# Properties

 Label 384.4.d.f.193.1 Level $384$ Weight $4$ Character 384.193 Analytic conductor $22.657$ 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] = [384,4,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.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: $$22.6567334422$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{22}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 193.1 Root $$0.0690906i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.4.d.f.193.8

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} -17.4288i q^{5} +2.99032 q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} -17.4288i q^{5} +2.99032 q^{7} -9.00000 q^{9} -10.6274i q^{11} -43.3156i q^{13} -52.2865 q^{15} -37.8823 q^{17} -79.8823i q^{19} -8.97095i q^{21} +191.204 q^{23} -178.765 q^{25} +27.0000i q^{27} +138.918i q^{29} -212.136 q^{31} -31.8823 q^{33} -52.1177i q^{35} +270.404i q^{37} -129.947 q^{39} -441.411 q^{41} +64.1177i q^{43} +156.860i q^{45} +436.234 q^{47} -334.058 q^{49} +113.647i q^{51} -278.348i q^{53} -185.224 q^{55} -239.647 q^{57} +830.039i q^{59} -724.580i q^{61} -26.9128 q^{63} -754.940 q^{65} -859.529i q^{67} -573.613i q^{69} -681.264 q^{71} -785.058 q^{73} +536.294i q^{75} -31.7793i q^{77} +1018.82 q^{79} +81.0000 q^{81} -467.334i q^{83} +660.244i q^{85} +416.753 q^{87} +510.706 q^{89} -129.527i q^{91} +636.409i q^{93} -1392.26 q^{95} -234.235 q^{97} +95.6468i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 72 q^{9}+O(q^{10})$$ 8 * q - 72 * q^9 $$8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100})$$ 8 * q - 72 * q^9 + 240 * q^17 - 344 * q^25 + 288 * q^33 - 816 * q^41 + 1672 * q^49 - 288 * q^57 - 1152 * q^65 - 1936 * q^73 + 648 * q^81 + 7344 * q^89 - 2960 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ − 3.00000i − 0.577350i
$$4$$ 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
$$9$$ −9.00000 −0.333333
$$10$$ 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 $$-0.847110\pi$$
0.886848 0.462061i $$-0.152890\pi$$
$$14$$ 0 0
$$15$$ −52.2865 −0.900021
$$16$$ 0 0
$$17$$ −37.8823 −0.540459 −0.270229 0.962796i $$-0.587100\pi$$
−0.270229 + 0.962796i $$0.587100\pi$$
$$18$$ 0 0
$$19$$ − 79.8823i − 0.964539i −0.876023 0.482270i $$-0.839813\pi$$
0.876023 0.482270i $$-0.160187\pi$$
$$20$$ 0 0
$$21$$ − 8.97095i − 0.0932201i
$$22$$ 0 0
$$23$$ 191.204 1.73343 0.866714 0.498806i $$-0.166228\pi$$
0.866714 + 0.498806i $$0.166228\pi$$
$$24$$ 0 0
$$25$$ −178.765 −1.43012
$$26$$ 0 0
$$27$$ 27.0000i 0.192450i
$$28$$ 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
$$33$$ −31.8823 −0.168181
$$34$$ 0 0
$$35$$ − 52.1177i − 0.251700i
$$36$$ 0 0
$$37$$ 270.404i 1.20146i 0.799451 + 0.600731i $$0.205124\pi$$
−0.799451 + 0.600731i $$0.794876\pi$$
$$38$$ 0 0
$$39$$ −129.947 −0.533542
$$40$$ 0 0
$$41$$ −441.411 −1.68139 −0.840693 0.541511i $$-0.817852\pi$$
−0.840693 + 0.541511i $$0.817852\pi$$
$$42$$ 0 0
$$43$$ 64.1177i 0.227392i 0.993516 + 0.113696i $$0.0362690\pi$$
−0.993516 + 0.113696i $$0.963731\pi$$
$$44$$ 0 0
$$45$$ 156.860i 0.519628i
$$46$$ 0 0
$$47$$ 436.234 1.35386 0.676929 0.736049i $$-0.263311\pi$$
0.676929 + 0.736049i $$0.263311\pi$$
$$48$$ 0 0
$$49$$ −334.058 −0.973930
$$50$$ 0 0
$$51$$ 113.647i 0.312034i
$$52$$ 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
$$57$$ −239.647 −0.556877
$$58$$ 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 $$-0.724986\pi$$
0.649416 0.760434i $$-0.275014\pi$$
$$62$$ 0 0
$$63$$ −26.9128 −0.0538206
$$64$$ 0 0
$$65$$ −754.940 −1.44060
$$66$$ 0 0
$$67$$ − 859.529i − 1.56729i −0.621211 0.783643i $$-0.713359\pi$$
0.621211 0.783643i $$-0.286641\pi$$
$$68$$ 0 0
$$69$$ − 573.613i − 1.00079i
$$70$$ 0 0
$$71$$ −681.264 −1.13875 −0.569374 0.822078i $$-0.692814\pi$$
−0.569374 + 0.822078i $$0.692814\pi$$
$$72$$ 0 0
$$73$$ −785.058 −1.25869 −0.629343 0.777128i $$-0.716676\pi$$
−0.629343 + 0.777128i $$0.716676\pi$$
$$74$$ 0 0
$$75$$ 536.294i 0.825678i
$$76$$ 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
$$81$$ 81.0000 0.111111
$$82$$ 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
$$87$$ 416.753 0.513570
$$88$$ 0 0
$$89$$ 510.706 0.608256 0.304128 0.952631i $$-0.401635\pi$$
0.304128 + 0.952631i $$0.401635\pi$$
$$90$$ 0 0
$$91$$ − 129.527i − 0.149210i
$$92$$ 0 0
$$93$$ 636.409i 0.709597i
$$94$$ 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
$$99$$ 95.6468i 0.0970996i
$$100$$ 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
$$105$$ −156.353 −0.145319
$$106$$ 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 $$0.0826174\pi$$
−0.966505 + 0.256646i $$0.917383\pi$$
$$110$$ 0 0
$$111$$ 811.211 0.693664
$$112$$ 0 0
$$113$$ −582.706 −0.485101 −0.242551 0.970139i $$-0.577984\pi$$
−0.242551 + 0.970139i $$0.577984\pi$$
$$114$$ 0 0
$$115$$ − 3332.47i − 2.70221i
$$116$$ 0 0
$$117$$ 389.840i 0.308040i
$$118$$ 0 0
$$119$$ −113.280 −0.0872635
$$120$$ 0 0
$$121$$ 1218.06 0.915145
$$122$$ 0 0
$$123$$ 1324.23i 0.970749i
$$124$$ 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
$$129$$ 192.353 0.131285
$$130$$ 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
$$135$$ 470.579 0.300007
$$136$$ 0 0
$$137$$ 2171.06 1.35391 0.676956 0.736024i $$-0.263299\pi$$
0.676956 + 0.736024i $$0.263299\pi$$
$$138$$ 0 0
$$139$$ − 1624.70i − 0.991407i −0.868492 0.495703i $$-0.834910\pi$$
0.868492 0.495703i $$-0.165090\pi$$
$$140$$ 0 0
$$141$$ − 1308.70i − 0.781650i
$$142$$ 0 0
$$143$$ −460.333 −0.269195
$$144$$ 0 0
$$145$$ 2421.17 1.38667
$$146$$ 0 0
$$147$$ 1002.17i 0.562299i
$$148$$ 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
$$153$$ 340.940 0.180153
$$154$$ 0 0
$$155$$ 3697.29i 1.91596i
$$156$$ 0 0
$$157$$ − 1656.50i − 0.842059i −0.907047 0.421029i $$-0.861669\pi$$
0.907047 0.421029i $$-0.138331\pi$$
$$158$$ 0 0
$$159$$ −835.045 −0.416499
$$160$$ 0 0
$$161$$ 571.761 0.279882
$$162$$ 0 0
$$163$$ − 2228.82i − 1.07101i −0.844532 0.535505i $$-0.820121\pi$$
0.844532 0.535505i $$-0.179879\pi$$
$$164$$ 0 0
$$165$$ 555.671i 0.262175i
$$166$$ 0 0
$$167$$ 1667.01 0.772439 0.386219 0.922407i $$-0.373781\pi$$
0.386219 + 0.922407i $$0.373781\pi$$
$$168$$ 0 0
$$169$$ 320.761 0.146000
$$170$$ 0 0
$$171$$ 718.940i 0.321513i
$$172$$ 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
$$177$$ 2490.12 1.05745
$$178$$ 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 $$-0.780891\pi$$
0.772294 0.635265i $$-0.219109\pi$$
$$182$$ 0 0
$$183$$ −2173.74 −0.878073
$$184$$ 0 0
$$185$$ 4712.82 1.87294
$$186$$ 0 0
$$187$$ 402.590i 0.157435i
$$188$$ 0 0
$$189$$ 80.7385i 0.0310734i
$$190$$ 0 0
$$191$$ 3656.98 1.38539 0.692695 0.721230i $$-0.256423\pi$$
0.692695 + 0.721230i $$0.256423\pi$$
$$192$$ 0 0
$$193$$ 2788.12 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$194$$ 0 0
$$195$$ 2264.82i 0.831729i
$$196$$ 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
$$201$$ −2578.59 −0.904873
$$202$$ 0 0
$$203$$ 415.408i 0.143625i
$$204$$ 0 0
$$205$$ 7693.29i 2.62109i
$$206$$ 0 0
$$207$$ −1720.84 −0.577809
$$208$$ 0 0
$$209$$ −848.942 −0.280969
$$210$$ 0 0
$$211$$ − 3222.35i − 1.05135i −0.850684 0.525677i $$-0.823812\pi$$
0.850684 0.525677i $$-0.176188\pi$$
$$212$$ 0 0
$$213$$ 2043.79i 0.657457i
$$214$$ 0 0
$$215$$ 1117.50 0.354478
$$216$$ 0 0
$$217$$ −634.355 −0.198446
$$218$$ 0 0
$$219$$ 2355.17i 0.726703i
$$220$$ 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
$$225$$ 1608.88 0.476705
$$226$$ 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 $$0.0140471\pi$$
−0.999026 + 0.0441160i $$0.985953\pi$$
$$230$$ 0 0
$$231$$ −95.3380 −0.0271549
$$232$$ 0 0
$$233$$ 639.648 0.179849 0.0899244 0.995949i $$-0.471337\pi$$
0.0899244 + 0.995949i $$0.471337\pi$$
$$234$$ 0 0
$$235$$ − 7603.05i − 2.11050i
$$236$$ 0 0
$$237$$ − 3056.45i − 0.837713i
$$238$$ 0 0
$$239$$ 1744.94 0.472262 0.236131 0.971721i $$-0.424121\pi$$
0.236131 + 0.971721i $$0.424121\pi$$
$$240$$ 0 0
$$241$$ 3357.29 0.897354 0.448677 0.893694i $$-0.351895\pi$$
0.448677 + 0.893694i $$0.351895\pi$$
$$242$$ 0 0
$$243$$ − 243.000i − 0.0641500i
$$244$$ 0 0
$$245$$ 5822.24i 1.51824i
$$246$$ 0 0
$$247$$ −3460.15 −0.891351
$$248$$ 0 0
$$249$$ −1402.00 −0.356820
$$250$$ 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
$$255$$ 1980.73 0.486424
$$256$$ 0 0
$$257$$ −3036.12 −0.736917 −0.368459 0.929644i $$-0.620114\pi$$
−0.368459 + 0.929644i $$0.620114\pi$$
$$258$$ 0 0
$$259$$ 808.592i 0.193990i
$$260$$ 0 0
$$261$$ − 1250.26i − 0.296510i
$$262$$ 0 0
$$263$$ −1655.76 −0.388206 −0.194103 0.980981i $$-0.562180\pi$$
−0.194103 + 0.980981i $$0.562180\pi$$
$$264$$ 0 0
$$265$$ −4851.29 −1.12458
$$266$$ 0 0
$$267$$ − 1532.12i − 0.351177i
$$268$$ 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
$$273$$ −388.582 −0.0861467
$$274$$ 0 0
$$275$$ 1899.80i 0.416591i
$$276$$ 0 0
$$277$$ − 5692.81i − 1.23483i −0.786638 0.617415i $$-0.788180\pi$$
0.786638 0.617415i $$-0.211820\pi$$
$$278$$ 0 0
$$279$$ 1909.23 0.409686
$$280$$ 0 0
$$281$$ 2024.83 0.429861 0.214931 0.976629i $$-0.431047\pi$$
0.214931 + 0.976629i $$0.431047\pi$$
$$282$$ 0 0
$$283$$ − 247.761i − 0.0520419i −0.999661 0.0260210i $$-0.991716\pi$$
0.999661 0.0260210i $$-0.00828366\pi$$
$$284$$ 0 0
$$285$$ 4176.77i 0.868106i
$$286$$ 0 0
$$287$$ −1319.96 −0.271480
$$288$$ 0 0
$$289$$ −3477.94 −0.707905
$$290$$ 0 0
$$291$$ 702.706i 0.141558i
$$292$$ 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
$$297$$ 286.940 0.0560605
$$298$$ 0 0
$$299$$ − 8282.12i − 1.60190i
$$300$$ 0 0
$$301$$ 191.732i 0.0367152i
$$302$$ 0 0
$$303$$ 616.664 0.116919
$$304$$ 0 0
$$305$$ −12628.6 −2.37085
$$306$$ 0 0
$$307$$ 2974.82i 0.553035i 0.961009 + 0.276518i $$0.0891805\pi$$
−0.961009 + 0.276518i $$0.910820\pi$$
$$308$$ 0 0
$$309$$ − 1174.14i − 0.216163i
$$310$$ 0 0
$$311$$ −4451.52 −0.811648 −0.405824 0.913951i $$-0.633015\pi$$
−0.405824 + 0.913951i $$0.633015\pi$$
$$312$$ 0 0
$$313$$ −8273.75 −1.49412 −0.747061 0.664755i $$-0.768536\pi$$
−0.747061 + 0.664755i $$0.768536\pi$$
$$314$$ 0 0
$$315$$ 469.060i 0.0839001i
$$316$$ 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
$$321$$ 2802.82 0.487347
$$322$$ 0 0
$$323$$ 3026.12i 0.521293i
$$324$$ 0 0
$$325$$ 7743.29i 1.32160i
$$326$$ 0 0
$$327$$ 1752.37 0.296349
$$328$$ 0 0
$$329$$ 1304.48 0.218596
$$330$$ 0 0
$$331$$ − 8196.71i − 1.36112i −0.732691 0.680562i $$-0.761736\pi$$
0.732691 0.680562i $$-0.238264\pi$$
$$332$$ 0 0
$$333$$ − 2433.63i − 0.400487i
$$334$$ 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
$$339$$ 1748.12i 0.280073i
$$340$$ 0 0
$$341$$ 2254.46i 0.358023i
$$342$$ 0 0
$$343$$ −2024.62 −0.318715
$$344$$ 0 0
$$345$$ −9997.40 −1.56012
$$346$$ 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 $$-0.733640\pi$$
0.669847 0.742499i $$-0.266360\pi$$
$$350$$ 0 0
$$351$$ 1169.52 0.177847
$$352$$ 0 0
$$353$$ −10540.3 −1.58925 −0.794626 0.607099i $$-0.792333\pi$$
−0.794626 + 0.607099i $$0.792333\pi$$
$$354$$ 0 0
$$355$$ 11873.6i 1.77518i
$$356$$ 0 0
$$357$$ 339.840i 0.0503816i
$$358$$ 0 0
$$359$$ 514.158 0.0755884 0.0377942 0.999286i $$-0.487967\pi$$
0.0377942 + 0.999286i $$0.487967\pi$$
$$360$$ 0 0
$$361$$ 477.826 0.0696641
$$362$$ 0 0
$$363$$ − 3654.17i − 0.528359i
$$364$$ 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
$$369$$ 3972.70 0.560462
$$370$$ 0 0
$$371$$ − 832.350i − 0.116478i
$$372$$ 0 0
$$373$$ 6956.92i 0.965726i 0.875696 + 0.482863i $$0.160403\pi$$
−0.875696 + 0.482863i $$0.839597\pi$$
$$374$$ 0 0
$$375$$ 2811.16 0.387114
$$376$$ 0 0
$$377$$ 6017.30 0.822034
$$378$$ 0 0
$$379$$ 10201.3i 1.38260i 0.722569 + 0.691299i $$0.242961\pi$$
−0.722569 + 0.691299i $$0.757039\pi$$
$$380$$ 0 0
$$381$$ 4383.10i 0.589378i
$$382$$ 0 0
$$383$$ 2461.56 0.328406 0.164203 0.986427i $$-0.447495\pi$$
0.164203 + 0.986427i $$0.447495\pi$$
$$384$$ 0 0
$$385$$ −553.877 −0.0733200
$$386$$ 0 0
$$387$$ − 577.060i − 0.0757974i
$$388$$ 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
$$393$$ −295.297 −0.0379027
$$394$$ 0 0
$$395$$ − 17756.8i − 2.26188i
$$396$$ 0 0
$$397$$ 2084.56i 0.263529i 0.991281 + 0.131764i $$0.0420642\pi$$
−0.991281 + 0.131764i $$0.957936\pi$$
$$398$$ 0 0
$$399$$ −716.620 −0.0899144
$$400$$ 0 0
$$401$$ −9710.59 −1.20929 −0.604643 0.796497i $$-0.706684\pi$$
−0.604643 + 0.796497i $$0.706684\pi$$
$$402$$ 0 0
$$403$$ 9188.81i 1.13580i
$$404$$ 0 0
$$405$$ − 1411.74i − 0.173209i
$$406$$ 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
$$411$$ − 6513.17i − 0.781681i
$$412$$ 0 0
$$413$$ 2482.08i 0.295727i
$$414$$ 0 0
$$415$$ −8145.09 −0.963438
$$416$$ 0 0
$$417$$ −4874.11 −0.572389
$$418$$ 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 $$0.0909814\pi$$
−0.959429 + 0.281951i $$0.909019\pi$$
$$422$$ 0 0
$$423$$ −3926.11 −0.451286
$$424$$ 0 0
$$425$$ 6772.00 0.772918
$$426$$ 0 0
$$427$$ − 2166.72i − 0.245562i
$$428$$ 0 0
$$429$$ 1381.00i 0.155420i
$$430$$ 0 0
$$431$$ 16916.7 1.89060 0.945302 0.326196i $$-0.105767\pi$$
0.945302 + 0.326196i $$0.105767\pi$$
$$432$$ 0 0
$$433$$ −1163.88 −0.129174 −0.0645870 0.997912i $$-0.520573\pi$$
−0.0645870 + 0.997912i $$0.520573\pi$$
$$434$$ 0 0
$$435$$ − 7263.52i − 0.800596i
$$436$$ 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
$$441$$ 3006.52 0.324643
$$442$$ 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
$$447$$ −1909.97 −0.202100
$$448$$ 0 0
$$449$$ 9620.94 1.01123 0.505613 0.862761i $$-0.331267\pi$$
0.505613 + 0.862761i $$0.331267\pi$$
$$450$$ 0 0
$$451$$ 4691.06i 0.489786i
$$452$$ 0 0
$$453$$ − 5458.03i − 0.566094i
$$454$$ 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
$$459$$ − 1022.82i − 0.104011i
$$460$$ 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
$$465$$ 11091.9 1.10618
$$466$$ 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
$$471$$ −4969.51 −0.486163
$$472$$ 0 0
$$473$$ 681.406 0.0662391
$$474$$ 0 0
$$475$$ 14280.1i 1.37940i
$$476$$ 0 0
$$477$$ 2505.14i 0.240466i
$$478$$ 0 0
$$479$$ 6458.37 0.616056 0.308028 0.951377i $$-0.400331\pi$$
0.308028 + 0.951377i $$0.400331\pi$$
$$480$$ 0 0
$$481$$ 11712.7 1.11030
$$482$$ 0 0
$$483$$ − 1715.28i − 0.161590i
$$484$$ 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
$$489$$ −6686.46 −0.618348
$$490$$ 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
$$495$$ 1667.01 0.151367
$$496$$ 0 0
$$497$$ −2037.19 −0.183865
$$498$$ 0 0
$$499$$ − 2631.77i − 0.236101i −0.993008 0.118051i $$-0.962336\pi$$
0.993008 0.118051i $$-0.0376645\pi$$
$$500$$ 0 0
$$501$$ − 5001.04i − 0.445968i
$$502$$ 0 0
$$503$$ 6907.45 0.612302 0.306151 0.951983i $$-0.400959\pi$$
0.306151 + 0.951983i $$0.400959\pi$$
$$504$$ 0 0
$$505$$ 3582.58 0.315689
$$506$$ 0 0
$$507$$ − 962.283i − 0.0842929i
$$508$$ 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
$$513$$ 2156.82 0.185626
$$514$$ 0 0
$$515$$ − 6821.29i − 0.583654i
$$516$$ 0 0
$$517$$ − 4636.04i − 0.394377i
$$518$$ 0 0
$$519$$ −7501.60 −0.634458
$$520$$ 0 0
$$521$$ 15846.1 1.33249 0.666247 0.745731i $$-0.267899\pi$$
0.666247 + 0.745731i $$0.267899\pi$$
$$522$$ 0 0
$$523$$ 8891.64i 0.743411i 0.928351 + 0.371706i $$0.121227\pi$$
−0.928351 + 0.371706i $$0.878773\pi$$
$$524$$ 0 0
$$525$$ 1603.69i 0.133316i
$$526$$ 0 0
$$527$$ 8036.20 0.664255
$$528$$ 0 0
$$529$$ 24392.0 2.00477
$$530$$ 0 0
$$531$$ − 7470.35i − 0.610519i
$$532$$ 0 0
$$533$$ 19120.0i 1.55381i
$$534$$ 0 0
$$535$$ 16283.3 1.31587
$$536$$ 0 0
$$537$$ 1136.23 0.0913068
$$538$$ 0 0
$$539$$ 3550.17i 0.283705i
$$540$$ 0 0
$$541$$ 12833.5i 1.01988i 0.860210 + 0.509940i $$0.170333\pi$$
−0.860210 + 0.509940i $$0.829667\pi$$
$$542$$ 0 0
$$543$$ −9281.63 −0.733541
$$544$$ 0 0
$$545$$ 10180.6 0.800162
$$546$$ 0 0
$$547$$ 16257.0i 1.27075i 0.772204 + 0.635375i $$0.219155\pi$$
−0.772204 + 0.635375i $$0.780845\pi$$
$$548$$ 0 0
$$549$$ 6521.22i 0.506956i
$$550$$ 0 0
$$551$$ 11097.1 0.857986
$$552$$ 0 0
$$553$$ 3046.59 0.234275
$$554$$ 0 0
$$555$$ − 14138.5i − 1.08134i
$$556$$ 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
$$561$$ 1207.77 0.0908951
$$562$$ 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
$$567$$ 242.216 0.0179402
$$568$$ 0 0
$$569$$ 7887.05 0.581094 0.290547 0.956861i $$-0.406163\pi$$
0.290547 + 0.956861i $$0.406163\pi$$
$$570$$ 0 0
$$571$$ 21819.3i 1.59914i 0.600573 + 0.799570i $$0.294939\pi$$
−0.600573 + 0.799570i $$0.705061\pi$$
$$572$$ 0 0
$$573$$ − 10970.9i − 0.799856i
$$574$$ 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
$$579$$ − 8364.35i − 0.600363i
$$580$$ 0 0
$$581$$ − 1397.48i − 0.0997884i
$$582$$ 0 0
$$583$$ −2958.12 −0.210142
$$584$$ 0 0
$$585$$ 6794.46 0.480199
$$586$$ 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
$$591$$ 3443.21 0.239653
$$592$$ 0 0
$$593$$ −8062.23 −0.558307 −0.279153 0.960246i $$-0.590054\pi$$
−0.279153 + 0.960246i $$0.590054\pi$$
$$594$$ 0 0
$$595$$ 1974.34i 0.136034i
$$596$$ 0 0
$$597$$ 14528.2i 0.995978i
$$598$$ 0 0
$$599$$ 2185.21 0.149058 0.0745288 0.997219i $$-0.476255\pi$$
0.0745288 + 0.997219i $$0.476255\pi$$
$$600$$ 0 0
$$601$$ −3542.25 −0.240418 −0.120209 0.992749i $$-0.538357\pi$$
−0.120209 + 0.992749i $$0.538357\pi$$
$$602$$ 0 0
$$603$$ 7735.76i 0.522429i
$$604$$ 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
$$609$$ 1246.22 0.0829220
$$610$$ 0 0
$$611$$ − 18895.7i − 1.25113i
$$612$$ 0 0
$$613$$ − 22514.2i − 1.48343i −0.670717 0.741713i $$-0.734014\pi$$
0.670717 0.741713i $$-0.265986\pi$$
$$614$$ 0 0
$$615$$ 23079.9 1.51328
$$616$$ 0 0
$$617$$ 4255.88 0.277691 0.138845 0.990314i $$-0.455661\pi$$
0.138845 + 0.990314i $$0.455661\pi$$
$$618$$ 0 0
$$619$$ − 228.949i − 0.0148663i −0.999972 0.00743315i $$-0.997634\pi$$
0.999972 0.00743315i $$-0.00236607\pi$$
$$620$$ 0 0
$$621$$ 5162.51i 0.333598i
$$622$$ 0 0
$$623$$ 1527.17 0.0982102
$$624$$ 0 0
$$625$$ −6013.82 −0.384884
$$626$$ 0 0
$$627$$ 2546.83i 0.162218i
$$628$$ 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
$$633$$ −9667.04 −0.606999
$$634$$ 0 0
$$635$$ 25464.1i 1.59136i
$$636$$ 0 0
$$637$$ 14469.9i 0.900030i
$$638$$ 0 0
$$639$$ 6131.38 0.379583
$$640$$ 0 0
$$641$$ 29381.4 1.81045 0.905223 0.424938i $$-0.139704\pi$$
0.905223 + 0.424938i $$0.139704\pi$$
$$642$$ 0 0
$$643$$ − 249.316i − 0.0152909i −0.999971 0.00764546i $$-0.997566\pi$$
0.999971 0.00764546i $$-0.00243365\pi$$
$$644$$ 0 0
$$645$$ − 3352.49i − 0.204658i
$$646$$ 0 0
$$647$$ −16025.8 −0.973785 −0.486893 0.873462i $$-0.661870\pi$$
−0.486893 + 0.873462i $$0.661870\pi$$
$$648$$ 0 0
$$649$$ 8821.17 0.533530
$$650$$ 0 0
$$651$$ 1903.06i 0.114573i
$$652$$ 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
$$657$$ 7065.52 0.419562
$$658$$ 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 $$-0.755439\pi$$
0.719084 0.694923i $$-0.244561\pi$$
$$662$$ 0 0
$$663$$ 4922.67 0.288357
$$664$$ 0 0
$$665$$ −4163.28 −0.242775
$$666$$ 0 0
$$667$$ 26561.6i 1.54194i
$$668$$ 0 0
$$669$$ − 14797.8i − 0.855183i
$$670$$ 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
$$675$$ − 4826.64i − 0.275226i
$$676$$ 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
$$681$$ 10858.5 0.611009
$$682$$ 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
$$687$$ 917.278 0.0509408
$$688$$ 0 0
$$689$$ −12056.8 −0.666659
$$690$$ 0 0
$$691$$ − 16521.5i − 0.909563i −0.890603 0.454782i $$-0.849717\pi$$
0.890603 0.454782i $$-0.150283\pi$$
$$692$$ 0 0
$$693$$ 286.014i 0.0156779i
$$694$$ 0 0
$$695$$ −28316.7 −1.54549
$$696$$ 0 0
$$697$$ 16721.7 0.908720
$$698$$ 0 0
$$699$$ − 1918.95i − 0.103836i
$$700$$ 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
$$705$$ −22809.2 −1.21850
$$706$$ 0 0
$$707$$ 614.674i 0.0326976i
$$708$$ 0 0
$$709$$ − 980.957i − 0.0519614i −0.999662 0.0259807i $$-0.991729\pi$$
0.999662 0.0259807i $$-0.00827084\pi$$
$$710$$ 0 0
$$711$$ −9169.36 −0.483654
$$712$$ 0 0
$$713$$ −40561.4 −2.13048
$$714$$ 0 0
$$715$$ 8023.06i 0.419644i
$$716$$ 0 0
$$717$$ − 5234.81i − 0.272660i
$$718$$ 0 0
$$719$$ 4115.73 0.213478 0.106739 0.994287i $$-0.465959\pi$$
0.106739 + 0.994287i $$0.465959\pi$$
$$720$$ 0 0
$$721$$ 1170.35 0.0604522
$$722$$ 0 0
$$723$$ − 10071.9i − 0.518088i
$$724$$ 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
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ − 2428.92i − 0.122896i
$$732$$ 0 0
$$733$$ 31517.2i 1.58815i 0.607821 + 0.794074i $$0.292044\pi$$
−0.607821 + 0.794074i $$0.707956\pi$$
$$734$$ 0 0
$$735$$ 17466.7 0.876558
$$736$$ 0 0
$$737$$ −9134.57 −0.456549
$$738$$ 0 0
$$739$$ − 11415.0i − 0.568213i −0.958793 0.284106i $$-0.908303\pi$$
0.958793 0.284106i $$-0.0916969\pi$$
$$740$$ 0 0
$$741$$ 10380.4i 0.514622i
$$742$$ 0 0
$$743$$ −5732.08 −0.283028 −0.141514 0.989936i $$-0.545197\pi$$
−0.141514 + 0.989936i $$0.545197\pi$$
$$744$$ 0 0
$$745$$ −11096.2 −0.545683
$$746$$ 0 0
$$747$$ 4206.01i 0.206010i
$$748$$ 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
$$753$$ −3952.34 −0.191277
$$754$$ 0 0
$$755$$ − 31709.0i − 1.52849i
$$756$$ 0 0
$$757$$ 29125.9i 1.39841i 0.714920 + 0.699206i $$0.246463\pi$$
−0.714920 + 0.699206i $$0.753537\pi$$
$$758$$ 0 0
$$759$$ −6096.02 −0.291530
$$760$$ 0 0
$$761$$ 14228.5 0.677768 0.338884 0.940828i $$-0.389951\pi$$
0.338884 + 0.940828i $$0.389951\pi$$
$$762$$ 0 0
$$763$$ 1746.71i 0.0828771i
$$764$$ 0 0
$$765$$ − 5942.19i − 0.280837i
$$766$$ 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
$$771$$ 9108.35i 0.425459i
$$772$$ 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
$$777$$ 2425.78 0.112000
$$778$$ 0 0
$$779$$ 35260.9i 1.62176i
$$780$$ 0 0
$$781$$ 7240.08i 0.331716i
$$782$$ 0 0
$$783$$ −3750.78 −0.171190
$$784$$ 0 0
$$785$$ −28870.9 −1.31267
$$786$$ 0 0
$$787$$ − 21001.0i − 0.951214i −0.879658 0.475607i $$-0.842228\pi$$
0.879658 0.475607i $$-0.157772\pi$$
$$788$$ 0 0
$$789$$ 4967.27i 0.224131i
$$790$$ 0 0
$$791$$ −1742.48 −0.0783253
$$792$$ 0 0
$$793$$ −31385.6 −1.40547
$$794$$ 0 0
$$795$$ 14553.9i 0.649274i
$$796$$ 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
$$801$$ −4596.36 −0.202752
$$802$$ 0 0
$$803$$ 8343.14i 0.366654i
$$804$$ 0 0
$$805$$ − 9965.13i − 0.436304i
$$806$$ 0 0
$$807$$ −15878.2 −0.692612
$$808$$ 0 0
$$809$$ −39481.6 −1.71582 −0.857911 0.513798i $$-0.828238\pi$$
−0.857911 + 0.513798i $$0.828238\pi$$
$$810$$ 0 0
$$811$$ − 31157.5i − 1.34906i −0.738247 0.674531i $$-0.764346\pi$$
0.738247 0.674531i $$-0.235654\pi$$
$$812$$ 0 0
$$813$$ 24031.6i 1.03668i
$$814$$ 0 0
$$815$$ −38845.8 −1.66958
$$816$$ 0 0
$$817$$ 5121.87 0.219329
$$818$$ 0 0
$$819$$ 1165.75i 0.0497368i
$$820$$ 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
$$825$$ 5699.41 0.240519
$$826$$ 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 $$0.195565\pi$$
−0.817127 + 0.576457i $$0.804435\pi$$
$$830$$ 0 0
$$831$$ −17078.4 −0.712929
$$832$$ 0 0
$$833$$ 12654.9 0.526369
$$834$$ 0 0
$$835$$ − 29054.1i − 1.20414i
$$836$$ 0 0
$$837$$ − 5727.68i − 0.236532i
$$838$$ 0 0
$$839$$ 29951.5 1.23247 0.616234 0.787563i $$-0.288657\pi$$
0.616234 + 0.787563i $$0.288657\pi$$
$$840$$ 0 0
$$841$$ 5090.88 0.208737
$$842$$ 0 0
$$843$$ − 6074.48i − 0.248180i
$$844$$ 0 0
$$845$$ − 5590.49i − 0.227596i
$$846$$ 0 0
$$847$$ 3642.38 0.147761
$$848$$ 0 0
$$849$$ −743.283 −0.0300464
$$850$$ 0 0
$$851$$ 51702.3i 2.08265i
$$852$$ 0 0
$$853$$ − 5174.61i − 0.207708i −0.994593 0.103854i $$-0.966882\pi$$
0.994593 0.103854i $$-0.0331175\pi$$
$$854$$ 0 0
$$855$$ 12530.3 0.501201
$$856$$ 0 0
$$857$$ 9258.34 0.369030 0.184515 0.982830i $$-0.440929\pi$$
0.184515 + 0.982830i $$0.440929\pi$$
$$858$$ 0 0
$$859$$ − 24353.0i − 0.967304i −0.875260 0.483652i $$-0.839310\pi$$
0.875260 0.483652i $$-0.160690\pi$$
$$860$$ 0 0
$$861$$ 3959.88i 0.156739i
$$862$$ 0 0
$$863$$ 42283.4 1.66784 0.833919 0.551887i $$-0.186092\pi$$
0.833919 + 0.551887i $$0.186092\pi$$
$$864$$ 0 0
$$865$$ −43581.4 −1.71308
$$866$$ 0 0
$$867$$ 10433.8i 0.408709i
$$868$$ 0 0
$$869$$ − 10827.4i − 0.422663i
$$870$$ 0 0
$$871$$ −37231.0 −1.44836
$$872$$ 0 0
$$873$$ 2108.12 0.0817286
$$874$$ 0 0
$$875$$ 2802.08i 0.108260i
$$876$$ 0 0
$$877$$ − 49843.1i − 1.91914i −0.281476 0.959568i $$-0.590824\pi$$
0.281476 0.959568i $$-0.409176\pi$$
$$878$$ 0 0
$$879$$ −24400.8 −0.936314
$$880$$ 0 0
$$881$$ 8986.94 0.343675 0.171837 0.985125i $$-0.445030\pi$$
0.171837 + 0.985125i $$0.445030\pi$$
$$882$$ 0 0
$$883$$ 3693.99i 0.140784i 0.997519 + 0.0703922i $$0.0224251\pi$$
−0.997519 + 0.0703922i $$0.977575\pi$$
$$884$$ 0 0
$$885$$ − 43399.8i − 1.64844i
$$886$$ 0 0
$$887$$ 51613.0 1.95377 0.976886 0.213763i $$-0.0685720\pi$$
0.976886 + 0.213763i $$0.0685720\pi$$
$$888$$ 0 0
$$889$$ −4368.95 −0.164825
$$890$$ 0 0
$$891$$ − 860.821i − 0.0323665i
$$892$$ 0 0
$$893$$ − 34847.4i − 1.30585i
$$894$$ 0 0
$$895$$ 6601.03 0.246534
$$896$$ 0 0
$$897$$ −24846.4 −0.924856
$$898$$ 0 0
$$899$$ − 29469.5i − 1.09328i
$$900$$ 0 0
$$901$$ 10544.5i 0.389886i
$$902$$ 0 0
$$903$$ 575.197 0.0211975
$$904$$ 0 0
$$905$$ −53922.7 −1.98061
$$906$$ 0 0
$$907$$ 17016.8i 0.622971i 0.950251 + 0.311486i $$0.100827\pi$$
−0.950251 + 0.311486i $$0.899173\pi$$
$$908$$ 0 0
$$909$$ − 1849.99i − 0.0675032i
$$910$$ 0 0
$$911$$ −2991.72 −0.108804 −0.0544019 0.998519i $$-0.517325\pi$$
−0.0544019 + 0.998519i $$0.517325\pi$$
$$912$$ 0 0
$$913$$ −4966.55 −0.180032
$$914$$ 0 0
$$915$$ 37885.7i 1.36881i
$$916$$ 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
$$921$$ 8924.46 0.319295
$$922$$ 0 0
$$923$$ 29509.3i 1.05234i
$$924$$ 0 0
$$925$$ − 48338.6i − 1.71823i
$$926$$ 0 0
$$927$$ −3522.41 −0.124802
$$928$$ 0 0
$$929$$ 49256.5 1.73956 0.869780 0.493439i $$-0.164260\pi$$
0.869780 + 0.493439i $$0.164260\pi$$
$$930$$ 0 0
$$931$$ 26685.3i 0.939394i
$$932$$ 0 0
$$933$$ 13354.6i 0.468605i
$$934$$ 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
$$939$$ 24821.3i 0.862632i
$$940$$ 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
$$945$$ 1407.18 0.0484397
$$946$$ 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
$$951$$ 1287.11 0.0438879
$$952$$ 0 0
$$953$$ −36779.7 −1.25017 −0.625085 0.780557i $$-0.714936\pi$$
−0.625085 + 0.780557i $$0.714936\pi$$
$$954$$ 0 0
$$955$$ − 63736.9i − 2.15966i
$$956$$ 0 0
$$957$$ − 4429.01i − 0.149602i
$$958$$ 0 0
$$959$$ 6492.15 0.218605
$$960$$ 0 0
$$961$$ 15210.9 0.510586
$$962$$ 0 0
$$963$$ − 8408.47i − 0.281370i
$$964$$ 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
$$969$$ 9078.36 0.300969
$$970$$ 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
$$975$$ 23229.9 0.763027
$$976$$ 0 0
$$977$$ 24074.3 0.788337 0.394168 0.919038i $$-0.371033\pi$$
0.394168 + 0.919038i $$0.371033\pi$$
$$978$$ 0 0
$$979$$ − 5427.49i − 0.177184i
$$980$$ 0 0
$$981$$ − 5257.10i − 0.171097i
$$982$$ 0 0
$$983$$ −15244.1 −0.494620 −0.247310 0.968936i $$-0.579547\pi$$
−0.247310 + 0.968936i $$0.579547\pi$$
$$984$$ 0 0
$$985$$ 20003.7 0.647079
$$986$$ 0 0
$$987$$ − 3913.43i − 0.126207i
$$988$$ 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
$$993$$ −24590.1 −0.785845
$$994$$ 0 0
$$995$$ 84403.1i 2.68920i
$$996$$ 0 0
$$997$$ 1717.01i 0.0545420i 0.999628 + 0.0272710i $$0.00868170\pi$$
−0.999628 + 0.0272710i $$0.991318\pi$$
$$998$$ 0 0
$$999$$ −7300.90 −0.231221
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 384.4.d.f.193.1 8
3.2 odd 2 1152.4.d.p.577.8 8
4.3 odd 2 inner 384.4.d.f.193.5 yes 8
8.3 odd 2 inner 384.4.d.f.193.4 yes 8
8.5 even 2 inner 384.4.d.f.193.8 yes 8
12.11 even 2 1152.4.d.p.577.7 8
16.3 odd 4 768.4.a.u.1.1 4
16.5 even 4 768.4.a.u.1.4 4
16.11 odd 4 768.4.a.v.1.4 4
16.13 even 4 768.4.a.v.1.1 4
24.5 odd 2 1152.4.d.p.577.2 8
24.11 even 2 1152.4.d.p.577.1 8
48.5 odd 4 2304.4.a.cb.1.1 4
48.11 even 4 2304.4.a.by.1.1 4
48.29 odd 4 2304.4.a.by.1.4 4
48.35 even 4 2304.4.a.cb.1.4 4

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