ 751.319i 0.335993i
$$172$$ 0 0
$$173$$ 1838.00i 0.807749i 0.914814 + 0.403874i $$0.132337\pi$$
−0.914814 + 0.403874i $$0.867663\pi$$
$$174$$ 0 0
$$175$$ 782.624 0.338062
$$176$$ 0 0
$$177$$ −1840.00 −0.781372
$$178$$ 0 0
$$179$$ 1842.52i 0.769365i 0.923049 + 0.384683i $$0.125689\pi$$
−0.923049 + 0.384683i $$0.874311\pi$$
$$180$$ 0 0
$$181$$ 1862.00i 0.764648i 0.924028 + 0.382324i $$0.124876\pi$$
−0.924028 + 0.382324i $$0.875124\pi$$
$$182$$ 0 0
$$183$$ 1118.03 0.451625
$$184$$ 0 0
$$185$$ −1070.00 −0.425232
$$186$$ 0 0
$$187$$ − 411.437i − 0.160894i
$$188$$ 0 0
$$189$$ − 4760.00i − 1.83195i
$$190$$ 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
$$195$$ 1386.36i 0.509125i
$$196$$ 0 0
$$197$$ 66.0000i 0.0238696i 0.999929 + 0.0119348i $$0.00379905\pi$$
−0.999929 + 0.0119348i $$0.996201\pi$$
$$198$$ 0 0
$$199$$ −1216.42 −0.433316 −0.216658 0.976248i $$-0.569516\pi$$
−0.216658 + 0.976248i $$0.569516\pi$$
$$200$$ 0 0
$$201$$ −220.000 −0.0772020
$$202$$ 0 0
$$203$$ − 2817.45i − 0.974118i
$$204$$ 0 0
$$205$$ − 50.0000i − 0.0170349i
$$206$$ 0 0
$$207$$ 1346.11 0.451987
$$208$$ 0 0
$$209$$ −960.000 −0.317725
$$210$$ 0 0
$$211$$ − 5286.06i − 1.72468i −0.506329 0.862341i $$-0.668998\pi$$
0.506329 0.862341i $$-0.331002\pi$$
$$212$$ 0 0
$$213$$ − 1640.00i − 0.527563i
$$214$$ 0 0
$$215$$ 335.410 0.106394
$$216$$ 0 0
$$217$$ −4760.00 −1.48908
$$218$$ 0 0
$$219$$ − 2334.45i − 0.720310i
$$220$$ 0 0
$$221$$ − 2852.00i − 0.868083i
$$222$$ 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
$$227$$ − 4369.28i − 1.27753i −0.769402 0.638765i $$-0.779446\pi$$
0.769402 0.638765i $$-0.220554\pi$$
$$228$$ 0 0
$$229$$ − 3250.00i − 0.937843i −0.883240 0.468921i $$-0.844643\pi$$
0.883240 0.468921i $$-0.155357\pi$$
$$230$$ 0 0
$$231$$ 1252.20 0.356661
$$232$$ 0 0
$$233$$ −3298.00 −0.927293 −0.463646 0.886020i $$-0.653459\pi$$
−0.463646 + 0.886020i $$0.653459\pi$$
$$234$$ 0 0
$$235$$ 1990.10i 0.552425i
$$236$$ 0 0
$$237$$ − 3920.00i − 1.07439i
$$238$$ 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
$$243$$ 1909.60i 0.504119i
$$244$$ 0 0
$$245$$ − 3185.00i − 0.830540i
$$246$$ 0 0
$$247$$ −6654.54 −1.71424
$$248$$ 0 0
$$249$$ −1700.00 −0.432663
$$250$$ 0 0
$$251$$ − 1386.36i − 0.348631i −0.984690 0.174316i $$-0.944229\pi$$
0.984690 0.174316i $$-0.0557713\pi$$
$$252$$ 0 0
$$253$$ 1720.00i 0.427413i
$$254$$ 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
$$259$$ 6699.26i 1.60723i
$$260$$ 0 0
$$261$$ 630.000i 0.149410i
$$262$$ 0 0
$$263$$ 961.509 0.225434 0.112717 0.993627i $$-0.464045\pi$$
0.112717 + 0.993627i $$0.464045\pi$$
$$264$$ 0 0
$$265$$ −3390.00 −0.785834
$$266$$ 0 0
$$267$$ − 4337.97i − 0.994305i
$$268$$ 0 0
$$269$$ 1494.00i 0.338627i 0.985562 + 0.169314i $$0.0541551\pi$$
−0.985562 + 0.169314i $$0.945845\pi$$
$$270$$ 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
$$275$$ − 223.607i − 0.0490327i
$$276$$ 0 0
$$277$$ − 1006.00i − 0.218212i −0.994030 0.109106i $$-0.965201\pi$$
0.994030 0.109106i $$-0.0347988\pi$$
$$278$$ 0 0
$$279$$ 1064.37 0.228395
$$280$$ 0 0
$$281$$ 3210.00 0.681468 0.340734 0.940160i $$-0.389324\pi$$
0.340734 + 0.940160i $$0.389324\pi$$
$$282$$ 0 0
$$283$$ − 3635.85i − 0.763705i −0.924223 0.381853i $$-0.875286\pi$$
0.924223 0.381853i $$-0.124714\pi$$
$$284$$ 0 0
$$285$$ 2400.00i 0.498820i
$$286$$ 0 0
$$287$$ −313.050 −0.0643858
$$288$$ 0 0
$$289$$ −2797.00 −0.569306
$$290$$ 0 0
$$291$$ − 4176.97i − 0.841439i
$$292$$ 0 0
$$293$$ − 3622.00i − 0.722183i −0.932530 0.361091i $$-0.882404\pi$$
0.932530 0.361091i $$-0.117596\pi$$
$$294$$ 0 0
$$295$$ 2057.18 0.406013
$$296$$ 0 0
$$297$$ −1360.00 −0.265708
$$298$$ 0 0
$$299$$ 11922.7i 2.30605i
$$300$$ 0 0
$$301$$ − 2100.00i − 0.402133i
$$302$$ 0 0
$$303$$ 2692.23 0.510443
$$304$$ 0 0
$$305$$ −1250.00 −0.234671
$$306$$ 0 0
$$307$$ 2088.49i 0.388261i 0.980976 + 0.194131i $$0.0621886\pi$$
−0.980976 + 0.194131i $$0.937811\pi$$
$$308$$ 0 0
$$309$$ − 8180.00i − 1.50597i
$$310$$ 0 0
$$311$$ −8899.55 −1.62266 −0.811330 0.584589i $$-0.801256\pi$$
−0.811330 + 0.584589i $$0.801256\pi$$
$$312$$ 0 0
$$313$$ −8778.00 −1.58518 −0.792591 0.609754i $$-0.791268\pi$$
−0.792591 + 0.609754i $$0.791268\pi$$
$$314$$ 0 0
$$315$$ 1095.67i 0.195982i
$$316$$ 0 0
$$317$$ 5046.00i 0.894043i 0.894523 + 0.447021i $$0.147515\pi$$
−0.894523 + 0.447021i $$0.852485\pi$$
$$318$$ 0 0
$$319$$ −804.984 −0.141287
$$320$$ 0 0
$$321$$ −6820.00 −1.18584
$$322$$ 0 0
$$323$$ − 4937.24i − 0.850512i
$$324$$ 0 0
$$325$$ − 1550.00i − 0.264549i
$$326$$ 0 0
$$327$$ 9632.98 1.62907
$$328$$ 0 0
$$329$$ 12460.0 2.08797
$$330$$ 0 0
$$331$$ 313.050i 0.0519842i 0.999662 + 0.0259921i $$0.00827447\pi$$
−0.999662 + 0.0259921i $$0.991726\pi$$
$$332$$ 0 0
$$333$$ − 1498.00i − 0.246516i
$$334$$ 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
$$339$$ − 9758.20i − 1.56340i
$$340$$ 0 0
$$341$$ 1360.00i 0.215977i
$$342$$ 0 0
$$343$$ −9203.66 −1.44884
$$344$$ 0 0
$$345$$ 4300.00 0.671027
$$346$$ 0 0
$$347$$ − 2643.03i − 0.408892i −0.978878 0.204446i $$-0.934461\pi$$
0.978878 0.204446i $$-0.0655392\pi$$
$$348$$ 0 0
$$349$$ 10170.0i 1.55985i 0.625873 + 0.779925i $$0.284743\pi$$
−0.625873 + 0.779925i $$0.715257\pi$$
$$350$$ 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
$$355$$ 1833.58i 0.274130i
$$356$$ 0 0
$$357$$ 6440.00i 0.954737i
$$358$$ 0 0
$$359$$ 12378.9 1.81987 0.909933 0.414755i $$-0.136133\pi$$
0.909933 + 0.414755i $$0.136133\pi$$
$$360$$ 0 0
$$361$$ −4661.00 −0.679545
$$362$$ 0 0
$$363$$ 5594.64i 0.808933i
$$364$$ 0 0
$$365$$ 2610.00i 0.374284i
$$366$$ 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
$$371$$ 21224.8i 2.97017i
$$372$$ 0 0
$$373$$ − 3278.00i − 0.455036i −0.973774 0.227518i $$-0.926939\pi$$
0.973774 0.227518i $$-0.0730610\pi$$
$$374$$ 0 0
$$375$$ −559.017 −0.0769800
$$376$$ 0 0
$$377$$ −5580.00 −0.762293
$$378$$ 0 0
$$379$$ − 5116.12i − 0.693397i −0.937977 0.346699i $$-0.887303\pi$$
0.937977 0.346699i $$-0.112697\pi$$
$$380$$ 0 0
$$381$$ − 5860.00i − 0.787971i
$$382$$ 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
$$387$$ 469.574i 0.0616791i
$$388$$ 0 0
$$389$$ 834.000i 0.108703i 0.998522 + 0.0543515i $$0.0173092\pi$$
−0.998522 + 0.0543515i $$0.982691\pi$$
$$390$$ 0 0
$$391$$ −8845.88 −1.14413
$$392$$ 0 0
$$393$$ 920.000 0.118086
$$394$$ 0 0
$$395$$ 4382.69i 0.558271i
$$396$$ 0 0
$$397$$ 8734.00i 1.10415i 0.833795 + 0.552074i $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ 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
$$403$$ 9427.26i 1.16527i
$$404$$ 0 0
$$405$$ 2455.00i 0.301210i
$$406$$ 0 0
$$407$$ 1914.07 0.233113
$$408$$ 0 0
$$409$$ 6514.00 0.787522 0.393761 0.919213i $$-0.371174\pi$$
0.393761 + 0.919213i $$0.371174\pi$$
$$410$$ 0 0
$$411$$ 9364.65i 1.12390i
$$412$$ 0 0
$$413$$ − 12880.0i − 1.53458i
$$414$$ 0 0
$$415$$ 1900.66 0.224818
$$416$$ 0 0
$$417$$ −6160.00 −0.723397
$$418$$ 0 0
$$419$$ 16081.8i 1.87505i 0.347913 + 0.937527i $$0.386890\pi$$
−0.347913 + 0.937527i $$0.613110\pi$$
$$420$$ 0 0
$$421$$ 7250.00i 0.839295i 0.907687 + 0.419648i $$0.137846\pi$$
−0.907687 + 0.419648i $$0.862154\pi$$
$$422$$ 0 0
$$423$$ −2786.14 −0.320252
$$424$$ 0 0
$$425$$ 1150.00 0.131255
$$426$$ 0 0
$$427$$ 7826.24i 0.886975i
$$428$$ 0 0
$$429$$ − 2480.00i − 0.279104i
$$430$$ 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
$$435$$ 2012.46i 0.221816i
$$436$$ 0 0
$$437$$ 20640.0i 2.25937i
$$438$$ 0 0
$$439$$ 3792.37 0.412301 0.206150 0.978520i $$-0.433906\pi$$
0.206150 + 0.978520i $$0.433906\pi$$
$$440$$ 0 0
$$441$$ 4459.00 0.481481
$$442$$ 0 0
$$443$$ 746.847i 0.0800988i 0.999198 + 0.0400494i $$0.0127515\pi$$
−0.999198 + 0.0400494i $$0.987248\pi$$
$$444$$ 0 0
$$445$$ 4850.00i 0.516656i
$$446$$ 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
$$451$$ 89.4427i 0.00933857i
$$452$$ 0 0
$$453$$ 14040.0i 1.45620i
$$454$$ 0 0
$$455$$ −9704.54 −0.999902
$$456$$ 0 0
$$457$$ 9526.00 0.975071 0.487536 0.873103i $$-0.337896\pi$$
0.487536 + 0.873103i $$0.337896\pi$$
$$458$$ 0 0
$$459$$ − 6994.42i − 0.711267i
$$460$$ 0 0
$$461$$ − 1518.00i − 0.153363i −0.997056 0.0766815i $$-0.975568\pi$$
0.997056 0.0766815i $$-0.0244325\pi$$
$$462$$ 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
$$467$$ − 16980.7i − 1.68260i −0.540570 0.841299i $$-0.681792\pi$$
0.540570 0.841299i $$-0.318208\pi$$
$$468$$ 0 0
$$469$$ − 1540.00i − 0.151622i
$$470$$ 0 0
$$471$$ 3729.76 0.364880
$$472$$ 0 0
$$473$$ −600.000 −0.0583256
$$474$$ 0 0
$$475$$ − 2683.28i − 0.259195i
$$476$$ 0 0
$$477$$ − 4746.00i − 0.455565i
$$478$$ 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
$$483$$ − 26922.3i − 2.53624i
$$484$$ 0 0
$$485$$ 4670.00i 0.437224i
$$486$$ 0 0
$$487$$ 1310.34 0.121924 0.0609620 0.998140i $$-0.480583\pi$$
0.0609620 + 0.998140i $$0.480583\pi$$
$$488$$ 0 0
$$489$$ 13820.0 1.27804
$$490$$ 0 0
$$491$$ − 2960.55i − 0.272114i −0.990701 0.136057i $$-0.956557\pi$$
0.990701 0.136057i $$-0.0434430\pi$$
$$492$$ 0 0
$$493$$ − 4140.00i − 0.378207i
$$494$$ 0 0
$$495$$ 313.050 0.0284253
$$496$$ 0 0
$$497$$ 11480.0 1.03611
$$498$$ 0 0
$$499$$ − 19319.6i − 1.73320i −0.499006 0.866598i $$-0.666301\pi$$
0.499006 0.866598i $$-0.333699\pi$$
$$500$$ 0 0
$$501$$ 20.0000i 0.00178350i
$$502$$ 0 0
$$503$$ 3072.36 0.272345 0.136173 0.990685i $$-0.456520\pi$$
0.136173 + 0.990685i $$0.456520\pi$$
$$504$$ 0 0
$$505$$ −3010.00 −0.265234
$$506$$ 0 0
$$507$$ − 7365.61i − 0.645203i
$$508$$ 0 0
$$509$$ − 18550.0i − 1.61535i −0.589626 0.807676i $$-0.700725\pi$$
0.589626 0.807676i $$-0.299275\pi$$
$$510$$ 0 0
$$511$$ 16341.2 1.41466
$$512$$ 0 0
$$513$$ −16320.0 −1.40457
$$514$$ 0 0
$$515$$ 9145.52i 0.782524i
$$516$$ 0 0
$$517$$ − 3560.00i − 0.302841i
$$518$$ 0 0
$$519$$ −8219.79 −0.695200
$$520$$ 0 0
$$521$$ 2102.00 0.176757 0.0883784 0.996087i $$-0.471832\pi$$
0.0883784 + 0.996087i $$0.471832\pi$$
$$522$$ 0 0
$$523$$ − 17696.2i − 1.47955i −0.672856 0.739773i $$-0.734933\pi$$
0.672856 0.739773i $$-0.265067\pi$$
$$524$$ 0 0
$$525$$ 3500.00i 0.290957i
$$526$$ 0 0
$$527$$ −6994.42 −0.578144
$$528$$ 0 0
$$529$$ 24813.0 2.03937
$$530$$ 0 0
$$531$$ 2880.06i 0.235374i
$$532$$ 0 0
$$533$$ 620.000i 0.0503850i
$$534$$ 0 0
$$535$$ 7624.99 0.616182
$$536$$ 0 0
$$537$$ −8240.00 −0.662164
$$538$$ 0 0
$$539$$ 5697.50i 0.455304i
$$540$$ 0 0
$$541$$ 9922.00i 0.788503i 0.919003 + 0.394251i $$0.128996\pi$$
−0.919003 + 0.394251i $$0.871004\pi$$
$$542$$ 0 0
$$543$$ −8327.12 −0.658105
$$544$$ 0 0
$$545$$ −10770.0 −0.846488
$$546$$ 0 0
$$547$$ 3716.34i 0.290493i 0.989396 + 0.145246i $$0.0463975\pi$$
−0.989396 + 0.145246i $$0.953603\pi$$
$$548$$ 0 0
$$549$$ − 1750.00i − 0.136044i
$$550$$ 0 0
$$551$$ −9659.81 −0.746864
$$552$$ 0 0
$$553$$ 27440.0 2.11007
$$554$$ 0 0
$$555$$ − 4785.19i − 0.365982i
$$556$$ 0 0
$$557$$ 15094.0i 1.14821i 0.818781 + 0.574105i $$0.194650\pi$$
−0.818781 + 0.574105i $$0.805350\pi$$
$$558$$ 0 0
$$559$$ −4159.09 −0.314688
$$560$$ 0 0
$$561$$ 1840.00 0.138476
$$562$$ 0 0
$$563$$ 5657.25i 0.423490i 0.977325 + 0.211745i $$0.0679146\pi$$
−0.977325 + 0.211745i $$0.932085\pi$$
$$564$$ 0 0
$$565$$ 10910.0i 0.812367i
$$566$$ 0 0
$$567$$ 15370.7 1.13847
$$568$$ 0 0
$$569$$ 5906.00 0.435136 0.217568 0.976045i $$-0.430188\pi$$
0.217568 + 0.976045i $$0.430188\pi$$
$$570$$ 0 0
$$571$$ 4892.52i 0.358573i 0.983797 + 0.179287i $$0.0573790\pi$$
−0.983797 + 0.179287i $$0.942621\pi$$
$$572$$ 0 0
$$573$$ 9240.00i 0.673659i
$$574$$ 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
$$579$$ 15106.9i 1.08432i
$$580$$ 0 0
$$581$$ − 11900.0i − 0.849734i
$$582$$ 0 0
$$583$$ 6064.22 0.430796
$$584$$ 0 0
$$585$$ 2170.00 0.153365
$$586$$ 0 0
$$587$$ − 9029.24i − 0.634884i −0.948278 0.317442i $$-0.897176\pi$$
0.948278 0.317442i $$-0.102824\pi$$
$$588$$ 0 0
$$589$$ 16320.0i 1.14169i
$$590$$ 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
$$595$$ − 7200.14i − 0.496096i
$$596$$ 0 0
$$597$$ − 5440.00i − 0.372939i
$$598$$ 0 0
$$599$$ −14149.8 −0.965187 −0.482593 0.875845i $$-0.660305\pi$$
−0.482593 + 0.875845i $$0.660305\pi$$
$$600$$ 0 0
$$601$$ −3110.00 −0.211081 −0.105540 0.994415i $$-0.533657\pi$$
−0.105540 + 0.994415i $$0.533657\pi$$
$$602$$ 0 0
$$603$$ 344.354i 0.0232557i
$$604$$ 0 0
$$605$$ − 6255.00i − 0.420334i
$$606$$ 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
$$611$$ − 24677.2i − 1.63394i
$$612$$ 0 0
$$613$$ − 5342.00i − 0.351976i −0.984392 0.175988i $$-0.943688\pi$$
0.984392 0.175988i $$-0.0563120\pi$$
$$614$$ 0 0
$$615$$ 223.607 0.0146613
$$616$$ 0 0
$$617$$ −19714.0 −1.28631 −0.643157 0.765734i $$-0.722376\pi$$
−0.643157 + 0.765734i $$0.722376\pi$$
$$618$$ 0 0
$$619$$ 13166.0i 0.854903i 0.904038 + 0.427451i $$0.140589\pi$$
−0.904038 + 0.427451i $$0.859411\pi$$
$$620$$ 0 0
$$621$$ 29240.0i 1.88947i
$$622$$ 0 0
$$623$$ 30365.8 1.95278
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ − 4293.25i − 0.273454i
$$628$$ 0 0
$$629$$ 9844.00i 0.624016i
$$630$$ 0 0
$$631$$ 12262.6 0.773639 0.386820 0.922155i $$-0.373574\pi$$
0.386820 + 0.922155i $$0.373574\pi$$
$$632$$ 0 0
$$633$$ 23640.0 1.48437
$$634$$ 0 0
$$635$$ 6551.68i 0.409442i
$$636$$ 0 0
$$637$$ 39494.0i 2.45653i
$$638$$ 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
$$643$$ 12240.2i 0.750712i 0.926881 + 0.375356i $$0.122480\pi$$
−0.926881 + 0.375356i $$0.877520\pi$$
$$644$$ 0 0
$$645$$ 1500.00i 0.0915697i
$$646$$ 0 0
$$647$$ −17973.5 −1.09214 −0.546068 0.837741i $$-0.683876\pi$$
−0.546068 + 0.837741i $$0.683876\pi$$
$$648$$ 0 0
$$649$$ −3680.00 −0.222577
$$650$$ 0 0
$$651$$ − 21287.4i − 1.28159i
$$652$$ 0 0
$$653$$ 3478.00i 0.208430i 0.994555 + 0.104215i $$0.0332330\pi$$
−0.994555 + 0.104215i $$0.966767\pi$$
$$654$$ 0 0
$$655$$ −1028.59 −0.0613594
$$656$$ 0 0
$$657$$ −3654.00 −0.216980
$$658$$ 0 0
$$659$$ − 10572.1i − 0.624934i −0.949929 0.312467i $$-0.898845\pi$$
0.949929 0.312467i $$-0.101155\pi$$
$$660$$ 0 0
$$661$$ − 110.000i − 0.00647277i −0.999995 0.00323639i $$-0.998970\pi$$
0.999995 0.00323639i $$-0.00103018\pi$$
$$662$$ 0 0
$$663$$ 12754.5 0.747127
$$664$$ 0 0
$$665$$ −16800.0 −0.979663
$$666$$ 0 0
$$667$$ 17307.2i 1.00470i
$$668$$ 0 0
$$669$$ − 13260.0i − 0.766310i
$$670$$ 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
$$675$$ − 3801.32i − 0.216760i
$$676$$ 0 0
$$677$$ 18386.0i 1.04377i 0.853016 + 0.521884i $$0.174771\pi$$
−0.853016 + 0.521884i $$0.825229\pi$$
$$678$$ 0 0
$$679$$ 29238.8 1.65255
$$680$$ 0 0
$$681$$ 19540.0 1.09952
$$682$$ 0 0
$$683$$ − 15317.1i − 0.858113i −0.903278 0.429057i $$-0.858846\pi$$
0.903278 0.429057i $$-0.141154\pi$$
$$684$$ 0 0
$$685$$ − 10470.0i − 0.583997i
$$686$$ 0 0
$$687$$ 14534.4 0.807167
$$688$$ 0 0
$$689$$ 42036.0 2.32430
$$690$$ 0 0
$$691$$ 9507.76i 0.523433i 0.965145 + 0.261717i $$0.0842886\pi$$
−0.965145 + 0.261717i $$0.915711\pi$$
$$692$$ 0 0
$$693$$ − 1960.00i − 0.107438i
$$694$$ 0 0
$$695$$ 6887.09 0.375888
$$696$$ 0 0
$$697$$ −460.000 −0.0249982
$$698$$ 0 0
$$699$$ − 14749.1i − 0.798086i
$$700$$ 0 0
$$701$$ 15830.0i 0.852911i 0.904508 + 0.426456i $$0.140238\pi$$
−0.904508 + 0.426456i $$0.859762\pi$$
$$702$$ 0 0
$$703$$ 22968.9 1.23227
$$704$$ 0 0
$$705$$ −8900.00 −0.475452
$$706$$ 0 0
$$707$$ 18845.6i 1.00249i
$$708$$ 0 0
$$709$$ − 20050.0i − 1.06205i −0.847356 0.531025i $$-0.821807\pi$$
0.847356 0.531025i $$-0.178193\pi$$
$$710$$ 0 0
$$711$$ −6135.77 −0.323642
$$712$$ 0 0
$$713$$ 29240.0 1.53583
$$714$$ 0 0
$$715$$ 2772.72i 0.145027i
$$716$$ 0 0
$$717$$ 2480.00i 0.129173i
$$718$$ 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
$$723$$ 23031.5i 1.18472i
$$724$$ 0 0
$$725$$ − 2250.00i − 0.115259i
$$726$$ 0 0
$$727$$ −11336.9 −0.578351 −0.289175 0.957276i $$-0.593381\pi$$
−0.289175 + 0.957276i $$0.593381\pi$$
$$728$$ 0 0
$$729$$ −21797.0 −1.10740
$$730$$ 0 0
$$731$$ − 3085.77i − 0.156131i
$$732$$ 0 0
$$733$$ 17198.0i 0.866607i 0.901248 + 0.433303i $$0.142652\pi$$
−0.901248 + 0.433303i $$0.857348\pi$$
$$734$$ 0 0
$$735$$ 14243.8 0.714815
$$736$$ 0 0
$$737$$ −440.000 −0.0219913
$$738$$ 0 0
$$739$$ 4597.36i 0.228845i 0.993432 + 0.114423i $$0.0365018\pi$$
−0.993432 + 0.114423i $$0.963498\pi$$
$$740$$ 0 0
$$741$$ − 29760.0i − 1.47539i
$$742$$ 0 0
$$743$$ −2419.43 −0.119462 −0.0597309 0.998215i $$-0.519024\pi$$
−0.0597309 + 0.998215i $$0.519024\pi$$
$$744$$ 0 0
$$745$$ −1670.00 −0.0821263
$$746$$ 0 0
$$747$$ 2660.92i 0.130332i
$$748$$ 0 0
$$749$$ − 47740.0i − 2.32895i
$$750$$ 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
$$755$$ − 15697.2i − 0.756662i
$$756$$ 0 0
$$757$$ 11474.0i 0.550898i 0.961316 + 0.275449i $$0.0888265\pi$$
−0.961316 + 0.275449i $$0.911174\pi$$
$$758$$ 0 0
$$759$$ −7692.07 −0.367858
$$760$$ 0 0
$$761$$ −31802.0 −1.51488 −0.757439 0.652906i $$-0.773550\pi$$
−0.757439 + 0.652906i $$0.773550\pi$$
$$762$$ 0 0
$$763$$ 67430.9i 3.19942i
$$764$$ 0 0
$$765$$ 1610.00i 0.0760911i
$$766$$ 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
$$771$$ − 18630.9i − 0.870267i
$$772$$ 0 0
$$773$$ 37938.0i 1.76525i 0.470082 + 0.882623i $$0.344224\pi$$
−0.470082 + 0.882623i $$0.655776\pi$$
$$774$$ 0 0
$$775$$ −3801.32 −0.176190
$$776$$ 0 0
$$777$$ −29960.0 −1.38328
$$778$$ 0 0
$$779$$ 1073.31i 0.0493651i
$$780$$ 0 0
$$781$$ − 3280.00i − 0.150279i
$$782$$ 0 0
$$783$$ −13684.7 −0.624588
$$784$$ 0 0
$$785$$ −4170.00 −0.189597
$$786$$ 0 0
$$787$$ 37633.0i 1.70454i 0.523103 + 0.852270i $$0.324774\pi$$
−0.523103 + 0.852270i $$0.675226\pi$$
$$788$$ 0 0
$$789$$ 4300.00i 0.194023i
$$790$$ 0 0
$$791$$ 68307.4 3.07046
$$792$$ 0 0
$$793$$ 15500.0 0.694100
$$794$$ 0 0
$$795$$ − 15160.5i − 0.676338i
$$796$$ 0 0
$$797$$ 17526.0i 0.778924i 0.921042 + 0.389462i $$0.127339\pi$$
−0.921042 + 0.389462i $$0.872661\pi$$
$$798$$ 0 0
$$799$$ 18308.9 0.810667
$$800$$ 0 0
$$801$$ −6790.00 −0.299517
$$802$$ 0 0
$$803$$ − 4668.91i − 0.205183i
$$804$$ 0 0
$$805$$ 30100.0i 1.31787i
$$806$$ 0 0
$$807$$ −6681.37 −0.291444
$$808$$ 0 0
$$809$$ −8970.00 −0.389825 −0.194912 0.980821i $$-0.562442\pi$$
−0.194912 + 0.980821i $$0.562442\pi$$
$$810$$ 0 0
$$811$$ 3550.88i 0.153746i 0.997041 + 0.0768731i $$0.0244936\pi$$
−0.997041 + 0.0768731i $$0.975506\pi$$
$$812$$ 0 0
$$813$$ − 22440.0i − 0.968026i
$$814$$ 0 0
$$815$$ −15451.2 −0.664090
$$816$$ 0 0
$$817$$ −7200.00 −0.308318
$$818$$ 0 0
$$819$$ − 13586.3i − 0.579665i
$$820$$ 0 0
$$821$$ − 15550.0i − 0.661022i −0.943802 0.330511i $$-0.892779\pi$$
0.943802 0.330511i $$-0.107221\pi$$
$$822$$ 0 0
$$823$$ −26712.1 −1.13138 −0.565689 0.824619i $$-0.691390\pi$$
−0.565689 + 0.824619i $$0.691390\pi$$
$$824$$ 0 0
$$825$$ 1000.00 0.0422006
$$826$$ 0 0
$$827$$ 863.122i 0.0362923i 0.999835 + 0.0181461i $$0.00577641\pi$$
−0.999835 + 0.0181461i $$0.994224\pi$$
$$828$$ 0 0
$$829$$ − 19066.0i − 0.798781i −0.916781 0.399391i $$-0.869222\pi$$
0.916781 0.399391i $$-0.130778\pi$$
$$830$$ 0 0
$$831$$ 4498.97 0.187807
$$832$$ 0 0
$$833$$ −29302.0 −1.21879
$$834$$ 0 0
$$835$$ − 22.3607i 0 0.000926734i
$$836$$ 0 0
$$837$$ 23120.0i 0.954772i
$$838$$ 0 0
$$839$$ −47744.5 −1.96463 −0.982315 0.187238i $$-0.940047\pi$$
−0.982315 + 0.187238i $$0.940047\pi$$
$$840$$ 0 0
$$841$$ 16289.0 0.667883
$$842$$ 0 0
$$843$$ 14355.6i 0.586514i
$$844$$ 0 0
$$845$$ 8235.00i 0.335258i
$$846$$ 0 0
$$847$$ −39162.5 −1.58871
$$848$$ 0 0
$$849$$ 16260.0 0.657293
$$850$$ 0 0
$$851$$ − 41152.6i − 1.65769i
$$852$$ 0 0
$$853$$ − 14462.0i − 0.580503i −0.956950 0.290252i $$-0.906261\pi$$
0.956950 0.290252i $$-0.0937390\pi$$
$$854$$ 0 0
$$855$$ 3756.59 0.150261
$$856$$ 0 0
$$857$$ −29346.0 −1.16971 −0.584854 0.811138i $$-0.698848\pi$$
−0.584854 + 0.811138i $$0.698848\pi$$
$$858$$ 0 0
$$859$$ − 22807.9i − 0.905932i −0.891528 0.452966i $$-0.850366\pi$$
0.891528 0.452966i $$-0.149634\pi$$
$$860$$ 0 0
$$861$$ − 1400.00i − 0.0554145i
$$862$$ 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
$$867$$ − 12508.6i − 0.489981i
$$868$$ 0 0
$$869$$ − 7840.00i − 0.306046i
$$870$$ 0 0
$$871$$ −3050.00 −0.118651
$$872$$ 0 0
$$873$$ −6538.00 −0.253468
$$874$$ 0 0
$$875$$ − 3913.12i − 0.151186i
$$876$$ 0 0
$$877$$ 32126.0i 1.23696i 0.785799 + 0.618482i $$0.212252\pi$$
−0.785799 + 0.618482i $$0.787748\pi$$
$$878$$ 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
$$883$$ − 6435.40i − 0.245265i −0.992452 0.122632i $$-0.960866\pi$$
0.992452 0.122632i $$-0.0391336\pi$$
$$884$$ 0 0
$$885$$ 9200.00i 0.349440i
$$886$$ 0 0
$$887$$ −46827.7 −1.77263 −0.886314 0.463084i $$-0.846743\pi$$
−0.886314 + 0.463084i $$0.846743\pi$$
$$888$$ 0 0
$$889$$ 41020.0 1.54754
$$890$$ 0 0
$$891$$ − 4391.64i − 0.165124i
$$892$$ 0 0
$$893$$ − 42720.0i − 1.60086i
$$894$$ 0 0
$$895$$ 9212.60 0.344071
$$896$$ 0 0
$$897$$ −53320.0 −1.98473
$$898$$ 0 0
$$899$$ 13684.7i 0.507688i
$$900$$ 0 0
$$901$$ 31188.0i 1.15319i
$$902$$ 0 0
$$903$$ 9391.49 0.346101
$$904$$ 0 0
$$905$$ 9310.00 0.341961
$$906$$ 0 0
$$907$$ − 11980.9i − 0.438608i −0.975657 0.219304i $$-0.929621\pi$$
0.975657 0.219304i $$-0.0703787\pi$$
$$908$$ 0 0
$$909$$ − 4214.00i − 0.153762i
$$910$$ 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
$$915$$ − 5590.17i − 0.201973i
$$916$$ 0 0
$$917$$ 6440.00i 0.231917i
$$918$$ 0 0
$$919$$ −37512.3 −1.34648 −0.673240 0.739424i $$-0.735098\pi$$
−0.673240 + 0.739424i $$0.735098\pi$$
$$920$$ 0 0
$$921$$ −9340.00 −0.334162
$$922$$ 0 0
$$923$$ − 22736.3i − 0.810808i
$$924$$ 0 0
$$925$$ 5350.00i 0.190170i
$$926$$ 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
$$931$$ 68370.0i 2.40681i
$$932$$ 0 0
$$933$$ − 39800.0i − 1.39656i
$$934$$ 0 0
$$935$$ −2057.18 −0.0719541
$$936$$ 0 0
$$937$$ 16286.0 0.567813 0.283906 0.958852i $$-0.408370\pi$$
0.283906 + 0.958852i $$0.408370\pi$$
$$938$$ 0 0
$$939$$ − 39256.4i − 1.36431i
$$940$$ 0 0
$$941$$ − 24302.0i − 0.841894i −0.907085 0.420947i $$-0.861698\pi$$
0.907085 0.420947i $$-0.138302\pi$$
$$942$$ 0 0
$$943$$ 1923.02 0.0664073
$$944$$ 0 0
$$945$$ −23800.0 −0.819274
$$946$$ 0 0
$$947$$ 19869.7i 0.681815i 0.940097 + 0.340907i $$0.110734\pi$$
−0.940097 + 0.340907i $$0.889266\pi$$
$$948$$ 0 0
$$949$$ − 32364.0i − 1.10704i
$$950$$ 0 0
$$951$$ −22566.4 −0.769470
$$952$$ 0 0
$$953$$ 22422.0 0.762140 0.381070 0.924546i $$-0.375556\pi$$
0.381070 + 0.924546i $$0.375556\pi$$
$$954$$ 0 0
$$955$$ − 10330.6i − 0.350043i
$$956$$ 0 0
$$957$$ − 3600.00i − 0.121600i
$$958$$ 0 0
$$959$$ −65552.6 −2.20730
$$960$$ 0 0
$$961$$ −6671.00 −0.223927
$$962$$ 0 0
$$963$$ 10675.0i 0.357214i
$$964$$ 0 0
$$965$$ − 16890.0i − 0.563428i
$$966$$ 0 0
$$967$$ 43777.7 1.45584 0.727920 0.685662i $$-0.240487\pi$$
0.727920 + 0.685662i $$0.240487\pi$$
$$968$$ 0 0
$$969$$ 22080.0 0.732004
$$970$$ 0 0
$$971$$ 25714.8i 0.849873i 0.905223 + 0.424936i $$0.139704\pi$$
−0.905223 + 0.424936i $$0.860296\pi$$
$$972$$ 0 0
$$973$$ − 43120.0i − 1.42072i
$$974$$ 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
$$979$$ − 8675.94i − 0.283232i
$$980$$ 0 0
$$981$$ − 15078.0i − 0.490727i
$$982$$ 0 0
$$983$$ 32123.4 1.04229 0.521147 0.853467i $$-0.325504\pi$$
0.521147 + 0.853467i $$0.325504\pi$$
$$984$$ 0 0
$$985$$ 330.000 0.0106748
$$986$$ 0 0
$$987$$ 55722.8i 1.79704i
$$988$$ 0 0
$$989$$ 12900.0i 0.414758i
$$990$$ 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
$$995$$ 6082.10i 0.193785i
$$996$$ 0 0
$$997$$ − 406.000i − 0.0128968i −0.999979 0.00644842i $$-0.997947\pi$$
0.999979 0.00644842i $$-0.00205261\pi$$
$$998$$ 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 1280.4.d.v.641.3 4
4.3 odd 2 inner 1280.4.d.v.641.1 4
8.3 odd 2 inner 1280.4.d.v.641.4 4
8.5 even 2 inner 1280.4.d.v.641.2 4
16.3 odd 4 160.4.a.d.1.2 yes 2
16.5 even 4 320.4.a.q.1.2 2
16.11 odd 4 320.4.a.q.1.1 2
16.13 even 4 160.4.a.d.1.1 2
48.29 odd 4 1440.4.a.bb.1.2 2
48.35 even 4 1440.4.a.bb.1.1 2
80.3 even 4 800.4.c.l.449.3 4
80.13 odd 4 800.4.c.l.449.2 4
80.19 odd 4 800.4.a.o.1.1 2
80.29 even 4 800.4.a.o.1.2 2
80.59 odd 4 1600.4.a.cg.1.2 2
80.67 even 4 800.4.c.l.449.1 4
80.69 even 4 1600.4.a.cg.1.1 2
80.77 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 16.13 even 4
160.4.a.d.1.2 yes 2 16.3 odd 4
320.4.a.q.1.1 2 16.11 odd 4
320.4.a.q.1.2 2 16.5 even 4
800.4.a.o.1.1 2 80.19 odd 4
800.4.a.o.1.2 2 80.29 even 4
800.4.c.l.449.1 4 80.67 even 4
800.4.c.l.449.2 4 80.13 odd 4
800.4.c.l.449.3 4 80.3 even 4
800.4.c.l.449.4 4 80.77 odd 4
1280.4.d.v.641.1 4 4.3 odd 2 inner
1280.4.d.v.641.2 4 8.5 even 2 inner
1280.4.d.v.641.3 4 1.1 even 1 trivial
1280.4.d.v.641.4 4 8.3 odd 2 inner
1440.4.a.bb.1.1 2 48.35 even 4
1440.4.a.bb.1.2 2 48.29 odd 4
1600.4.a.cg.1.1 2 80.69 even 4
1600.4.a.cg.1.2 2 80.59 odd 4