# Properties

 Label 1280.4.d.c.641.2 Level $1280$ Weight $4$ Character 1280.641 Analytic conductor $75.522$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 641.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1280.641 Dual form 1280.4.d.c.641.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000i q^{3} +5.00000i q^{5} -16.0000 q^{7} +11.0000 q^{9} +O(q^{10})$$ $$q+4.00000i q^{3} +5.00000i q^{5} -16.0000 q^{7} +11.0000 q^{9} +60.0000i q^{11} -86.0000i q^{13} -20.0000 q^{15} +18.0000 q^{17} +44.0000i q^{19} -64.0000i q^{21} +48.0000 q^{23} -25.0000 q^{25} +152.000i q^{27} +186.000i q^{29} -176.000 q^{31} -240.000 q^{33} -80.0000i q^{35} +254.000i q^{37} +344.000 q^{39} -186.000 q^{41} +100.000i q^{43} +55.0000i q^{45} -168.000 q^{47} -87.0000 q^{49} +72.0000i q^{51} -498.000i q^{53} -300.000 q^{55} -176.000 q^{57} +252.000i q^{59} +58.0000i q^{61} -176.000 q^{63} +430.000 q^{65} -1036.00i q^{67} +192.000i q^{69} +168.000 q^{71} -506.000 q^{73} -100.000i q^{75} -960.000i q^{77} -272.000 q^{79} -311.000 q^{81} +948.000i q^{83} +90.0000i q^{85} -744.000 q^{87} +1014.00 q^{89} +1376.00i q^{91} -704.000i q^{93} -220.000 q^{95} -766.000 q^{97} +660.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{7} + 22 q^{9}+O(q^{10})$$ 2 * q - 32 * q^7 + 22 * q^9 $$2 q - 32 q^{7} + 22 q^{9} - 40 q^{15} + 36 q^{17} + 96 q^{23} - 50 q^{25} - 352 q^{31} - 480 q^{33} + 688 q^{39} - 372 q^{41} - 336 q^{47} - 174 q^{49} - 600 q^{55} - 352 q^{57} - 352 q^{63} + 860 q^{65} + 336 q^{71} - 1012 q^{73} - 544 q^{79} - 622 q^{81} - 1488 q^{87} + 2028 q^{89} - 440 q^{95} - 1532 q^{97}+O(q^{100})$$ 2 * q - 32 * q^7 + 22 * q^9 - 40 * q^15 + 36 * q^17 + 96 * q^23 - 50 * q^25 - 352 * q^31 - 480 * q^33 + 688 * q^39 - 372 * q^41 - 336 * q^47 - 174 * q^49 - 600 * q^55 - 352 * q^57 - 352 * q^63 + 860 * q^65 + 336 * q^71 - 1012 * q^73 - 544 * q^79 - 622 * q^81 - 1488 * q^87 + 2028 * q^89 - 440 * q^95 - 1532 * q^97

## 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
$$3$$ 4.00000i 0.769800i 0.922958 + 0.384900i $$0.125764\pi$$
−0.922958 + 0.384900i $$0.874236\pi$$
$$4$$ 0 0
$$5$$ 5.00000i 0.447214i
$$6$$ 0 0
$$7$$ −16.0000 −0.863919 −0.431959 0.901893i $$-0.642178\pi$$
−0.431959 + 0.901893i $$0.642178\pi$$
$$8$$ 0 0
$$9$$ 11.0000 0.407407
$$10$$ 0 0
$$11$$ 60.0000i 1.64461i 0.569049 + 0.822304i $$0.307311\pi$$
−0.569049 + 0.822304i $$0.692689\pi$$
$$12$$ 0 0
$$13$$ − 86.0000i − 1.83478i −0.397992 0.917389i $$-0.630293\pi$$
0.397992 0.917389i $$-0.369707\pi$$
$$14$$ 0 0
$$15$$ −20.0000 −0.344265
$$16$$ 0 0
$$17$$ 18.0000 0.256802 0.128401 0.991722i $$-0.459015\pi$$
0.128401 + 0.991722i $$0.459015\pi$$
$$18$$ 0 0
$$19$$ 44.0000i 0.531279i 0.964072 + 0.265639i $$0.0855830\pi$$
−0.964072 + 0.265639i $$0.914417\pi$$
$$20$$ 0 0
$$21$$ − 64.0000i − 0.665045i
$$22$$ 0 0
$$23$$ 48.0000 0.435161 0.217580 0.976042i $$-0.430184\pi$$
0.217580 + 0.976042i $$0.430184\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ 152.000i 1.08342i
$$28$$ 0 0
$$29$$ 186.000i 1.19101i 0.803351 + 0.595506i $$0.203048\pi$$
−0.803351 + 0.595506i $$0.796952\pi$$
$$30$$ 0 0
$$31$$ −176.000 −1.01969 −0.509847 0.860265i $$-0.670298\pi$$
−0.509847 + 0.860265i $$0.670298\pi$$
$$32$$ 0 0
$$33$$ −240.000 −1.26602
$$34$$ 0 0
$$35$$ − 80.0000i − 0.386356i
$$36$$ 0 0
$$37$$ 254.000i 1.12858i 0.825578 + 0.564288i $$0.190849\pi$$
−0.825578 + 0.564288i $$0.809151\pi$$
$$38$$ 0 0
$$39$$ 344.000 1.41241
$$40$$ 0 0
$$41$$ −186.000 −0.708496 −0.354248 0.935152i $$-0.615263\pi$$
−0.354248 + 0.935152i $$0.615263\pi$$
$$42$$ 0 0
$$43$$ 100.000i 0.354648i 0.984153 + 0.177324i $$0.0567440\pi$$
−0.984153 + 0.177324i $$0.943256\pi$$
$$44$$ 0 0
$$45$$ 55.0000i 0.182198i
$$46$$ 0 0
$$47$$ −168.000 −0.521390 −0.260695 0.965421i $$-0.583952\pi$$
−0.260695 + 0.965421i $$0.583952\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ 72.0000i 0.197687i
$$52$$ 0 0
$$53$$ − 498.000i − 1.29067i −0.763899 0.645335i $$-0.776718\pi$$
0.763899 0.645335i $$-0.223282\pi$$
$$54$$ 0 0
$$55$$ −300.000 −0.735491
$$56$$ 0 0
$$57$$ −176.000 −0.408978
$$58$$ 0 0
$$59$$ 252.000i 0.556061i 0.960572 + 0.278031i $$0.0896817\pi$$
−0.960572 + 0.278031i $$0.910318\pi$$
$$60$$ 0 0
$$61$$ 58.0000i 0.121740i 0.998146 + 0.0608700i $$0.0193875\pi$$
−0.998146 + 0.0608700i $$0.980612\pi$$
$$62$$ 0 0
$$63$$ −176.000 −0.351967
$$64$$ 0 0
$$65$$ 430.000 0.820537
$$66$$ 0 0
$$67$$ − 1036.00i − 1.88907i −0.328414 0.944534i $$-0.606514\pi$$
0.328414 0.944534i $$-0.393486\pi$$
$$68$$ 0 0
$$69$$ 192.000i 0.334987i
$$70$$ 0 0
$$71$$ 168.000 0.280816 0.140408 0.990094i $$-0.455159\pi$$
0.140408 + 0.990094i $$0.455159\pi$$
$$72$$ 0 0
$$73$$ −506.000 −0.811272 −0.405636 0.914035i $$-0.632950\pi$$
−0.405636 + 0.914035i $$0.632950\pi$$
$$74$$ 0 0
$$75$$ − 100.000i − 0.153960i
$$76$$ 0 0
$$77$$ − 960.000i − 1.42081i
$$78$$ 0 0
$$79$$ −272.000 −0.387372 −0.193686 0.981064i $$-0.562044\pi$$
−0.193686 + 0.981064i $$0.562044\pi$$
$$80$$ 0 0
$$81$$ −311.000 −0.426612
$$82$$ 0 0
$$83$$ 948.000i 1.25369i 0.779143 + 0.626846i $$0.215655\pi$$
−0.779143 + 0.626846i $$0.784345\pi$$
$$84$$ 0 0
$$85$$ 90.0000i 0.114846i
$$86$$ 0 0
$$87$$ −744.000 −0.916841
$$88$$ 0 0
$$89$$ 1014.00 1.20768 0.603841 0.797104i $$-0.293636\pi$$
0.603841 + 0.797104i $$0.293636\pi$$
$$90$$ 0 0
$$91$$ 1376.00i 1.58510i
$$92$$ 0 0
$$93$$ − 704.000i − 0.784961i
$$94$$ 0 0
$$95$$ −220.000 −0.237595
$$96$$ 0 0
$$97$$ −766.000 −0.801809 −0.400905 0.916120i $$-0.631304\pi$$
−0.400905 + 0.916120i $$0.631304\pi$$
$$98$$ 0 0
$$99$$ 660.000i 0.670025i
$$100$$ 0 0
$$101$$ − 1314.00i − 1.29453i −0.762264 0.647267i $$-0.775912\pi$$
0.762264 0.647267i $$-0.224088\pi$$
$$102$$ 0 0
$$103$$ −448.000 −0.428570 −0.214285 0.976771i $$-0.568742\pi$$
−0.214285 + 0.976771i $$0.568742\pi$$
$$104$$ 0 0
$$105$$ 320.000 0.297417
$$106$$ 0 0
$$107$$ − 1548.00i − 1.39861i −0.714826 0.699303i $$-0.753494\pi$$
0.714826 0.699303i $$-0.246506\pi$$
$$108$$ 0 0
$$109$$ − 278.000i − 0.244290i −0.992512 0.122145i $$-0.961023\pi$$
0.992512 0.122145i $$-0.0389772\pi$$
$$110$$ 0 0
$$111$$ −1016.00 −0.868779
$$112$$ 0 0
$$113$$ −558.000 −0.464533 −0.232266 0.972652i $$-0.574614\pi$$
−0.232266 + 0.972652i $$0.574614\pi$$
$$114$$ 0 0
$$115$$ 240.000i 0.194610i
$$116$$ 0 0
$$117$$ − 946.000i − 0.747502i
$$118$$ 0 0
$$119$$ −288.000 −0.221856
$$120$$ 0 0
$$121$$ −2269.00 −1.70473
$$122$$ 0 0
$$123$$ − 744.000i − 0.545400i
$$124$$ 0 0
$$125$$ − 125.000i − 0.0894427i
$$126$$ 0 0
$$127$$ −344.000 −0.240355 −0.120177 0.992752i $$-0.538346\pi$$
−0.120177 + 0.992752i $$0.538346\pi$$
$$128$$ 0 0
$$129$$ −400.000 −0.273008
$$130$$ 0 0
$$131$$ 780.000i 0.520221i 0.965579 + 0.260110i $$0.0837590\pi$$
−0.965579 + 0.260110i $$0.916241\pi$$
$$132$$ 0 0
$$133$$ − 704.000i − 0.458982i
$$134$$ 0 0
$$135$$ −760.000 −0.484521
$$136$$ 0 0
$$137$$ −666.000 −0.415330 −0.207665 0.978200i $$-0.566586\pi$$
−0.207665 + 0.978200i $$0.566586\pi$$
$$138$$ 0 0
$$139$$ − 884.000i − 0.539424i −0.962941 0.269712i $$-0.913072\pi$$
0.962941 0.269712i $$-0.0869285\pi$$
$$140$$ 0 0
$$141$$ − 672.000i − 0.401366i
$$142$$ 0 0
$$143$$ 5160.00 3.01749
$$144$$ 0 0
$$145$$ −930.000 −0.532637
$$146$$ 0 0
$$147$$ − 348.000i − 0.195255i
$$148$$ 0 0
$$149$$ − 114.000i − 0.0626795i −0.999509 0.0313397i $$-0.990023\pi$$
0.999509 0.0313397i $$-0.00997738\pi$$
$$150$$ 0 0
$$151$$ −40.0000 −0.0215573 −0.0107787 0.999942i $$-0.503431\pi$$
−0.0107787 + 0.999942i $$0.503431\pi$$
$$152$$ 0 0
$$153$$ 198.000 0.104623
$$154$$ 0 0
$$155$$ − 880.000i − 0.456021i
$$156$$ 0 0
$$157$$ 154.000i 0.0782837i 0.999234 + 0.0391418i $$0.0124624\pi$$
−0.999234 + 0.0391418i $$0.987538\pi$$
$$158$$ 0 0
$$159$$ 1992.00 0.993559
$$160$$ 0 0
$$161$$ −768.000 −0.375943
$$162$$ 0 0
$$163$$ 2180.00i 1.04755i 0.851856 + 0.523775i $$0.175477\pi$$
−0.851856 + 0.523775i $$0.824523\pi$$
$$164$$ 0 0
$$165$$ − 1200.00i − 0.566181i
$$166$$ 0 0
$$167$$ 3696.00 1.71261 0.856303 0.516474i $$-0.172756\pi$$
0.856303 + 0.516474i $$0.172756\pi$$
$$168$$ 0 0
$$169$$ −5199.00 −2.36641
$$170$$ 0 0
$$171$$ 484.000i 0.216447i
$$172$$ 0 0
$$173$$ − 1302.00i − 0.572192i −0.958201 0.286096i $$-0.907642\pi$$
0.958201 0.286096i $$-0.0923576\pi$$
$$174$$ 0 0
$$175$$ 400.000 0.172784
$$176$$ 0 0
$$177$$ −1008.00 −0.428056
$$178$$ 0 0
$$179$$ − 4308.00i − 1.79885i −0.437070 0.899427i $$-0.643984\pi$$
0.437070 0.899427i $$-0.356016\pi$$
$$180$$ 0 0
$$181$$ 1550.00i 0.636523i 0.948003 + 0.318261i $$0.103099\pi$$
−0.948003 + 0.318261i $$0.896901\pi$$
$$182$$ 0 0
$$183$$ −232.000 −0.0937155
$$184$$ 0 0
$$185$$ −1270.00 −0.504715
$$186$$ 0 0
$$187$$ 1080.00i 0.422339i
$$188$$ 0 0
$$189$$ − 2432.00i − 0.935989i
$$190$$ 0 0
$$191$$ −48.0000 −0.0181841 −0.00909204 0.999959i $$-0.502894\pi$$
−0.00909204 + 0.999959i $$0.502894\pi$$
$$192$$ 0 0
$$193$$ 1058.00 0.394593 0.197297 0.980344i $$-0.436784\pi$$
0.197297 + 0.980344i $$0.436784\pi$$
$$194$$ 0 0
$$195$$ 1720.00i 0.631650i
$$196$$ 0 0
$$197$$ − 3714.00i − 1.34321i −0.740911 0.671603i $$-0.765606\pi$$
0.740911 0.671603i $$-0.234394\pi$$
$$198$$ 0 0
$$199$$ −1768.00 −0.629800 −0.314900 0.949125i $$-0.601971\pi$$
−0.314900 + 0.949125i $$0.601971\pi$$
$$200$$ 0 0
$$201$$ 4144.00 1.45421
$$202$$ 0 0
$$203$$ − 2976.00i − 1.02894i
$$204$$ 0 0
$$205$$ − 930.000i − 0.316849i
$$206$$ 0 0
$$207$$ 528.000 0.177288
$$208$$ 0 0
$$209$$ −2640.00 −0.873745
$$210$$ 0 0
$$211$$ − 4036.00i − 1.31682i −0.752658 0.658412i $$-0.771229\pi$$
0.752658 0.658412i $$-0.228771\pi$$
$$212$$ 0 0
$$213$$ 672.000i 0.216172i
$$214$$ 0 0
$$215$$ −500.000 −0.158603
$$216$$ 0 0
$$217$$ 2816.00 0.880933
$$218$$ 0 0
$$219$$ − 2024.00i − 0.624517i
$$220$$ 0 0
$$221$$ − 1548.00i − 0.471175i
$$222$$ 0 0
$$223$$ −680.000 −0.204198 −0.102099 0.994774i $$-0.532556\pi$$
−0.102099 + 0.994774i $$0.532556\pi$$
$$224$$ 0 0
$$225$$ −275.000 −0.0814815
$$226$$ 0 0
$$227$$ 2388.00i 0.698225i 0.937081 + 0.349113i $$0.113517\pi$$
−0.937081 + 0.349113i $$0.886483\pi$$
$$228$$ 0 0
$$229$$ − 3874.00i − 1.11791i −0.829198 0.558954i $$-0.811203\pi$$
0.829198 0.558954i $$-0.188797\pi$$
$$230$$ 0 0
$$231$$ 3840.00 1.09374
$$232$$ 0 0
$$233$$ −3162.00 −0.889054 −0.444527 0.895766i $$-0.646628\pi$$
−0.444527 + 0.895766i $$0.646628\pi$$
$$234$$ 0 0
$$235$$ − 840.000i − 0.233173i
$$236$$ 0 0
$$237$$ − 1088.00i − 0.298199i
$$238$$ 0 0
$$239$$ −5424.00 −1.46799 −0.733995 0.679155i $$-0.762346\pi$$
−0.733995 + 0.679155i $$0.762346\pi$$
$$240$$ 0 0
$$241$$ −3886.00 −1.03867 −0.519335 0.854571i $$-0.673820\pi$$
−0.519335 + 0.854571i $$0.673820\pi$$
$$242$$ 0 0
$$243$$ 2860.00i 0.755017i
$$244$$ 0 0
$$245$$ − 435.000i − 0.113433i
$$246$$ 0 0
$$247$$ 3784.00 0.974778
$$248$$ 0 0
$$249$$ −3792.00 −0.965093
$$250$$ 0 0
$$251$$ 5100.00i 1.28251i 0.767329 + 0.641253i $$0.221585\pi$$
−0.767329 + 0.641253i $$0.778415\pi$$
$$252$$ 0 0
$$253$$ 2880.00i 0.715668i
$$254$$ 0 0
$$255$$ −360.000 −0.0884081
$$256$$ 0 0
$$257$$ 2178.00 0.528638 0.264319 0.964435i $$-0.414853\pi$$
0.264319 + 0.964435i $$0.414853\pi$$
$$258$$ 0 0
$$259$$ − 4064.00i − 0.974999i
$$260$$ 0 0
$$261$$ 2046.00i 0.485227i
$$262$$ 0 0
$$263$$ −6144.00 −1.44051 −0.720257 0.693707i $$-0.755976\pi$$
−0.720257 + 0.693707i $$0.755976\pi$$
$$264$$ 0 0
$$265$$ 2490.00 0.577206
$$266$$ 0 0
$$267$$ 4056.00i 0.929675i
$$268$$ 0 0
$$269$$ − 822.000i − 0.186313i −0.995651 0.0931566i $$-0.970304\pi$$
0.995651 0.0931566i $$-0.0296957\pi$$
$$270$$ 0 0
$$271$$ −8480.00 −1.90082 −0.950412 0.310994i $$-0.899338\pi$$
−0.950412 + 0.310994i $$0.899338\pi$$
$$272$$ 0 0
$$273$$ −5504.00 −1.22021
$$274$$ 0 0
$$275$$ − 1500.00i − 0.328921i
$$276$$ 0 0
$$277$$ − 1138.00i − 0.246844i −0.992354 0.123422i $$-0.960613\pi$$
0.992354 0.123422i $$-0.0393869\pi$$
$$278$$ 0 0
$$279$$ −1936.00 −0.415431
$$280$$ 0 0
$$281$$ −5706.00 −1.21136 −0.605679 0.795709i $$-0.707098\pi$$
−0.605679 + 0.795709i $$0.707098\pi$$
$$282$$ 0 0
$$283$$ 3028.00i 0.636028i 0.948086 + 0.318014i $$0.103016\pi$$
−0.948086 + 0.318014i $$0.896984\pi$$
$$284$$ 0 0
$$285$$ − 880.000i − 0.182901i
$$286$$ 0 0
$$287$$ 2976.00 0.612083
$$288$$ 0 0
$$289$$ −4589.00 −0.934053
$$290$$ 0 0
$$291$$ − 3064.00i − 0.617233i
$$292$$ 0 0
$$293$$ 3390.00i 0.675925i 0.941160 + 0.337962i $$0.109738\pi$$
−0.941160 + 0.337962i $$0.890262\pi$$
$$294$$ 0 0
$$295$$ −1260.00 −0.248678
$$296$$ 0 0
$$297$$ −9120.00 −1.78180
$$298$$ 0 0
$$299$$ − 4128.00i − 0.798423i
$$300$$ 0 0
$$301$$ − 1600.00i − 0.306387i
$$302$$ 0 0
$$303$$ 5256.00 0.996532
$$304$$ 0 0
$$305$$ −290.000 −0.0544438
$$306$$ 0 0
$$307$$ − 4156.00i − 0.772624i −0.922368 0.386312i $$-0.873749\pi$$
0.922368 0.386312i $$-0.126251\pi$$
$$308$$ 0 0
$$309$$ − 1792.00i − 0.329914i
$$310$$ 0 0
$$311$$ 6552.00 1.19463 0.597315 0.802007i $$-0.296234\pi$$
0.597315 + 0.802007i $$0.296234\pi$$
$$312$$ 0 0
$$313$$ 1366.00 0.246680 0.123340 0.992364i $$-0.460639\pi$$
0.123340 + 0.992364i $$0.460639\pi$$
$$314$$ 0 0
$$315$$ − 880.000i − 0.157404i
$$316$$ 0 0
$$317$$ − 2598.00i − 0.460310i −0.973154 0.230155i $$-0.926077\pi$$
0.973154 0.230155i $$-0.0739233\pi$$
$$318$$ 0 0
$$319$$ −11160.0 −1.95875
$$320$$ 0 0
$$321$$ 6192.00 1.07665
$$322$$ 0 0
$$323$$ 792.000i 0.136434i
$$324$$ 0 0
$$325$$ 2150.00i 0.366956i
$$326$$ 0 0
$$327$$ 1112.00 0.188054
$$328$$ 0 0
$$329$$ 2688.00 0.450438
$$330$$ 0 0
$$331$$ 3292.00i 0.546661i 0.961920 + 0.273330i $$0.0881252\pi$$
−0.961920 + 0.273330i $$0.911875\pi$$
$$332$$ 0 0
$$333$$ 2794.00i 0.459791i
$$334$$ 0 0
$$335$$ 5180.00 0.844817
$$336$$ 0 0
$$337$$ 6194.00 1.00121 0.500606 0.865675i $$-0.333110\pi$$
0.500606 + 0.865675i $$0.333110\pi$$
$$338$$ 0 0
$$339$$ − 2232.00i − 0.357598i
$$340$$ 0 0
$$341$$ − 10560.0i − 1.67700i
$$342$$ 0 0
$$343$$ 6880.00 1.08305
$$344$$ 0 0
$$345$$ −960.000 −0.149811
$$346$$ 0 0
$$347$$ 10020.0i 1.55015i 0.631870 + 0.775075i $$0.282288\pi$$
−0.631870 + 0.775075i $$0.717712\pi$$
$$348$$ 0 0
$$349$$ 3130.00i 0.480072i 0.970764 + 0.240036i $$0.0771592\pi$$
−0.970764 + 0.240036i $$0.922841\pi$$
$$350$$ 0 0
$$351$$ 13072.0 1.98784
$$352$$ 0 0
$$353$$ 4194.00 0.632363 0.316181 0.948699i $$-0.397599\pi$$
0.316181 + 0.948699i $$0.397599\pi$$
$$354$$ 0 0
$$355$$ 840.000i 0.125585i
$$356$$ 0 0
$$357$$ − 1152.00i − 0.170785i
$$358$$ 0 0
$$359$$ −4104.00 −0.603345 −0.301672 0.953412i $$-0.597545\pi$$
−0.301672 + 0.953412i $$0.597545\pi$$
$$360$$ 0 0
$$361$$ 4923.00 0.717743
$$362$$ 0 0
$$363$$ − 9076.00i − 1.31230i
$$364$$ 0 0
$$365$$ − 2530.00i − 0.362812i
$$366$$ 0 0
$$367$$ −7496.00 −1.06618 −0.533090 0.846059i $$-0.678969\pi$$
−0.533090 + 0.846059i $$0.678969\pi$$
$$368$$ 0 0
$$369$$ −2046.00 −0.288646
$$370$$ 0 0
$$371$$ 7968.00i 1.11503i
$$372$$ 0 0
$$373$$ − 5842.00i − 0.810958i −0.914104 0.405479i $$-0.867105\pi$$
0.914104 0.405479i $$-0.132895\pi$$
$$374$$ 0 0
$$375$$ 500.000 0.0688530
$$376$$ 0 0
$$377$$ 15996.0 2.18524
$$378$$ 0 0
$$379$$ 412.000i 0.0558391i 0.999610 + 0.0279195i $$0.00888822\pi$$
−0.999610 + 0.0279195i $$0.991112\pi$$
$$380$$ 0 0
$$381$$ − 1376.00i − 0.185025i
$$382$$ 0 0
$$383$$ −2568.00 −0.342607 −0.171304 0.985218i $$-0.554798\pi$$
−0.171304 + 0.985218i $$0.554798\pi$$
$$384$$ 0 0
$$385$$ 4800.00 0.635404
$$386$$ 0 0
$$387$$ 1100.00i 0.144486i
$$388$$ 0 0
$$389$$ 13086.0i 1.70562i 0.522221 + 0.852810i $$0.325104\pi$$
−0.522221 + 0.852810i $$0.674896\pi$$
$$390$$ 0 0
$$391$$ 864.000 0.111750
$$392$$ 0 0
$$393$$ −3120.00 −0.400466
$$394$$ 0 0
$$395$$ − 1360.00i − 0.173238i
$$396$$ 0 0
$$397$$ − 10454.0i − 1.32159i −0.750566 0.660795i $$-0.770219\pi$$
0.750566 0.660795i $$-0.229781\pi$$
$$398$$ 0 0
$$399$$ 2816.00 0.353324
$$400$$ 0 0
$$401$$ −10830.0 −1.34869 −0.674345 0.738417i $$-0.735574\pi$$
−0.674345 + 0.738417i $$0.735574\pi$$
$$402$$ 0 0
$$403$$ 15136.0i 1.87091i
$$404$$ 0 0
$$405$$ − 1555.00i − 0.190787i
$$406$$ 0 0
$$407$$ −15240.0 −1.85607
$$408$$ 0 0
$$409$$ 8566.00 1.03560 0.517801 0.855501i $$-0.326751\pi$$
0.517801 + 0.855501i $$0.326751\pi$$
$$410$$ 0 0
$$411$$ − 2664.00i − 0.319721i
$$412$$ 0 0
$$413$$ − 4032.00i − 0.480392i
$$414$$ 0 0
$$415$$ −4740.00 −0.560669
$$416$$ 0 0
$$417$$ 3536.00 0.415249
$$418$$ 0 0
$$419$$ 13884.0i 1.61880i 0.587257 + 0.809401i $$0.300208\pi$$
−0.587257 + 0.809401i $$0.699792\pi$$
$$420$$ 0 0
$$421$$ 4286.00i 0.496168i 0.968738 + 0.248084i $$0.0798010\pi$$
−0.968738 + 0.248084i $$0.920199\pi$$
$$422$$ 0 0
$$423$$ −1848.00 −0.212418
$$424$$ 0 0
$$425$$ −450.000 −0.0513605
$$426$$ 0 0
$$427$$ − 928.000i − 0.105173i
$$428$$ 0 0
$$429$$ 20640.0i 2.32286i
$$430$$ 0 0
$$431$$ −6336.00 −0.708108 −0.354054 0.935225i $$-0.615197\pi$$
−0.354054 + 0.935225i $$0.615197\pi$$
$$432$$ 0 0
$$433$$ −8974.00 −0.995988 −0.497994 0.867180i $$-0.665930\pi$$
−0.497994 + 0.867180i $$0.665930\pi$$
$$434$$ 0 0
$$435$$ − 3720.00i − 0.410024i
$$436$$ 0 0
$$437$$ 2112.00i 0.231191i
$$438$$ 0 0
$$439$$ −2968.00 −0.322676 −0.161338 0.986899i $$-0.551581\pi$$
−0.161338 + 0.986899i $$0.551581\pi$$
$$440$$ 0 0
$$441$$ −957.000 −0.103337
$$442$$ 0 0
$$443$$ 12372.0i 1.32689i 0.748226 + 0.663444i $$0.230906\pi$$
−0.748226 + 0.663444i $$0.769094\pi$$
$$444$$ 0 0
$$445$$ 5070.00i 0.540092i
$$446$$ 0 0
$$447$$ 456.000 0.0482507
$$448$$ 0 0
$$449$$ 11394.0 1.19759 0.598793 0.800904i $$-0.295647\pi$$
0.598793 + 0.800904i $$0.295647\pi$$
$$450$$ 0 0
$$451$$ − 11160.0i − 1.16520i
$$452$$ 0 0
$$453$$ − 160.000i − 0.0165948i
$$454$$ 0 0
$$455$$ −6880.00 −0.708878
$$456$$ 0 0
$$457$$ 358.000 0.0366445 0.0183222 0.999832i $$-0.494168\pi$$
0.0183222 + 0.999832i $$0.494168\pi$$
$$458$$ 0 0
$$459$$ 2736.00i 0.278226i
$$460$$ 0 0
$$461$$ 7530.00i 0.760753i 0.924832 + 0.380376i $$0.124206\pi$$
−0.924832 + 0.380376i $$0.875794\pi$$
$$462$$ 0 0
$$463$$ 13768.0 1.38197 0.690986 0.722868i $$-0.257177\pi$$
0.690986 + 0.722868i $$0.257177\pi$$
$$464$$ 0 0
$$465$$ 3520.00 0.351045
$$466$$ 0 0
$$467$$ 13380.0i 1.32581i 0.748704 + 0.662904i $$0.230676\pi$$
−0.748704 + 0.662904i $$0.769324\pi$$
$$468$$ 0 0
$$469$$ 16576.0i 1.63200i
$$470$$ 0 0
$$471$$ −616.000 −0.0602628
$$472$$ 0 0
$$473$$ −6000.00 −0.583256
$$474$$ 0 0
$$475$$ − 1100.00i − 0.106256i
$$476$$ 0 0
$$477$$ − 5478.00i − 0.525829i
$$478$$ 0 0
$$479$$ 6336.00 0.604383 0.302191 0.953247i $$-0.402282\pi$$
0.302191 + 0.953247i $$0.402282\pi$$
$$480$$ 0 0
$$481$$ 21844.0 2.07069
$$482$$ 0 0
$$483$$ − 3072.00i − 0.289401i
$$484$$ 0 0
$$485$$ − 3830.00i − 0.358580i
$$486$$ 0 0
$$487$$ −5008.00 −0.465984 −0.232992 0.972479i $$-0.574852\pi$$
−0.232992 + 0.972479i $$0.574852\pi$$
$$488$$ 0 0
$$489$$ −8720.00 −0.806405
$$490$$ 0 0
$$491$$ − 12900.0i − 1.18568i −0.805320 0.592840i $$-0.798007\pi$$
0.805320 0.592840i $$-0.201993\pi$$
$$492$$ 0 0
$$493$$ 3348.00i 0.305855i
$$494$$ 0 0
$$495$$ −3300.00 −0.299644
$$496$$ 0 0
$$497$$ −2688.00 −0.242602
$$498$$ 0 0
$$499$$ − 8116.00i − 0.728100i −0.931379 0.364050i $$-0.881394\pi$$
0.931379 0.364050i $$-0.118606\pi$$
$$500$$ 0 0
$$501$$ 14784.0i 1.31836i
$$502$$ 0 0
$$503$$ −4944.00 −0.438255 −0.219127 0.975696i $$-0.570321\pi$$
−0.219127 + 0.975696i $$0.570321\pi$$
$$504$$ 0 0
$$505$$ 6570.00 0.578933
$$506$$ 0 0
$$507$$ − 20796.0i − 1.82166i
$$508$$ 0 0
$$509$$ 5466.00i 0.475985i 0.971267 + 0.237992i $$0.0764893\pi$$
−0.971267 + 0.237992i $$0.923511\pi$$
$$510$$ 0 0
$$511$$ 8096.00 0.700873
$$512$$ 0 0
$$513$$ −6688.00 −0.575599
$$514$$ 0 0
$$515$$ − 2240.00i − 0.191663i
$$516$$ 0 0
$$517$$ − 10080.0i − 0.857481i
$$518$$ 0 0
$$519$$ 5208.00 0.440474
$$520$$ 0 0
$$521$$ −10074.0 −0.847121 −0.423560 0.905868i $$-0.639220\pi$$
−0.423560 + 0.905868i $$0.639220\pi$$
$$522$$ 0 0
$$523$$ 13828.0i 1.15613i 0.815991 + 0.578065i $$0.196192\pi$$
−0.815991 + 0.578065i $$0.803808\pi$$
$$524$$ 0 0
$$525$$ 1600.00i 0.133009i
$$526$$ 0 0
$$527$$ −3168.00 −0.261860
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ 0 0
$$531$$ 2772.00i 0.226543i
$$532$$ 0 0
$$533$$ 15996.0i 1.29993i
$$534$$ 0 0
$$535$$ 7740.00 0.625475
$$536$$ 0 0
$$537$$ 17232.0 1.38476
$$538$$ 0 0
$$539$$ − 5220.00i − 0.417145i
$$540$$ 0 0
$$541$$ 15226.0i 1.21001i 0.796221 + 0.605006i $$0.206829\pi$$
−0.796221 + 0.605006i $$0.793171\pi$$
$$542$$ 0 0
$$543$$ −6200.00 −0.489995
$$544$$ 0 0
$$545$$ 1390.00 0.109250
$$546$$ 0 0
$$547$$ − 13228.0i − 1.03398i −0.855991 0.516991i $$-0.827052\pi$$
0.855991 0.516991i $$-0.172948\pi$$
$$548$$ 0 0
$$549$$ 638.000i 0.0495978i
$$550$$ 0 0
$$551$$ −8184.00 −0.632759
$$552$$ 0 0
$$553$$ 4352.00 0.334658
$$554$$ 0 0
$$555$$ − 5080.00i − 0.388530i
$$556$$ 0 0
$$557$$ 8490.00i 0.645840i 0.946426 + 0.322920i $$0.104664\pi$$
−0.946426 + 0.322920i $$0.895336\pi$$
$$558$$ 0 0
$$559$$ 8600.00 0.650700
$$560$$ 0 0
$$561$$ −4320.00 −0.325117
$$562$$ 0 0
$$563$$ − 10284.0i − 0.769838i −0.922950 0.384919i $$-0.874229\pi$$
0.922950 0.384919i $$-0.125771\pi$$
$$564$$ 0 0
$$565$$ − 2790.00i − 0.207745i
$$566$$ 0 0
$$567$$ 4976.00 0.368558
$$568$$ 0 0
$$569$$ −1770.00 −0.130408 −0.0652041 0.997872i $$-0.520770\pi$$
−0.0652041 + 0.997872i $$0.520770\pi$$
$$570$$ 0 0
$$571$$ − 6068.00i − 0.444725i −0.974964 0.222362i $$-0.928623\pi$$
0.974964 0.222362i $$-0.0713768\pi$$
$$572$$ 0 0
$$573$$ − 192.000i − 0.0139981i
$$574$$ 0 0
$$575$$ −1200.00 −0.0870321
$$576$$ 0 0
$$577$$ 21506.0 1.55166 0.775829 0.630943i $$-0.217332\pi$$
0.775829 + 0.630943i $$0.217332\pi$$
$$578$$ 0 0
$$579$$ 4232.00i 0.303758i
$$580$$ 0 0
$$581$$ − 15168.0i − 1.08309i
$$582$$ 0 0
$$583$$ 29880.0 2.12265
$$584$$ 0 0
$$585$$ 4730.00 0.334293
$$586$$ 0 0
$$587$$ − 12108.0i − 0.851364i −0.904873 0.425682i $$-0.860034\pi$$
0.904873 0.425682i $$-0.139966\pi$$
$$588$$ 0 0
$$589$$ − 7744.00i − 0.541742i
$$590$$ 0 0
$$591$$ 14856.0 1.03400
$$592$$ 0 0
$$593$$ 15474.0 1.07157 0.535785 0.844354i $$-0.320016\pi$$
0.535785 + 0.844354i $$0.320016\pi$$
$$594$$ 0 0
$$595$$ − 1440.00i − 0.0992172i
$$596$$ 0 0
$$597$$ − 7072.00i − 0.484820i
$$598$$ 0 0
$$599$$ −2520.00 −0.171894 −0.0859469 0.996300i $$-0.527392\pi$$
−0.0859469 + 0.996300i $$0.527392\pi$$
$$600$$ 0 0
$$601$$ 12790.0 0.868078 0.434039 0.900894i $$-0.357088\pi$$
0.434039 + 0.900894i $$0.357088\pi$$
$$602$$ 0 0
$$603$$ − 11396.0i − 0.769620i
$$604$$ 0 0
$$605$$ − 11345.0i − 0.762380i
$$606$$ 0 0
$$607$$ −11576.0 −0.774062 −0.387031 0.922067i $$-0.626499\pi$$
−0.387031 + 0.922067i $$0.626499\pi$$
$$608$$ 0 0
$$609$$ 11904.0 0.792076
$$610$$ 0 0
$$611$$ 14448.0i 0.956634i
$$612$$ 0 0
$$613$$ 20126.0i 1.32607i 0.748588 + 0.663035i $$0.230732\pi$$
−0.748588 + 0.663035i $$0.769268\pi$$
$$614$$ 0 0
$$615$$ 3720.00 0.243910
$$616$$ 0 0
$$617$$ 27942.0 1.82318 0.911590 0.411100i $$-0.134855\pi$$
0.911590 + 0.411100i $$0.134855\pi$$
$$618$$ 0 0
$$619$$ 22540.0i 1.46358i 0.681528 + 0.731792i $$0.261316\pi$$
−0.681528 + 0.731792i $$0.738684\pi$$
$$620$$ 0 0
$$621$$ 7296.00i 0.471463i
$$622$$ 0 0
$$623$$ −16224.0 −1.04334
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ − 10560.0i − 0.672609i
$$628$$ 0 0
$$629$$ 4572.00i 0.289821i
$$630$$ 0 0
$$631$$ −5128.00 −0.323522 −0.161761 0.986830i $$-0.551717\pi$$
−0.161761 + 0.986830i $$0.551717\pi$$
$$632$$ 0 0
$$633$$ 16144.0 1.01369
$$634$$ 0 0
$$635$$ − 1720.00i − 0.107490i
$$636$$ 0 0
$$637$$ 7482.00i 0.465381i
$$638$$ 0 0
$$639$$ 1848.00 0.114406
$$640$$ 0 0
$$641$$ −12798.0 −0.788597 −0.394298 0.918982i $$-0.629012\pi$$
−0.394298 + 0.918982i $$0.629012\pi$$
$$642$$ 0 0
$$643$$ − 21148.0i − 1.29704i −0.761198 0.648519i $$-0.775389\pi$$
0.761198 0.648519i $$-0.224611\pi$$
$$644$$ 0 0
$$645$$ − 2000.00i − 0.122093i
$$646$$ 0 0
$$647$$ 16464.0 1.00041 0.500206 0.865906i $$-0.333258\pi$$
0.500206 + 0.865906i $$0.333258\pi$$
$$648$$ 0 0
$$649$$ −15120.0 −0.914502
$$650$$ 0 0
$$651$$ 11264.0i 0.678143i
$$652$$ 0 0
$$653$$ 24234.0i 1.45230i 0.687538 + 0.726148i $$0.258691\pi$$
−0.687538 + 0.726148i $$0.741309\pi$$
$$654$$ 0 0
$$655$$ −3900.00 −0.232650
$$656$$ 0 0
$$657$$ −5566.00 −0.330518
$$658$$ 0 0
$$659$$ − 22836.0i − 1.34987i −0.737877 0.674935i $$-0.764172\pi$$
0.737877 0.674935i $$-0.235828\pi$$
$$660$$ 0 0
$$661$$ 26318.0i 1.54864i 0.632794 + 0.774320i $$0.281908\pi$$
−0.632794 + 0.774320i $$0.718092\pi$$
$$662$$ 0 0
$$663$$ 6192.00 0.362711
$$664$$ 0 0
$$665$$ 3520.00 0.205263
$$666$$ 0 0
$$667$$ 8928.00i 0.518281i
$$668$$ 0 0
$$669$$ − 2720.00i − 0.157192i
$$670$$ 0 0
$$671$$ −3480.00 −0.200214
$$672$$ 0 0
$$673$$ 28802.0 1.64968 0.824841 0.565365i $$-0.191265\pi$$
0.824841 + 0.565365i $$0.191265\pi$$
$$674$$ 0 0
$$675$$ − 3800.00i − 0.216685i
$$676$$ 0 0
$$677$$ 2526.00i 0.143400i 0.997426 + 0.0717002i $$0.0228425\pi$$
−0.997426 + 0.0717002i $$0.977158\pi$$
$$678$$ 0 0
$$679$$ 12256.0 0.692698
$$680$$ 0 0
$$681$$ −9552.00 −0.537494
$$682$$ 0 0
$$683$$ 23076.0i 1.29279i 0.763001 + 0.646397i $$0.223725\pi$$
−0.763001 + 0.646397i $$0.776275\pi$$
$$684$$ 0 0
$$685$$ − 3330.00i − 0.185741i
$$686$$ 0 0
$$687$$ 15496.0 0.860567
$$688$$ 0 0
$$689$$ −42828.0 −2.36809
$$690$$ 0 0
$$691$$ 7868.00i 0.433159i 0.976265 + 0.216579i $$0.0694900\pi$$
−0.976265 + 0.216579i $$0.930510\pi$$
$$692$$ 0 0
$$693$$ − 10560.0i − 0.578847i
$$694$$ 0 0
$$695$$ 4420.00 0.241238
$$696$$ 0 0
$$697$$ −3348.00 −0.181943
$$698$$ 0 0
$$699$$ − 12648.0i − 0.684394i
$$700$$ 0 0
$$701$$ − 21510.0i − 1.15895i −0.814991 0.579473i $$-0.803258\pi$$
0.814991 0.579473i $$-0.196742\pi$$
$$702$$ 0 0
$$703$$ −11176.0 −0.599589
$$704$$ 0 0
$$705$$ 3360.00 0.179496
$$706$$ 0 0
$$707$$ 21024.0i 1.11837i
$$708$$ 0 0
$$709$$ 30014.0i 1.58984i 0.606712 + 0.794922i $$0.292488\pi$$
−0.606712 + 0.794922i $$0.707512\pi$$
$$710$$ 0 0
$$711$$ −2992.00 −0.157818
$$712$$ 0 0
$$713$$ −8448.00 −0.443731
$$714$$ 0 0
$$715$$ 25800.0i 1.34946i
$$716$$ 0 0
$$717$$ − 21696.0i − 1.13006i
$$718$$ 0 0
$$719$$ 816.000 0.0423250 0.0211625 0.999776i $$-0.493263\pi$$
0.0211625 + 0.999776i $$0.493263\pi$$
$$720$$ 0 0
$$721$$ 7168.00 0.370250
$$722$$ 0 0
$$723$$ − 15544.0i − 0.799568i
$$724$$ 0 0
$$725$$ − 4650.00i − 0.238202i
$$726$$ 0 0
$$727$$ −9952.00 −0.507702 −0.253851 0.967243i $$-0.581697\pi$$
−0.253851 + 0.967243i $$0.581697\pi$$
$$728$$ 0 0
$$729$$ −19837.0 −1.00782
$$730$$ 0 0
$$731$$ 1800.00i 0.0910744i
$$732$$ 0 0
$$733$$ 33946.0i 1.71054i 0.518185 + 0.855269i $$0.326608\pi$$
−0.518185 + 0.855269i $$0.673392\pi$$
$$734$$ 0 0
$$735$$ 1740.00 0.0873209
$$736$$ 0 0
$$737$$ 62160.0 3.10677
$$738$$ 0 0
$$739$$ 23420.0i 1.16579i 0.812548 + 0.582895i $$0.198080\pi$$
−0.812548 + 0.582895i $$0.801920\pi$$
$$740$$ 0 0
$$741$$ 15136.0i 0.750384i
$$742$$ 0 0
$$743$$ −14592.0 −0.720496 −0.360248 0.932857i $$-0.617308\pi$$
−0.360248 + 0.932857i $$0.617308\pi$$
$$744$$ 0 0
$$745$$ 570.000 0.0280311
$$746$$ 0 0
$$747$$ 10428.0i 0.510764i
$$748$$ 0 0
$$749$$ 24768.0i 1.20828i
$$750$$ 0 0
$$751$$ −9056.00 −0.440024 −0.220012 0.975497i $$-0.570610\pi$$
−0.220012 + 0.975497i $$0.570610\pi$$
$$752$$ 0 0
$$753$$ −20400.0 −0.987274
$$754$$ 0 0
$$755$$ − 200.000i − 0.00964072i
$$756$$ 0 0
$$757$$ − 17554.0i − 0.842815i −0.906871 0.421408i $$-0.861536\pi$$
0.906871 0.421408i $$-0.138464\pi$$
$$758$$ 0 0
$$759$$ −11520.0 −0.550922
$$760$$ 0 0
$$761$$ 36438.0 1.73571 0.867856 0.496816i $$-0.165498\pi$$
0.867856 + 0.496816i $$0.165498\pi$$
$$762$$ 0 0
$$763$$ 4448.00i 0.211046i
$$764$$ 0 0
$$765$$ 990.000i 0.0467889i
$$766$$ 0 0
$$767$$ 21672.0 1.02025
$$768$$ 0 0
$$769$$ −9022.00 −0.423071 −0.211536 0.977370i $$-0.567846\pi$$
−0.211536 + 0.977370i $$0.567846\pi$$
$$770$$ 0 0
$$771$$ 8712.00i 0.406946i
$$772$$ 0 0
$$773$$ 1470.00i 0.0683987i 0.999415 + 0.0341994i $$0.0108881\pi$$
−0.999415 + 0.0341994i $$0.989112\pi$$
$$774$$ 0 0
$$775$$ 4400.00 0.203939
$$776$$ 0 0
$$777$$ 16256.0 0.750554
$$778$$ 0 0
$$779$$ − 8184.00i − 0.376409i
$$780$$ 0 0
$$781$$ 10080.0i 0.461832i
$$782$$ 0 0
$$783$$ −28272.0 −1.29037
$$784$$ 0 0
$$785$$ −770.000 −0.0350095
$$786$$ 0 0
$$787$$ 5252.00i 0.237883i 0.992901 + 0.118941i $$0.0379500\pi$$
−0.992901 + 0.118941i $$0.962050\pi$$
$$788$$ 0 0
$$789$$ − 24576.0i − 1.10891i
$$790$$ 0 0
$$791$$ 8928.00 0.401319
$$792$$ 0 0
$$793$$ 4988.00 0.223366
$$794$$ 0 0
$$795$$ 9960.00i 0.444333i
$$796$$ 0 0
$$797$$ − 12294.0i − 0.546394i −0.961958 0.273197i $$-0.911919\pi$$
0.961958 0.273197i $$-0.0880810\pi$$
$$798$$ 0 0
$$799$$ −3024.00 −0.133894
$$800$$ 0 0
$$801$$ 11154.0 0.492019
$$802$$ 0 0
$$803$$ − 30360.0i − 1.33422i
$$804$$ 0 0
$$805$$ − 3840.00i − 0.168127i
$$806$$ 0 0
$$807$$ 3288.00 0.143424
$$808$$ 0 0
$$809$$ −15546.0 −0.675610 −0.337805 0.941216i $$-0.609684\pi$$
−0.337805 + 0.941216i $$0.609684\pi$$
$$810$$ 0 0
$$811$$ − 19364.0i − 0.838424i −0.907888 0.419212i $$-0.862306\pi$$
0.907888 0.419212i $$-0.137694\pi$$
$$812$$ 0 0
$$813$$ − 33920.0i − 1.46326i
$$814$$ 0 0
$$815$$ −10900.0 −0.468479
$$816$$ 0 0
$$817$$ −4400.00 −0.188417
$$818$$ 0 0
$$819$$ 15136.0i 0.645781i
$$820$$ 0 0
$$821$$ − 7314.00i − 0.310914i −0.987843 0.155457i $$-0.950315\pi$$
0.987843 0.155457i $$-0.0496850\pi$$
$$822$$ 0 0
$$823$$ 11984.0 0.507577 0.253789 0.967260i $$-0.418323\pi$$
0.253789 + 0.967260i $$0.418323\pi$$
$$824$$ 0 0
$$825$$ 6000.00 0.253204
$$826$$ 0 0
$$827$$ − 13500.0i − 0.567643i −0.958877 0.283822i $$-0.908398\pi$$
0.958877 0.283822i $$-0.0916024\pi$$
$$828$$ 0 0
$$829$$ 44602.0i 1.86863i 0.356453 + 0.934313i $$0.383986\pi$$
−0.356453 + 0.934313i $$0.616014\pi$$
$$830$$ 0 0
$$831$$ 4552.00 0.190021
$$832$$ 0 0
$$833$$ −1566.00 −0.0651365
$$834$$ 0 0
$$835$$ 18480.0i 0.765900i
$$836$$ 0 0
$$837$$ − 26752.0i − 1.10476i
$$838$$ 0 0
$$839$$ 35448.0 1.45864 0.729321 0.684172i $$-0.239836\pi$$
0.729321 + 0.684172i $$0.239836\pi$$
$$840$$ 0 0
$$841$$ −10207.0 −0.418508
$$842$$ 0 0
$$843$$ − 22824.0i − 0.932503i
$$844$$ 0 0
$$845$$ − 25995.0i − 1.05829i
$$846$$ 0 0
$$847$$ 36304.0 1.47275
$$848$$ 0 0
$$849$$ −12112.0 −0.489615
$$850$$ 0 0
$$851$$ 12192.0i 0.491112i
$$852$$ 0 0
$$853$$ 12590.0i 0.505362i 0.967550 + 0.252681i $$0.0813122\pi$$
−0.967550 + 0.252681i $$0.918688\pi$$
$$854$$ 0 0
$$855$$ −2420.00 −0.0967980
$$856$$ 0 0
$$857$$ −24906.0 −0.992734 −0.496367 0.868113i $$-0.665333\pi$$
−0.496367 + 0.868113i $$0.665333\pi$$
$$858$$ 0 0
$$859$$ − 23204.0i − 0.921665i −0.887487 0.460833i $$-0.847551\pi$$
0.887487 0.460833i $$-0.152449\pi$$
$$860$$ 0 0
$$861$$ 11904.0i 0.471181i
$$862$$ 0 0
$$863$$ −19848.0 −0.782890 −0.391445 0.920202i $$-0.628025\pi$$
−0.391445 + 0.920202i $$0.628025\pi$$
$$864$$ 0 0
$$865$$ 6510.00 0.255892
$$866$$ 0 0
$$867$$ − 18356.0i − 0.719034i
$$868$$ 0 0
$$869$$ − 16320.0i − 0.637075i
$$870$$ 0 0
$$871$$ −89096.0 −3.46602
$$872$$ 0 0
$$873$$ −8426.00 −0.326663
$$874$$ 0 0
$$875$$ 2000.00i 0.0772712i
$$876$$ 0 0
$$877$$ − 27542.0i − 1.06046i −0.847852 0.530232i $$-0.822105\pi$$
0.847852 0.530232i $$-0.177895\pi$$
$$878$$ 0 0
$$879$$ −13560.0 −0.520327
$$880$$ 0 0
$$881$$ −20718.0 −0.792290 −0.396145 0.918188i $$-0.629652\pi$$
−0.396145 + 0.918188i $$0.629652\pi$$
$$882$$ 0 0
$$883$$ 25172.0i 0.959349i 0.877446 + 0.479675i $$0.159245\pi$$
−0.877446 + 0.479675i $$0.840755\pi$$
$$884$$ 0 0
$$885$$ − 5040.00i − 0.191432i
$$886$$ 0 0
$$887$$ −12864.0 −0.486957 −0.243478 0.969906i $$-0.578289\pi$$
−0.243478 + 0.969906i $$0.578289\pi$$
$$888$$ 0 0
$$889$$ 5504.00 0.207647
$$890$$ 0 0
$$891$$ − 18660.0i − 0.701609i
$$892$$ 0 0
$$893$$ − 7392.00i − 0.277003i
$$894$$ 0 0
$$895$$ 21540.0 0.804472
$$896$$ 0 0
$$897$$ 16512.0 0.614626
$$898$$ 0 0
$$899$$ − 32736.0i − 1.21447i
$$900$$ 0 0
$$901$$ − 8964.00i − 0.331447i
$$902$$ 0 0
$$903$$ 6400.00 0.235857
$$904$$ 0 0
$$905$$ −7750.00 −0.284662
$$906$$ 0 0
$$907$$ 23092.0i 0.845377i 0.906275 + 0.422689i $$0.138914\pi$$
−0.906275 + 0.422689i $$0.861086\pi$$
$$908$$ 0 0
$$909$$ − 14454.0i − 0.527403i
$$910$$ 0 0
$$911$$ 14208.0 0.516720 0.258360 0.966049i $$-0.416818\pi$$
0.258360 + 0.966049i $$0.416818\pi$$
$$912$$ 0 0
$$913$$ −56880.0 −2.06183
$$914$$ 0 0
$$915$$ − 1160.00i − 0.0419108i
$$916$$ 0 0
$$917$$ − 12480.0i − 0.449428i
$$918$$ 0 0
$$919$$ −26584.0 −0.954217 −0.477108 0.878844i $$-0.658315\pi$$
−0.477108 + 0.878844i $$0.658315\pi$$
$$920$$ 0 0
$$921$$ 16624.0 0.594766
$$922$$ 0 0
$$923$$ − 14448.0i − 0.515235i
$$924$$ 0 0
$$925$$ − 6350.00i − 0.225715i
$$926$$ 0 0
$$927$$ −4928.00 −0.174603
$$928$$ 0 0
$$929$$ 162.000 0.00572126 0.00286063 0.999996i $$-0.499089\pi$$
0.00286063 + 0.999996i $$0.499089\pi$$
$$930$$ 0 0
$$931$$ − 3828.00i − 0.134756i
$$932$$ 0 0
$$933$$ 26208.0i 0.919626i
$$934$$ 0 0
$$935$$ −5400.00 −0.188876
$$936$$ 0 0
$$937$$ 29734.0 1.03668 0.518339 0.855175i $$-0.326551\pi$$
0.518339 + 0.855175i $$0.326551\pi$$
$$938$$ 0 0
$$939$$ 5464.00i 0.189894i
$$940$$ 0 0
$$941$$ − 17142.0i − 0.593850i −0.954901 0.296925i $$-0.904039\pi$$
0.954901 0.296925i $$-0.0959612\pi$$
$$942$$ 0 0
$$943$$ −8928.00 −0.308309
$$944$$ 0 0
$$945$$ 12160.0 0.418587
$$946$$ 0 0
$$947$$ 26436.0i 0.907133i 0.891223 + 0.453566i $$0.149848\pi$$
−0.891223 + 0.453566i $$0.850152\pi$$
$$948$$ 0 0
$$949$$ 43516.0i 1.48850i
$$950$$ 0 0
$$951$$ 10392.0 0.354347
$$952$$ 0 0
$$953$$ −27882.0 −0.947730 −0.473865 0.880598i $$-0.657142\pi$$
−0.473865 + 0.880598i $$0.657142\pi$$
$$954$$ 0 0
$$955$$ − 240.000i − 0.00813217i
$$956$$ 0 0
$$957$$ − 44640.0i − 1.50784i
$$958$$ 0 0
$$959$$ 10656.0 0.358811
$$960$$ 0 0
$$961$$ 1185.00 0.0397771
$$962$$ 0 0
$$963$$ − 17028.0i − 0.569802i
$$964$$ 0 0
$$965$$ 5290.00i 0.176467i
$$966$$ 0 0
$$967$$ 12656.0 0.420879 0.210439 0.977607i $$-0.432511\pi$$
0.210439 + 0.977607i $$0.432511\pi$$
$$968$$ 0 0
$$969$$ −3168.00 −0.105027
$$970$$ 0 0
$$971$$ − 2916.00i − 0.0963737i −0.998838 0.0481869i $$-0.984656\pi$$
0.998838 0.0481869i $$-0.0153443\pi$$
$$972$$ 0 0
$$973$$ 14144.0i 0.466018i
$$974$$ 0 0
$$975$$ −8600.00 −0.282482
$$976$$ 0 0
$$977$$ −6894.00 −0.225751 −0.112875 0.993609i $$-0.536006\pi$$
−0.112875 + 0.993609i $$0.536006\pi$$
$$978$$ 0 0
$$979$$ 60840.0i 1.98616i
$$980$$ 0 0
$$981$$ − 3058.00i − 0.0995254i
$$982$$ 0 0
$$983$$ −45264.0 −1.46866 −0.734332 0.678790i $$-0.762505\pi$$
−0.734332 + 0.678790i $$0.762505\pi$$
$$984$$ 0 0
$$985$$ 18570.0 0.600700
$$986$$ 0 0
$$987$$ 10752.0i 0.346748i
$$988$$ 0 0
$$989$$ 4800.00i 0.154329i
$$990$$ 0 0
$$991$$ −52016.0 −1.66735 −0.833674 0.552256i $$-0.813767\pi$$
−0.833674 + 0.552256i $$0.813767\pi$$
$$992$$ 0 0
$$993$$ −13168.0 −0.420820
$$994$$ 0 0
$$995$$ − 8840.00i − 0.281655i
$$996$$ 0 0
$$997$$ − 13858.0i − 0.440208i −0.975476 0.220104i $$-0.929360\pi$$
0.975476 0.220104i $$-0.0706397\pi$$
$$998$$ 0 0
$$999$$ −38608.0 −1.22273
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.c.641.2 2
4.3 odd 2 1280.4.d.n.641.1 2
8.3 odd 2 1280.4.d.n.641.2 2
8.5 even 2 inner 1280.4.d.c.641.1 2
16.3 odd 4 20.4.a.a.1.1 1
16.5 even 4 320.4.a.k.1.1 1
16.11 odd 4 320.4.a.d.1.1 1
16.13 even 4 80.4.a.c.1.1 1
48.29 odd 4 720.4.a.k.1.1 1
48.35 even 4 180.4.a.a.1.1 1
80.3 even 4 100.4.c.a.49.2 2
80.13 odd 4 400.4.c.j.49.1 2
80.19 odd 4 100.4.a.a.1.1 1
80.29 even 4 400.4.a.o.1.1 1
80.59 odd 4 1600.4.a.bl.1.1 1
80.67 even 4 100.4.c.a.49.1 2
80.69 even 4 1600.4.a.p.1.1 1
80.77 odd 4 400.4.c.j.49.2 2
112.3 even 12 980.4.i.n.961.1 2
112.19 even 12 980.4.i.n.361.1 2
112.51 odd 12 980.4.i.e.361.1 2
112.67 odd 12 980.4.i.e.961.1 2
112.83 even 4 980.4.a.c.1.1 1
144.67 odd 12 1620.4.i.d.541.1 2
144.83 even 12 1620.4.i.j.1081.1 2
144.115 odd 12 1620.4.i.d.1081.1 2
144.131 even 12 1620.4.i.j.541.1 2
176.131 even 4 2420.4.a.d.1.1 1
240.83 odd 4 900.4.d.k.649.2 2
240.179 even 4 900.4.a.m.1.1 1
240.227 odd 4 900.4.d.k.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
20.4.a.a.1.1 1 16.3 odd 4
80.4.a.c.1.1 1 16.13 even 4
100.4.a.a.1.1 1 80.19 odd 4
100.4.c.a.49.1 2 80.67 even 4
100.4.c.a.49.2 2 80.3 even 4
180.4.a.a.1.1 1 48.35 even 4
320.4.a.d.1.1 1 16.11 odd 4
320.4.a.k.1.1 1 16.5 even 4
400.4.a.o.1.1 1 80.29 even 4
400.4.c.j.49.1 2 80.13 odd 4
400.4.c.j.49.2 2 80.77 odd 4
720.4.a.k.1.1 1 48.29 odd 4
900.4.a.m.1.1 1 240.179 even 4
900.4.d.k.649.1 2 240.227 odd 4
900.4.d.k.649.2 2 240.83 odd 4
980.4.a.c.1.1 1 112.83 even 4
980.4.i.e.361.1 2 112.51 odd 12
980.4.i.e.961.1 2 112.67 odd 12
980.4.i.n.361.1 2 112.19 even 12
980.4.i.n.961.1 2 112.3 even 12
1280.4.d.c.641.1 2 8.5 even 2 inner
1280.4.d.c.641.2 2 1.1 even 1 trivial
1280.4.d.n.641.1 2 4.3 odd 2
1280.4.d.n.641.2 2 8.3 odd 2
1600.4.a.p.1.1 1 80.69 even 4
1600.4.a.bl.1.1 1 80.59 odd 4
1620.4.i.d.541.1 2 144.67 odd 12
1620.4.i.d.1081.1 2 144.115 odd 12
1620.4.i.j.541.1 2 144.131 even 12
1620.4.i.j.1081.1 2 144.83 even 12
2420.4.a.d.1.1 1 176.131 even 4