Properties

 Label 1152.4.d.p.577.2 Level $1152$ Weight $4$ Character 1152.577 Analytic conductor $67.970$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(577,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1152.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$67.9702003266$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1534132224.8 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1$$ x^8 + 18*x^6 + 107*x^4 + 210*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{26}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 577.2 Root $$0.0690906i$$ of defining polynomial Character $$\chi$$ $$=$$ 1152.577 Dual form 1152.4.d.p.577.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-17.4288i q^{5} +2.99032 q^{7} +O(q^{10})$$ $$q-17.4288i q^{5} +2.99032 q^{7} -10.6274i q^{11} +43.3156i q^{13} +37.8823 q^{17} +79.8823i q^{19} -191.204 q^{23} -178.765 q^{25} +138.918i q^{29} -212.136 q^{31} -52.1177i q^{35} -270.404i q^{37} +441.411 q^{41} -64.1177i q^{43} -436.234 q^{47} -334.058 q^{49} -278.348i q^{53} -185.224 q^{55} +830.039i q^{59} +724.580i q^{61} +754.940 q^{65} +859.529i q^{67} +681.264 q^{71} -785.058 q^{73} -31.7793i q^{77} +1018.82 q^{79} -467.334i q^{83} -660.244i q^{85} -510.706 q^{89} +129.527i q^{91} +1392.26 q^{95} -234.235 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 240 q^{17} - 344 q^{25} + 816 q^{41} + 1672 q^{49} + 1152 q^{65} - 1936 q^{73} - 7344 q^{89} - 2960 q^{97}+O(q^{100})$$ 8 * q - 240 * q^17 - 344 * q^25 + 816 * q^41 + 1672 * q^49 + 1152 * q^65 - 1936 * q^73 - 7344 * q^89 - 2960 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 17.4288i − 1.55888i −0.626475 0.779441i $$-0.715503\pi$$
0.626475 0.779441i $$-0.284497\pi$$
$$6$$ 0 0
$$7$$ 2.99032 0.161462 0.0807310 0.996736i $$-0.474275\pi$$
0.0807310 + 0.996736i $$0.474275\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 10.6274i − 0.291299i −0.989336 0.145649i $$-0.953473\pi$$
0.989336 0.145649i $$-0.0465271\pi$$
$$12$$ 0 0
$$13$$ 43.3156i 0.924121i 0.886848 + 0.462061i $$0.152890\pi$$
−0.886848 + 0.462061i $$0.847110\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 37.8823 0.540459 0.270229 0.962796i $$-0.412900\pi$$
0.270229 + 0.962796i $$0.412900\pi$$
$$18$$ 0 0
$$19$$ 79.8823i 0.964539i 0.876023 + 0.482270i $$0.160187\pi$$
−0.876023 + 0.482270i $$0.839813\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −191.204 −1.73343 −0.866714 0.498806i $$-0.833772\pi$$
−0.866714 + 0.498806i $$0.833772\pi$$
$$24$$ 0 0
$$25$$ −178.765 −1.43012
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 138.918i 0.889530i 0.895647 + 0.444765i $$0.146713\pi$$
−0.895647 + 0.444765i $$0.853287\pi$$
$$30$$ 0 0
$$31$$ −212.136 −1.22906 −0.614529 0.788894i $$-0.710654\pi$$
−0.614529 + 0.788894i $$0.710654\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 52.1177i − 0.251700i
$$36$$ 0 0
$$37$$ − 270.404i − 1.20146i −0.799451 0.600731i $$-0.794876\pi$$
0.799451 0.600731i $$-0.205124\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 441.411 1.68139 0.840693 0.541511i $$-0.182148\pi$$
0.840693 + 0.541511i $$0.182148\pi$$
$$42$$ 0 0
$$43$$ − 64.1177i − 0.227392i −0.993516 0.113696i $$-0.963731\pi$$
0.993516 0.113696i $$-0.0362690\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −436.234 −1.35386 −0.676929 0.736049i $$-0.736689\pi$$
−0.676929 + 0.736049i $$0.736689\pi$$
$$48$$ 0 0
$$49$$ −334.058 −0.973930
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 278.348i − 0.721398i −0.932682 0.360699i $$-0.882538\pi$$
0.932682 0.360699i $$-0.117462\pi$$
$$54$$ 0 0
$$55$$ −185.224 −0.454101
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 830.039i 1.83156i 0.401684 + 0.915778i $$0.368425\pi$$
−0.401684 + 0.915778i $$0.631575\pi$$
$$60$$ 0 0
$$61$$ 724.580i 1.52087i 0.649416 + 0.760434i $$0.275014\pi$$
−0.649416 + 0.760434i $$0.724986\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 754.940 1.44060
$$66$$ 0 0
$$67$$ 859.529i 1.56729i 0.621211 + 0.783643i $$0.286641\pi$$
−0.621211 + 0.783643i $$0.713359\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 681.264 1.13875 0.569374 0.822078i $$-0.307186\pi$$
0.569374 + 0.822078i $$0.307186\pi$$
$$72$$ 0 0
$$73$$ −785.058 −1.25869 −0.629343 0.777128i $$-0.716676\pi$$
−0.629343 + 0.777128i $$0.716676\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 31.7793i − 0.0470337i
$$78$$ 0 0
$$79$$ 1018.82 1.45096 0.725481 0.688243i $$-0.241618\pi$$
0.725481 + 0.688243i $$0.241618\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 467.334i − 0.618031i −0.951057 0.309015i $$-0.900001\pi$$
0.951057 0.309015i $$-0.0999995\pi$$
$$84$$ 0 0
$$85$$ − 660.244i − 0.842512i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −510.706 −0.608256 −0.304128 0.952631i $$-0.598365\pi$$
−0.304128 + 0.952631i $$0.598365\pi$$
$$90$$ 0 0
$$91$$ 129.527i 0.149210i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1392.26 1.50360
$$96$$ 0 0
$$97$$ −234.235 −0.245186 −0.122593 0.992457i $$-0.539121\pi$$
−0.122593 + 0.992457i $$0.539121\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 205.555i 0.202509i 0.994861 + 0.101255i $$0.0322857\pi$$
−0.994861 + 0.101255i $$0.967714\pi$$
$$102$$ 0 0
$$103$$ 391.379 0.374405 0.187203 0.982321i $$-0.440058\pi$$
0.187203 + 0.982321i $$0.440058\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 934.274i 0.844109i 0.906570 + 0.422055i $$0.138691\pi$$
−0.906570 + 0.422055i $$0.861309\pi$$
$$108$$ 0 0
$$109$$ − 584.123i − 0.513292i −0.966505 0.256646i $$-0.917383\pi$$
0.966505 0.256646i $$-0.0826174\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 582.706 0.485101 0.242551 0.970139i $$-0.422016\pi$$
0.242551 + 0.970139i $$0.422016\pi$$
$$114$$ 0 0
$$115$$ 3332.47i 2.70221i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 113.280 0.0872635
$$120$$ 0 0
$$121$$ 1218.06 0.915145
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 937.053i 0.670501i
$$126$$ 0 0
$$127$$ −1461.03 −1.02083 −0.510416 0.859928i $$-0.670509\pi$$
−0.510416 + 0.859928i $$0.670509\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 98.4323i − 0.0656494i −0.999461 0.0328247i $$-0.989550\pi$$
0.999461 0.0328247i $$-0.0104503\pi$$
$$132$$ 0 0
$$133$$ 238.873i 0.155736i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2171.06 −1.35391 −0.676956 0.736024i $$-0.736701\pi$$
−0.676956 + 0.736024i $$0.736701\pi$$
$$138$$ 0 0
$$139$$ 1624.70i 0.991407i 0.868492 + 0.495703i $$0.165090\pi$$
−0.868492 + 0.495703i $$0.834910\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 460.333 0.269195
$$144$$ 0 0
$$145$$ 2421.17 1.38667
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 636.658i − 0.350047i −0.984564 0.175024i $$-0.944000\pi$$
0.984564 0.175024i $$-0.0560002\pi$$
$$150$$ 0 0
$$151$$ 1819.34 0.980503 0.490252 0.871581i $$-0.336905\pi$$
0.490252 + 0.871581i $$0.336905\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3697.29i 1.91596i
$$156$$ 0 0
$$157$$ 1656.50i 0.842059i 0.907047 + 0.421029i $$0.138331\pi$$
−0.907047 + 0.421029i $$0.861669\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −571.761 −0.279882
$$162$$ 0 0
$$163$$ 2228.82i 1.07101i 0.844532 + 0.535505i $$0.179879\pi$$
−0.844532 + 0.535505i $$0.820121\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1667.01 −0.772439 −0.386219 0.922407i $$-0.626219\pi$$
−0.386219 + 0.922407i $$0.626219\pi$$
$$168$$ 0 0
$$169$$ 320.761 0.146000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 2500.53i − 1.09891i −0.835522 0.549457i $$-0.814835\pi$$
0.835522 0.549457i $$-0.185165\pi$$
$$174$$ 0 0
$$175$$ −534.562 −0.230909
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 378.742i 0.158148i 0.996869 + 0.0790740i $$0.0251963\pi$$
−0.996869 + 0.0790740i $$0.974804\pi$$
$$180$$ 0 0
$$181$$ 3093.88i 1.27053i 0.772294 + 0.635265i $$0.219109\pi$$
−0.772294 + 0.635265i $$0.780891\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −4712.82 −1.87294
$$186$$ 0 0
$$187$$ − 402.590i − 0.157435i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3656.98 −1.38539 −0.692695 0.721230i $$-0.743577\pi$$
−0.692695 + 0.721230i $$0.743577\pi$$
$$192$$ 0 0
$$193$$ 2788.12 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1147.74i 0.415091i 0.978225 + 0.207546i $$0.0665475\pi$$
−0.978225 + 0.207546i $$0.933452\pi$$
$$198$$ 0 0
$$199$$ −4842.73 −1.72508 −0.862542 0.505986i $$-0.831129\pi$$
−0.862542 + 0.505986i $$0.831129\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 415.408i 0.143625i
$$204$$ 0 0
$$205$$ − 7693.29i − 2.62109i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 848.942 0.280969
$$210$$ 0 0
$$211$$ 3222.35i 1.05135i 0.850684 + 0.525677i $$0.176188\pi$$
−0.850684 + 0.525677i $$0.823812\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1117.50 −0.354478
$$216$$ 0 0
$$217$$ −634.355 −0.198446
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1640.89i 0.499449i
$$222$$ 0 0
$$223$$ 4932.61 1.48122 0.740610 0.671935i $$-0.234537\pi$$
0.740610 + 0.671935i $$0.234537\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3619.49i 1.05830i 0.848529 + 0.529150i $$0.177489\pi$$
−0.848529 + 0.529150i $$0.822511\pi$$
$$228$$ 0 0
$$229$$ − 305.759i − 0.0882320i −0.999026 0.0441160i $$-0.985953\pi$$
0.999026 0.0441160i $$-0.0140471\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −639.648 −0.179849 −0.0899244 0.995949i $$-0.528663\pi$$
−0.0899244 + 0.995949i $$0.528663\pi$$
$$234$$ 0 0
$$235$$ 7603.05i 2.11050i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1744.94 −0.472262 −0.236131 0.971721i $$-0.575879\pi$$
−0.236131 + 0.971721i $$0.575879\pi$$
$$240$$ 0 0
$$241$$ 3357.29 0.897354 0.448677 0.893694i $$-0.351895\pi$$
0.448677 + 0.893694i $$0.351895\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 5822.24i 1.51824i
$$246$$ 0 0
$$247$$ −3460.15 −0.891351
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 1317.45i − 0.331301i −0.986185 0.165650i $$-0.947028\pi$$
0.986185 0.165650i $$-0.0529723\pi$$
$$252$$ 0 0
$$253$$ 2032.01i 0.504945i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3036.12 0.736917 0.368459 0.929644i $$-0.379886\pi$$
0.368459 + 0.929644i $$0.379886\pi$$
$$258$$ 0 0
$$259$$ − 808.592i − 0.193990i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1655.76 0.388206 0.194103 0.980981i $$-0.437820\pi$$
0.194103 + 0.980981i $$0.437820\pi$$
$$264$$ 0 0
$$265$$ −4851.29 −1.12458
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 5292.72i − 1.19964i −0.800135 0.599820i $$-0.795239\pi$$
0.800135 0.599820i $$-0.204761\pi$$
$$270$$ 0 0
$$271$$ −8010.52 −1.79559 −0.897795 0.440414i $$-0.854832\pi$$
−0.897795 + 0.440414i $$0.854832\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1899.80i 0.416591i
$$276$$ 0 0
$$277$$ 5692.81i 1.23483i 0.786638 + 0.617415i $$0.211820\pi$$
−0.786638 + 0.617415i $$0.788180\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2024.83 −0.429861 −0.214931 0.976629i $$-0.568953\pi$$
−0.214931 + 0.976629i $$0.568953\pi$$
$$282$$ 0 0
$$283$$ 247.761i 0.0520419i 0.999661 + 0.0260210i $$0.00828366\pi$$
−0.999661 + 0.0260210i $$0.991716\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1319.96 0.271480
$$288$$ 0 0
$$289$$ −3477.94 −0.707905
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 8133.61i − 1.62174i −0.585225 0.810871i $$-0.698994\pi$$
0.585225 0.810871i $$-0.301006\pi$$
$$294$$ 0 0
$$295$$ 14466.6 2.85518
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 8282.12i − 1.60190i
$$300$$ 0 0
$$301$$ − 191.732i − 0.0367152i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12628.6 2.37085
$$306$$ 0 0
$$307$$ − 2974.82i − 0.553035i −0.961009 0.276518i $$-0.910820\pi$$
0.961009 0.276518i $$-0.0891805\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4451.52 0.811648 0.405824 0.913951i $$-0.366985\pi$$
0.405824 + 0.913951i $$0.366985\pi$$
$$312$$ 0 0
$$313$$ −8273.75 −1.49412 −0.747061 0.664755i $$-0.768536\pi$$
−0.747061 + 0.664755i $$0.768536\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 429.036i 0.0760160i 0.999277 + 0.0380080i $$0.0121012\pi$$
−0.999277 + 0.0380080i $$0.987899\pi$$
$$318$$ 0 0
$$319$$ 1476.34 0.259119
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3026.12i 0.521293i
$$324$$ 0 0
$$325$$ − 7743.29i − 1.32160i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −1304.48 −0.218596
$$330$$ 0 0
$$331$$ 8196.71i 1.36112i 0.732691 + 0.680562i $$0.238264\pi$$
−0.732691 + 0.680562i $$0.761736\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 14980.6 2.44322
$$336$$ 0 0
$$337$$ −2000.35 −0.323341 −0.161670 0.986845i $$-0.551688\pi$$
−0.161670 + 0.986845i $$0.551688\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2254.46i 0.358023i
$$342$$ 0 0
$$343$$ −2024.62 −0.318715
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7707.48i 1.19239i 0.802840 + 0.596195i $$0.203321\pi$$
−0.802840 + 0.596195i $$0.796679\pi$$
$$348$$ 0 0
$$349$$ 9681.98i 1.48500i 0.669847 + 0.742499i $$0.266360\pi$$
−0.669847 + 0.742499i $$0.733640\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 10540.3 1.58925 0.794626 0.607099i $$-0.207667\pi$$
0.794626 + 0.607099i $$0.207667\pi$$
$$354$$ 0 0
$$355$$ − 11873.6i − 1.77518i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −514.158 −0.0755884 −0.0377942 0.999286i $$-0.512033\pi$$
−0.0377942 + 0.999286i $$0.512033\pi$$
$$360$$ 0 0
$$361$$ 477.826 0.0696641
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 13682.7i 1.96214i
$$366$$ 0 0
$$367$$ 11272.4 1.60331 0.801657 0.597785i $$-0.203952\pi$$
0.801657 + 0.597785i $$0.203952\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 832.350i − 0.116478i
$$372$$ 0 0
$$373$$ − 6956.92i − 0.965726i −0.875696 0.482863i $$-0.839597\pi$$
0.875696 0.482863i $$-0.160403\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6017.30 −0.822034
$$378$$ 0 0
$$379$$ − 10201.3i − 1.38260i −0.722569 0.691299i $$-0.757039\pi$$
0.722569 0.691299i $$-0.242961\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2461.56 −0.328406 −0.164203 0.986427i $$-0.552505\pi$$
−0.164203 + 0.986427i $$0.552505\pi$$
$$384$$ 0 0
$$385$$ −553.877 −0.0733200
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 546.451i 0.0712240i 0.999366 + 0.0356120i $$0.0113381\pi$$
−0.999366 + 0.0356120i $$0.988662\pi$$
$$390$$ 0 0
$$391$$ −7243.25 −0.936846
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 17756.8i − 2.26188i
$$396$$ 0 0
$$397$$ − 2084.56i − 0.263529i −0.991281 0.131764i $$-0.957936\pi$$
0.991281 0.131764i $$-0.0420642\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 9710.59 1.20929 0.604643 0.796497i $$-0.293316\pi$$
0.604643 + 0.796497i $$0.293316\pi$$
$$402$$ 0 0
$$403$$ − 9188.81i − 1.13580i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2873.69 −0.349984
$$408$$ 0 0
$$409$$ −6659.89 −0.805160 −0.402580 0.915385i $$-0.631886\pi$$
−0.402580 + 0.915385i $$0.631886\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 2482.08i 0.295727i
$$414$$ 0 0
$$415$$ −8145.09 −0.963438
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 10576.7i 1.23318i 0.787283 + 0.616592i $$0.211487\pi$$
−0.787283 + 0.616592i $$0.788513\pi$$
$$420$$ 0 0
$$421$$ − 4871.09i − 0.563901i −0.959429 0.281951i $$-0.909019\pi$$
0.959429 0.281951i $$-0.0909814\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −6772.00 −0.772918
$$426$$ 0 0
$$427$$ 2166.72i 0.245562i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −16916.7 −1.89060 −0.945302 0.326196i $$-0.894233\pi$$
−0.945302 + 0.326196i $$0.894233\pi$$
$$432$$ 0 0
$$433$$ −1163.88 −0.129174 −0.0645870 0.997912i $$-0.520573\pi$$
−0.0645870 + 0.997912i $$0.520573\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 15273.8i − 1.67196i
$$438$$ 0 0
$$439$$ −1856.28 −0.201812 −0.100906 0.994896i $$-0.532174\pi$$
−0.100906 + 0.994896i $$0.532174\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1472.21i 0.157893i 0.996879 + 0.0789465i $$0.0251556\pi$$
−0.996879 + 0.0789465i $$0.974844\pi$$
$$444$$ 0 0
$$445$$ 8901.02i 0.948200i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9620.94 −1.01123 −0.505613 0.862761i $$-0.668733\pi$$
−0.505613 + 0.862761i $$0.668733\pi$$
$$450$$ 0 0
$$451$$ − 4691.06i − 0.489786i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2257.51 0.232602
$$456$$ 0 0
$$457$$ 3613.53 0.369877 0.184938 0.982750i $$-0.440791\pi$$
0.184938 + 0.982750i $$0.440791\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 17710.7i − 1.78931i −0.446759 0.894654i $$-0.647422\pi$$
0.446759 0.894654i $$-0.352578\pi$$
$$462$$ 0 0
$$463$$ −1674.57 −0.168087 −0.0840433 0.996462i $$-0.526783\pi$$
−0.0840433 + 0.996462i $$0.526783\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 15208.3i − 1.50697i −0.657466 0.753484i $$-0.728372\pi$$
0.657466 0.753484i $$-0.271628\pi$$
$$468$$ 0 0
$$469$$ 2570.26i 0.253057i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −681.406 −0.0662391
$$474$$ 0 0
$$475$$ − 14280.1i − 1.37940i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −6458.37 −0.616056 −0.308028 0.951377i $$-0.599669\pi$$
−0.308028 + 0.951377i $$0.599669\pi$$
$$480$$ 0 0
$$481$$ 11712.7 1.11030
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4082.45i 0.382216i
$$486$$ 0 0
$$487$$ −11337.7 −1.05495 −0.527474 0.849571i $$-0.676861\pi$$
−0.527474 + 0.849571i $$0.676861\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 14946.7i − 1.37380i −0.726754 0.686898i $$-0.758972\pi$$
0.726754 0.686898i $$-0.241028\pi$$
$$492$$ 0 0
$$493$$ 5262.51i 0.480754i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2037.19 0.183865
$$498$$ 0 0
$$499$$ 2631.77i 0.236101i 0.993008 + 0.118051i $$0.0376645\pi$$
−0.993008 + 0.118051i $$0.962336\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6907.45 −0.612302 −0.306151 0.951983i $$-0.599041\pi$$
−0.306151 + 0.951983i $$0.599041\pi$$
$$504$$ 0 0
$$505$$ 3582.58 0.315689
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 12020.3i 1.04674i 0.852107 + 0.523368i $$0.175325\pi$$
−0.852107 + 0.523368i $$0.824675\pi$$
$$510$$ 0 0
$$511$$ −2347.57 −0.203230
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 6821.29i − 0.583654i
$$516$$ 0 0
$$517$$ 4636.04i 0.394377i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15846.1 −1.33249 −0.666247 0.745731i $$-0.732101\pi$$
−0.666247 + 0.745731i $$0.732101\pi$$
$$522$$ 0 0
$$523$$ − 8891.64i − 0.743411i −0.928351 0.371706i $$-0.878773\pi$$
0.928351 0.371706i $$-0.121227\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8036.20 −0.664255
$$528$$ 0 0
$$529$$ 24392.0 2.00477
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 19120.0i 1.55381i
$$534$$ 0 0
$$535$$ 16283.3 1.31587
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3550.17i 0.283705i
$$540$$ 0 0
$$541$$ − 12833.5i − 1.01988i −0.860210 0.509940i $$-0.829667\pi$$
0.860210 0.509940i $$-0.170333\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −10180.6 −0.800162
$$546$$ 0 0
$$547$$ − 16257.0i − 1.27075i −0.772204 0.635375i $$-0.780845\pi$$
0.772204 0.635375i $$-0.219155\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −11097.1 −0.857986
$$552$$ 0 0
$$553$$ 3046.59 0.234275
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1558.32i 0.118542i 0.998242 + 0.0592712i $$0.0188777\pi$$
−0.998242 + 0.0592712i $$0.981122\pi$$
$$558$$ 0 0
$$559$$ 2777.30 0.210138
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 9782.16i − 0.732272i −0.930561 0.366136i $$-0.880681\pi$$
0.930561 0.366136i $$-0.119319\pi$$
$$564$$ 0 0
$$565$$ − 10155.9i − 0.756216i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7887.05 −0.581094 −0.290547 0.956861i $$-0.593837\pi$$
−0.290547 + 0.956861i $$0.593837\pi$$
$$570$$ 0 0
$$571$$ − 21819.3i − 1.59914i −0.600573 0.799570i $$-0.705061\pi$$
0.600573 0.799570i $$-0.294939\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 34180.5 2.47900
$$576$$ 0 0
$$577$$ 7190.22 0.518774 0.259387 0.965773i $$-0.416479\pi$$
0.259387 + 0.965773i $$0.416479\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 1397.48i − 0.0997884i
$$582$$ 0 0
$$583$$ −2958.12 −0.210142
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 13305.6i 0.935569i 0.883843 + 0.467785i $$0.154948\pi$$
−0.883843 + 0.467785i $$0.845052\pi$$
$$588$$ 0 0
$$589$$ − 16945.9i − 1.18548i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 8062.23 0.558307 0.279153 0.960246i $$-0.409946\pi$$
0.279153 + 0.960246i $$0.409946\pi$$
$$594$$ 0 0
$$595$$ − 1974.34i − 0.136034i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −2185.21 −0.149058 −0.0745288 0.997219i $$-0.523745\pi$$
−0.0745288 + 0.997219i $$0.523745\pi$$
$$600$$ 0 0
$$601$$ −3542.25 −0.240418 −0.120209 0.992749i $$-0.538357\pi$$
−0.120209 + 0.992749i $$0.538357\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 21229.3i − 1.42660i
$$606$$ 0 0
$$607$$ −6050.64 −0.404593 −0.202296 0.979324i $$-0.564840\pi$$
−0.202296 + 0.979324i $$0.564840\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 18895.7i − 1.25113i
$$612$$ 0 0
$$613$$ 22514.2i 1.48343i 0.670717 + 0.741713i $$0.265986\pi$$
−0.670717 + 0.741713i $$0.734014\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −4255.88 −0.277691 −0.138845 0.990314i $$-0.544339\pi$$
−0.138845 + 0.990314i $$0.544339\pi$$
$$618$$ 0 0
$$619$$ 228.949i 0.0148663i 0.999972 + 0.00743315i $$0.00236607\pi$$
−0.999972 + 0.00743315i $$0.997634\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −1527.17 −0.0982102
$$624$$ 0 0
$$625$$ −6013.82 −0.384884
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 10243.5i − 0.649340i
$$630$$ 0 0
$$631$$ 11429.2 0.721058 0.360529 0.932748i $$-0.382596\pi$$
0.360529 + 0.932748i $$0.382596\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 25464.1i 1.59136i
$$636$$ 0 0
$$637$$ − 14469.9i − 0.900030i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −29381.4 −1.81045 −0.905223 0.424938i $$-0.860296\pi$$
−0.905223 + 0.424938i $$0.860296\pi$$
$$642$$ 0 0
$$643$$ 249.316i 0.0152909i 0.999971 + 0.00764546i $$0.00243365\pi$$
−0.999971 + 0.00764546i $$0.997566\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16025.8 0.973785 0.486893 0.873462i $$-0.338130\pi$$
0.486893 + 0.873462i $$0.338130\pi$$
$$648$$ 0 0
$$649$$ 8821.17 0.533530
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14008.6i 0.839511i 0.907637 + 0.419755i $$0.137884\pi$$
−0.907637 + 0.419755i $$0.862116\pi$$
$$654$$ 0 0
$$655$$ −1715.56 −0.102340
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1011.91i 0.0598152i 0.999553 + 0.0299076i $$0.00952131\pi$$
−0.999553 + 0.0299076i $$0.990479\pi$$
$$660$$ 0 0
$$661$$ 23619.4i 1.38985i 0.719084 + 0.694923i $$0.244561\pi$$
−0.719084 + 0.694923i $$0.755439\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4163.28 0.242775
$$666$$ 0 0
$$667$$ − 26561.6i − 1.54194i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 7700.41 0.443027
$$672$$ 0 0
$$673$$ −25811.9 −1.47842 −0.739208 0.673477i $$-0.764800\pi$$
−0.739208 + 0.673477i $$0.764800\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18255.2i 1.03634i 0.855277 + 0.518172i $$0.173387\pi$$
−0.855277 + 0.518172i $$0.826613\pi$$
$$678$$ 0 0
$$679$$ −700.438 −0.0395881
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 20090.4i − 1.12553i −0.826617 0.562765i $$-0.809738\pi$$
0.826617 0.562765i $$-0.190262\pi$$
$$684$$ 0 0
$$685$$ 37839.0i 2.11059i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 12056.8 0.666659
$$690$$ 0 0
$$691$$ 16521.5i 0.909563i 0.890603 + 0.454782i $$0.150283\pi$$
−0.890603 + 0.454782i $$0.849717\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 28316.7 1.54549
$$696$$ 0 0
$$697$$ 16721.7 0.908720
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 12431.4i − 0.669795i −0.942255 0.334897i $$-0.891298\pi$$
0.942255 0.334897i $$-0.108702\pi$$
$$702$$ 0 0
$$703$$ 21600.4 1.15886
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 614.674i 0.0326976i
$$708$$ 0 0
$$709$$ 980.957i 0.0519614i 0.999662 + 0.0259807i $$0.00827084\pi$$
−0.999662 + 0.0259807i $$0.991729\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 40561.4 2.13048
$$714$$ 0 0
$$715$$ − 8023.06i − 0.419644i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −4115.73 −0.213478 −0.106739 0.994287i $$-0.534041\pi$$
−0.106739 + 0.994287i $$0.534041\pi$$
$$720$$ 0 0
$$721$$ 1170.35 0.0604522
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 24833.5i − 1.27213i
$$726$$ 0 0
$$727$$ −20850.1 −1.06367 −0.531833 0.846849i $$-0.678497\pi$$
−0.531833 + 0.846849i $$0.678497\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 2428.92i − 0.122896i
$$732$$ 0 0
$$733$$ − 31517.2i − 1.58815i −0.607821 0.794074i $$-0.707956\pi$$
0.607821 0.794074i $$-0.292044\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 9134.57 0.456549
$$738$$ 0 0
$$739$$ 11415.0i 0.568213i 0.958793 + 0.284106i $$0.0916969\pi$$
−0.958793 + 0.284106i $$0.908303\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 5732.08 0.283028 0.141514 0.989936i $$-0.454803\pi$$
0.141514 + 0.989936i $$0.454803\pi$$
$$744$$ 0 0
$$745$$ −11096.2 −0.545683
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 2793.78i 0.136291i
$$750$$ 0 0
$$751$$ 7843.07 0.381089 0.190544 0.981679i $$-0.438975\pi$$
0.190544 + 0.981679i $$0.438975\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 31709.0i − 1.52849i
$$756$$ 0 0
$$757$$ − 29125.9i − 1.39841i −0.714920 0.699206i $$-0.753537\pi$$
0.714920 0.699206i $$-0.246463\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14228.5 −0.677768 −0.338884 0.940828i $$-0.610049\pi$$
−0.338884 + 0.940828i $$0.610049\pi$$
$$762$$ 0 0
$$763$$ − 1746.71i − 0.0828771i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −35953.6 −1.69258
$$768$$ 0 0
$$769$$ −28133.7 −1.31928 −0.659641 0.751581i $$-0.729292\pi$$
−0.659641 + 0.751581i $$0.729292\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14686.9i 0.683377i 0.939813 + 0.341689i $$0.110999\pi$$
−0.939813 + 0.341689i $$0.889001\pi$$
$$774$$ 0 0
$$775$$ 37922.5 1.75770
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 35260.9i 1.62176i
$$780$$ 0 0
$$781$$ − 7240.08i − 0.331716i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 28870.9 1.31267
$$786$$ 0 0
$$787$$ 21001.0i 0.951214i 0.879658 + 0.475607i $$0.157772\pi$$
−0.879658 + 0.475607i $$0.842228\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1742.48 0.0783253
$$792$$ 0 0
$$793$$ −31385.6 −1.40547
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 13362.7i 0.593892i 0.954894 + 0.296946i $$0.0959682\pi$$
−0.954894 + 0.296946i $$0.904032\pi$$
$$798$$ 0 0
$$799$$ −16525.5 −0.731704
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8343.14i 0.366654i
$$804$$ 0 0
$$805$$ 9965.13i 0.436304i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 39481.6 1.71582 0.857911 0.513798i $$-0.171762\pi$$
0.857911 + 0.513798i $$0.171762\pi$$
$$810$$ 0 0
$$811$$ 31157.5i 1.34906i 0.738247 + 0.674531i $$0.235654\pi$$
−0.738247 + 0.674531i $$0.764346\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 38845.8 1.66958
$$816$$ 0 0
$$817$$ 5121.87 0.219329
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 21229.9i 0.902470i 0.892405 + 0.451235i $$0.149016\pi$$
−0.892405 + 0.451235i $$0.850984\pi$$
$$822$$ 0 0
$$823$$ −24603.8 −1.04208 −0.521041 0.853532i $$-0.674456\pi$$
−0.521041 + 0.853532i $$0.674456\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 13668.1i − 0.574710i −0.957824 0.287355i $$-0.907224\pi$$
0.957824 0.287355i $$-0.0927759\pi$$
$$828$$ 0 0
$$829$$ − 27518.8i − 1.15291i −0.817127 0.576457i $$-0.804435\pi$$
0.817127 0.576457i $$-0.195565\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −12654.9 −0.526369
$$834$$ 0 0
$$835$$ 29054.1i 1.20414i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −29951.5 −1.23247 −0.616234 0.787563i $$-0.711343\pi$$
−0.616234 + 0.787563i $$0.711343\pi$$
$$840$$ 0 0
$$841$$ 5090.88 0.208737
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 5590.49i − 0.227596i
$$846$$ 0 0
$$847$$ 3642.38 0.147761
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 51702.3i 2.08265i
$$852$$ 0 0
$$853$$ 5174.61i 0.207708i 0.994593 + 0.103854i $$0.0331175\pi$$
−0.994593 + 0.103854i $$0.966882\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −9258.34 −0.369030 −0.184515 0.982830i $$-0.559071\pi$$
−0.184515 + 0.982830i $$0.559071\pi$$
$$858$$ 0 0
$$859$$ 24353.0i 0.967304i 0.875260 + 0.483652i $$0.160690\pi$$
−0.875260 + 0.483652i $$0.839310\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −42283.4 −1.66784 −0.833919 0.551887i $$-0.813908\pi$$
−0.833919 + 0.551887i $$0.813908\pi$$
$$864$$ 0 0
$$865$$ −43581.4 −1.71308
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 10827.4i − 0.422663i
$$870$$ 0 0
$$871$$ −37231.0 −1.44836
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2802.08i 0.108260i
$$876$$ 0 0
$$877$$ 49843.1i 1.91914i 0.281476 + 0.959568i $$0.409176\pi$$
−0.281476 + 0.959568i $$0.590824\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −8986.94 −0.343675 −0.171837 0.985125i $$-0.554970\pi$$
−0.171837 + 0.985125i $$0.554970\pi$$
$$882$$ 0 0
$$883$$ − 3693.99i − 0.140784i −0.997519 0.0703922i $$-0.977575\pi$$
0.997519 0.0703922i $$-0.0224251\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −51613.0 −1.95377 −0.976886 0.213763i $$-0.931428\pi$$
−0.976886 + 0.213763i $$0.931428\pi$$
$$888$$ 0 0
$$889$$ −4368.95 −0.164825
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 34847.4i − 1.30585i
$$894$$ 0 0
$$895$$ 6601.03 0.246534
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 29469.5i − 1.09328i
$$900$$ 0 0
$$901$$ − 10544.5i − 0.389886i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 53922.7 1.98061
$$906$$ 0 0
$$907$$ − 17016.8i − 0.622971i −0.950251 0.311486i $$-0.899173\pi$$
0.950251 0.311486i $$-0.100827\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2991.72 0.108804 0.0544019 0.998519i $$-0.482675\pi$$
0.0544019 + 0.998519i $$0.482675\pi$$
$$912$$ 0 0
$$913$$ −4966.55 −0.180032
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 294.344i − 0.0105999i
$$918$$ 0 0
$$919$$ 5174.82 0.185747 0.0928736 0.995678i $$-0.470395\pi$$
0.0928736 + 0.995678i $$0.470395\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 29509.3i 1.05234i
$$924$$ 0 0
$$925$$ 48338.6i 1.71823i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −49256.5 −1.73956 −0.869780 0.493439i $$-0.835740\pi$$
−0.869780 + 0.493439i $$0.835740\pi$$
$$930$$ 0 0
$$931$$ − 26685.3i − 0.939394i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −7016.69 −0.245423
$$936$$ 0 0
$$937$$ 31566.5 1.10057 0.550283 0.834978i $$-0.314520\pi$$
0.550283 + 0.834978i $$0.314520\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 37575.6i 1.30173i 0.759193 + 0.650866i $$0.225594\pi$$
−0.759193 + 0.650866i $$0.774406\pi$$
$$942$$ 0 0
$$943$$ −84399.7 −2.91456
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 10289.3i 0.353070i 0.984294 + 0.176535i $$0.0564889\pi$$
−0.984294 + 0.176535i $$0.943511\pi$$
$$948$$ 0 0
$$949$$ − 34005.2i − 1.16318i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 36779.7 1.25017 0.625085 0.780557i $$-0.285064\pi$$
0.625085 + 0.780557i $$0.285064\pi$$
$$954$$ 0 0
$$955$$ 63736.9i 2.15966i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −6492.15 −0.218605
$$960$$ 0 0
$$961$$ 15210.9 0.510586
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 48593.6i − 1.62102i
$$966$$ 0 0
$$967$$ 35228.9 1.17155 0.585773 0.810475i $$-0.300791\pi$$
0.585773 + 0.810475i $$0.300791\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 15314.8i − 0.506155i −0.967446 0.253077i $$-0.918557\pi$$
0.967446 0.253077i $$-0.0814428\pi$$
$$972$$ 0 0
$$973$$ 4858.38i 0.160074i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −24074.3 −0.788337 −0.394168 0.919038i $$-0.628967\pi$$
−0.394168 + 0.919038i $$0.628967\pi$$
$$978$$ 0 0
$$979$$ 5427.49i 0.177184i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 15244.1 0.494620 0.247310 0.968936i $$-0.420453\pi$$
0.247310 + 0.968936i $$0.420453\pi$$
$$984$$ 0 0
$$985$$ 20003.7 0.647079
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 12259.6i 0.394168i
$$990$$ 0 0
$$991$$ 37518.6 1.20264 0.601321 0.799008i $$-0.294641\pi$$
0.601321 + 0.799008i $$0.294641\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 84403.1i 2.68920i
$$996$$ 0 0
$$997$$ − 1717.01i − 0.0545420i −0.999628 0.0272710i $$-0.991318\pi$$
0.999628 0.0272710i $$-0.00868170\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1152.4.d.p.577.2 8
3.2 odd 2 384.4.d.f.193.8 yes 8
4.3 odd 2 inner 1152.4.d.p.577.1 8
8.3 odd 2 inner 1152.4.d.p.577.7 8
8.5 even 2 inner 1152.4.d.p.577.8 8
12.11 even 2 384.4.d.f.193.4 yes 8
16.3 odd 4 2304.4.a.by.1.1 4
16.5 even 4 2304.4.a.by.1.4 4
16.11 odd 4 2304.4.a.cb.1.4 4
16.13 even 4 2304.4.a.cb.1.1 4
24.5 odd 2 384.4.d.f.193.1 8
24.11 even 2 384.4.d.f.193.5 yes 8
48.5 odd 4 768.4.a.v.1.1 4
48.11 even 4 768.4.a.u.1.1 4
48.29 odd 4 768.4.a.u.1.4 4
48.35 even 4 768.4.a.v.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.f.193.1 8 24.5 odd 2
384.4.d.f.193.4 yes 8 12.11 even 2
384.4.d.f.193.5 yes 8 24.11 even 2
384.4.d.f.193.8 yes 8 3.2 odd 2
768.4.a.u.1.1 4 48.11 even 4
768.4.a.u.1.4 4 48.29 odd 4
768.4.a.v.1.1 4 48.5 odd 4
768.4.a.v.1.4 4 48.35 even 4
1152.4.d.p.577.1 8 4.3 odd 2 inner
1152.4.d.p.577.2 8 1.1 even 1 trivial
1152.4.d.p.577.7 8 8.3 odd 2 inner
1152.4.d.p.577.8 8 8.5 even 2 inner
2304.4.a.by.1.1 4 16.3 odd 4
2304.4.a.by.1.4 4 16.5 even 4
2304.4.a.cb.1.1 4 16.13 even 4
2304.4.a.cb.1.4 4 16.11 odd 4