# Properties

 Label 768.4.a.u.1.4 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## 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: yes Analytic conductor: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.9792.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 2x + 7$$ x^4 - 2*x^3 - 7*x^2 + 2*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.48330$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +17.4288 q^{5} -2.99032 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +17.4288 q^{5} -2.99032 q^{7} +9.00000 q^{9} +10.6274 q^{11} -43.3156 q^{13} -52.2865 q^{15} -37.8823 q^{17} -79.8823 q^{19} +8.97095 q^{21} -191.204 q^{23} +178.765 q^{25} -27.0000 q^{27} +138.918 q^{29} -212.136 q^{31} -31.8823 q^{33} -52.1177 q^{35} -270.404 q^{37} +129.947 q^{39} +441.411 q^{41} -64.1177 q^{43} +156.860 q^{45} +436.234 q^{47} -334.058 q^{49} +113.647 q^{51} +278.348 q^{53} +185.224 q^{55} +239.647 q^{57} -830.039 q^{59} -724.580 q^{61} -26.9128 q^{63} -754.940 q^{65} -859.529 q^{67} +573.613 q^{69} +681.264 q^{71} +785.058 q^{73} -536.294 q^{75} -31.7793 q^{77} +1018.82 q^{79} +81.0000 q^{81} -467.334 q^{83} -660.244 q^{85} -416.753 q^{87} -510.706 q^{89} +129.527 q^{91} +636.409 q^{93} -1392.26 q^{95} -234.235 q^{97} +95.6468 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 + 36 * q^9 $$4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99}+O(q^{100})$$ 4 * q - 12 * q^3 + 36 * q^9 - 48 * q^11 + 120 * q^17 - 48 * q^19 + 172 * q^25 - 108 * q^27 + 144 * q^33 - 480 * q^35 + 408 * q^41 - 528 * q^43 + 836 * q^49 - 360 * q^51 + 144 * q^57 - 1872 * q^59 - 576 * q^65 - 2352 * q^67 + 968 * q^73 - 516 * q^75 + 324 * q^81 - 3408 * q^83 - 3672 * q^89 - 5184 * q^91 - 1480 * q^97 - 432 * q^99

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ 17.4288 1.55888 0.779441 0.626475i $$-0.215503\pi$$
0.779441 + 0.626475i $$0.215503\pi$$
$$6$$ 0 0
$$7$$ −2.99032 −0.161462 −0.0807310 0.996736i $$-0.525725\pi$$
−0.0807310 + 0.996736i $$0.525725\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 10.6274 0.291299 0.145649 0.989336i $$-0.453473\pi$$
0.145649 + 0.989336i $$0.453473\pi$$
$$12$$ 0 0
$$13$$ −43.3156 −0.924121 −0.462061 0.886848i $$-0.652890\pi$$
−0.462061 + 0.886848i $$0.652890\pi$$
$$14$$ 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
$$19$$ −79.8823 −0.964539 −0.482270 0.876023i $$-0.660187\pi$$
−0.482270 + 0.876023i $$0.660187\pi$$
$$20$$ 0 0
$$21$$ 8.97095 0.0932201
$$22$$ 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
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 138.918 0.889530 0.444765 0.895647i $$-0.353287\pi$$
0.444765 + 0.895647i $$0.353287\pi$$
$$30$$ 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
$$35$$ −52.1177 −0.251700
$$36$$ 0 0
$$37$$ −270.404 −1.20146 −0.600731 0.799451i $$-0.705124\pi$$
−0.600731 + 0.799451i $$0.705124\pi$$
$$38$$ 0 0
$$39$$ 129.947 0.533542
$$40$$ 0 0
$$41$$ 441.411 1.68139 0.840693 0.541511i $$-0.182148\pi$$
0.840693 + 0.541511i $$0.182148\pi$$
$$42$$ 0 0
$$43$$ −64.1177 −0.227392 −0.113696 0.993516i $$-0.536269\pi$$
−0.113696 + 0.993516i $$0.536269\pi$$
$$44$$ 0 0
$$45$$ 156.860 0.519628
$$46$$ 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
$$51$$ 113.647 0.312034
$$52$$ 0 0
$$53$$ 278.348 0.721398 0.360699 0.932682i $$-0.382538\pi$$
0.360699 + 0.932682i $$0.382538\pi$$
$$54$$ 0 0
$$55$$ 185.224 0.454101
$$56$$ 0 0
$$57$$ 239.647 0.556877
$$58$$ 0 0
$$59$$ −830.039 −1.83156 −0.915778 0.401684i $$-0.868425\pi$$
−0.915778 + 0.401684i $$0.868425\pi$$
$$60$$ 0 0
$$61$$ −724.580 −1.52087 −0.760434 0.649416i $$-0.775014\pi$$
−0.760434 + 0.649416i $$0.775014\pi$$
$$62$$ 0 0
$$63$$ −26.9128 −0.0538206
$$64$$ 0 0
$$65$$ −754.940 −1.44060
$$66$$ 0 0
$$67$$ −859.529 −1.56729 −0.783643 0.621211i $$-0.786641\pi$$
−0.783643 + 0.621211i $$0.786641\pi$$
$$68$$ 0 0
$$69$$ 573.613 1.00079
$$70$$ 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 $$-0.283324\pi$$
0.629343 + 0.777128i $$0.283324\pi$$
$$74$$ 0 0
$$75$$ −536.294 −0.825678
$$76$$ 0 0
$$77$$ −31.7793 −0.0470337
$$78$$ 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
$$83$$ −467.334 −0.618031 −0.309015 0.951057i $$-0.599999\pi$$
−0.309015 + 0.951057i $$0.599999\pi$$
$$84$$ 0 0
$$85$$ −660.244 −0.842512
$$86$$ 0 0
$$87$$ −416.753 −0.513570
$$88$$ 0 0
$$89$$ −510.706 −0.608256 −0.304128 0.952631i $$-0.598365\pi$$
−0.304128 + 0.952631i $$0.598365\pi$$
$$90$$ 0 0
$$91$$ 129.527 0.149210
$$92$$ 0 0
$$93$$ 636.409 0.709597
$$94$$ 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
$$99$$ 95.6468 0.0970996
$$100$$ 0 0
$$101$$ −205.555 −0.202509 −0.101255 0.994861i $$-0.532286\pi$$
−0.101255 + 0.994861i $$0.532286\pi$$
$$102$$ 0 0
$$103$$ −391.379 −0.374405 −0.187203 0.982321i $$-0.559942\pi$$
−0.187203 + 0.982321i $$0.559942\pi$$
$$104$$ 0 0
$$105$$ 156.353 0.145319
$$106$$ 0 0
$$107$$ −934.274 −0.844109 −0.422055 0.906570i $$-0.638691\pi$$
−0.422055 + 0.906570i $$0.638691\pi$$
$$108$$ 0 0
$$109$$ 584.123 0.513292 0.256646 0.966505i $$-0.417383\pi$$
0.256646 + 0.966505i $$0.417383\pi$$
$$110$$ 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
$$115$$ −3332.47 −2.70221
$$116$$ 0 0
$$117$$ −389.840 −0.308040
$$118$$ 0 0
$$119$$ 113.280 0.0872635
$$120$$ 0 0
$$121$$ −1218.06 −0.915145
$$122$$ 0 0
$$123$$ −1324.23 −0.970749
$$124$$ 0 0
$$125$$ 937.053 0.670501
$$126$$ 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
$$131$$ −98.4323 −0.0656494 −0.0328247 0.999461i $$-0.510450\pi$$
−0.0328247 + 0.999461i $$0.510450\pi$$
$$132$$ 0 0
$$133$$ 238.873 0.155736
$$134$$ 0 0
$$135$$ −470.579 −0.300007
$$136$$ 0 0
$$137$$ −2171.06 −1.35391 −0.676956 0.736024i $$-0.736701\pi$$
−0.676956 + 0.736024i $$0.736701\pi$$
$$138$$ 0 0
$$139$$ 1624.70 0.991407 0.495703 0.868492i $$-0.334910\pi$$
0.495703 + 0.868492i $$0.334910\pi$$
$$140$$ 0 0
$$141$$ −1308.70 −0.781650
$$142$$ 0 0
$$143$$ −460.333 −0.269195
$$144$$ 0 0
$$145$$ 2421.17 1.38667
$$146$$ 0 0
$$147$$ 1002.17 0.562299
$$148$$ 0 0
$$149$$ 636.658 0.350047 0.175024 0.984564i $$-0.444000\pi$$
0.175024 + 0.984564i $$0.444000\pi$$
$$150$$ 0 0
$$151$$ −1819.34 −0.980503 −0.490252 0.871581i $$-0.663095\pi$$
−0.490252 + 0.871581i $$0.663095\pi$$
$$152$$ 0 0
$$153$$ −340.940 −0.180153
$$154$$ 0 0
$$155$$ −3697.29 −1.91596
$$156$$ 0 0
$$157$$ −1656.50 −0.842059 −0.421029 0.907047i $$-0.638331\pi$$
−0.421029 + 0.907047i $$0.638331\pi$$
$$158$$ 0 0
$$159$$ −835.045 −0.416499
$$160$$ 0 0
$$161$$ 571.761 0.279882
$$162$$ 0 0
$$163$$ −2228.82 −1.07101 −0.535505 0.844532i $$-0.679879\pi$$
−0.535505 + 0.844532i $$0.679879\pi$$
$$164$$ 0 0
$$165$$ −555.671 −0.262175
$$166$$ 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
$$171$$ −718.940 −0.321513
$$172$$ 0 0
$$173$$ −2500.53 −1.09891 −0.549457 0.835522i $$-0.685165\pi$$
−0.549457 + 0.835522i $$0.685165\pi$$
$$174$$ 0 0
$$175$$ −534.562 −0.230909
$$176$$ 0 0
$$177$$ 2490.12 1.05745
$$178$$ 0 0
$$179$$ 378.742 0.158148 0.0790740 0.996869i $$-0.474804\pi$$
0.0790740 + 0.996869i $$0.474804\pi$$
$$180$$ 0 0
$$181$$ 3093.88 1.27053 0.635265 0.772294i $$-0.280891\pi$$
0.635265 + 0.772294i $$0.280891\pi$$
$$182$$ 0 0
$$183$$ 2173.74 0.878073
$$184$$ 0 0
$$185$$ −4712.82 −1.87294
$$186$$ 0 0
$$187$$ −402.590 −0.157435
$$188$$ 0 0
$$189$$ 80.7385 0.0310734
$$190$$ 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
$$195$$ 2264.82 0.831729
$$196$$ 0 0
$$197$$ −1147.74 −0.415091 −0.207546 0.978225i $$-0.566548\pi$$
−0.207546 + 0.978225i $$0.566548\pi$$
$$198$$ 0 0
$$199$$ 4842.73 1.72508 0.862542 0.505986i $$-0.168871\pi$$
0.862542 + 0.505986i $$0.168871\pi$$
$$200$$ 0 0
$$201$$ 2578.59 0.904873
$$202$$ 0 0
$$203$$ −415.408 −0.143625
$$204$$ 0 0
$$205$$ 7693.29 2.62109
$$206$$ 0 0
$$207$$ −1720.84 −0.577809
$$208$$ 0 0
$$209$$ −848.942 −0.280969
$$210$$ 0 0
$$211$$ −3222.35 −1.05135 −0.525677 0.850684i $$-0.676188\pi$$
−0.525677 + 0.850684i $$0.676188\pi$$
$$212$$ 0 0
$$213$$ −2043.79 −0.657457
$$214$$ 0 0
$$215$$ −1117.50 −0.354478
$$216$$ 0 0
$$217$$ 634.355 0.198446
$$218$$ 0 0
$$219$$ −2355.17 −0.726703
$$220$$ 0 0
$$221$$ 1640.89 0.499449
$$222$$ 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
$$227$$ 3619.49 1.05830 0.529150 0.848529i $$-0.322511\pi$$
0.529150 + 0.848529i $$0.322511\pi$$
$$228$$ 0 0
$$229$$ −305.759 −0.0882320 −0.0441160 0.999026i $$-0.514047\pi$$
−0.0441160 + 0.999026i $$0.514047\pi$$
$$230$$ 0 0
$$231$$ 95.3380 0.0271549
$$232$$ 0 0
$$233$$ −639.648 −0.179849 −0.0899244 0.995949i $$-0.528663\pi$$
−0.0899244 + 0.995949i $$0.528663\pi$$
$$234$$ 0 0
$$235$$ 7603.05 2.11050
$$236$$ 0 0
$$237$$ −3056.45 −0.837713
$$238$$ 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
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −5822.24 −1.51824
$$246$$ 0 0
$$247$$ 3460.15 0.891351
$$248$$ 0 0
$$249$$ 1402.00 0.356820
$$250$$ 0 0
$$251$$ 1317.45 0.331301 0.165650 0.986185i $$-0.447028\pi$$
0.165650 + 0.986185i $$0.447028\pi$$
$$252$$ 0 0
$$253$$ −2032.01 −0.504945
$$254$$ 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
$$259$$ 808.592 0.193990
$$260$$ 0 0
$$261$$ 1250.26 0.296510
$$262$$ 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
$$267$$ 1532.12 0.351177
$$268$$ 0 0
$$269$$ −5292.72 −1.19964 −0.599820 0.800135i $$-0.704761\pi$$
−0.599820 + 0.800135i $$0.704761\pi$$
$$270$$ 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
$$275$$ 1899.80 0.416591
$$276$$ 0 0
$$277$$ 5692.81 1.23483 0.617415 0.786638i $$-0.288180\pi$$
0.617415 + 0.786638i $$0.288180\pi$$
$$278$$ 0 0
$$279$$ −1909.23 −0.409686
$$280$$ 0 0
$$281$$ −2024.83 −0.429861 −0.214931 0.976629i $$-0.568953\pi$$
−0.214931 + 0.976629i $$0.568953\pi$$
$$282$$ 0 0
$$283$$ 247.761 0.0520419 0.0260210 0.999661i $$-0.491716\pi$$
0.0260210 + 0.999661i $$0.491716\pi$$
$$284$$ 0 0
$$285$$ 4176.77 0.868106
$$286$$ 0 0
$$287$$ −1319.96 −0.271480
$$288$$ 0 0
$$289$$ −3477.94 −0.707905
$$290$$ 0 0
$$291$$ 702.706 0.141558
$$292$$ 0 0
$$293$$ 8133.61 1.62174 0.810871 0.585225i $$-0.198994\pi$$
0.810871 + 0.585225i $$0.198994\pi$$
$$294$$ 0 0
$$295$$ −14466.6 −2.85518
$$296$$ 0 0
$$297$$ −286.940 −0.0560605
$$298$$ 0 0
$$299$$ 8282.12 1.60190
$$300$$ 0 0
$$301$$ 191.732 0.0367152
$$302$$ 0 0
$$303$$ 616.664 0.116919
$$304$$ 0 0
$$305$$ −12628.6 −2.37085
$$306$$ 0 0
$$307$$ 2974.82 0.553035 0.276518 0.961009i $$-0.410820\pi$$
0.276518 + 0.961009i $$0.410820\pi$$
$$308$$ 0 0
$$309$$ 1174.14 0.216163
$$310$$ 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 $$-0.231464\pi$$
0.747061 + 0.664755i $$0.231464\pi$$
$$314$$ 0 0
$$315$$ −469.060 −0.0839001
$$316$$ 0 0
$$317$$ 429.036 0.0760160 0.0380080 0.999277i $$-0.487899\pi$$
0.0380080 + 0.999277i $$0.487899\pi$$
$$318$$ 0 0
$$319$$ 1476.34 0.259119
$$320$$ 0 0
$$321$$ 2802.82 0.487347
$$322$$ 0 0
$$323$$ 3026.12 0.521293
$$324$$ 0 0
$$325$$ −7743.29 −1.32160
$$326$$ 0 0
$$327$$ −1752.37 −0.296349
$$328$$ 0 0
$$329$$ −1304.48 −0.218596
$$330$$ 0 0
$$331$$ 8196.71 1.36112 0.680562 0.732691i $$-0.261736\pi$$
0.680562 + 0.732691i $$0.261736\pi$$
$$332$$ 0 0
$$333$$ −2433.63 −0.400487
$$334$$ 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
$$339$$ 1748.12 0.280073
$$340$$ 0 0
$$341$$ −2254.46 −0.358023
$$342$$ 0 0
$$343$$ 2024.62 0.318715
$$344$$ 0 0
$$345$$ 9997.40 1.56012
$$346$$ 0 0
$$347$$ −7707.48 −1.19239 −0.596195 0.802840i $$-0.703321\pi$$
−0.596195 + 0.802840i $$0.703321\pi$$
$$348$$ 0 0
$$349$$ −9681.98 −1.48500 −0.742499 0.669847i $$-0.766360\pi$$
−0.742499 + 0.669847i $$0.766360\pi$$
$$350$$ 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
$$355$$ 11873.6 1.77518
$$356$$ 0 0
$$357$$ −339.840 −0.0503816
$$358$$ 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
$$363$$ 3654.17 0.528359
$$364$$ 0 0
$$365$$ 13682.7 1.96214
$$366$$ 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
$$371$$ −832.350 −0.116478
$$372$$ 0 0
$$373$$ −6956.92 −0.965726 −0.482863 0.875696i $$-0.660403\pi$$
−0.482863 + 0.875696i $$0.660403\pi$$
$$374$$ 0 0
$$375$$ −2811.16 −0.387114
$$376$$ 0 0
$$377$$ −6017.30 −0.822034
$$378$$ 0 0
$$379$$ −10201.3 −1.38260 −0.691299 0.722569i $$-0.742961\pi$$
−0.691299 + 0.722569i $$0.742961\pi$$
$$380$$ 0 0
$$381$$ 4383.10 0.589378
$$382$$ 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
$$387$$ −577.060 −0.0757974
$$388$$ 0 0
$$389$$ −546.451 −0.0712240 −0.0356120 0.999366i $$-0.511338\pi$$
−0.0356120 + 0.999366i $$0.511338\pi$$
$$390$$ 0 0
$$391$$ 7243.25 0.936846
$$392$$ 0 0
$$393$$ 295.297 0.0379027
$$394$$ 0 0
$$395$$ 17756.8 2.26188
$$396$$ 0 0
$$397$$ 2084.56 0.263529 0.131764 0.991281i $$-0.457936\pi$$
0.131764 + 0.991281i $$0.457936\pi$$
$$398$$ 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
$$403$$ 9188.81 1.13580
$$404$$ 0 0
$$405$$ 1411.74 0.173209
$$406$$ 0 0
$$407$$ −2873.69 −0.349984
$$408$$ 0 0
$$409$$ 6659.89 0.805160 0.402580 0.915385i $$-0.368114\pi$$
0.402580 + 0.915385i $$0.368114\pi$$
$$410$$ 0 0
$$411$$ 6513.17 0.781681
$$412$$ 0 0
$$413$$ 2482.08 0.295727
$$414$$ 0 0
$$415$$ −8145.09 −0.963438
$$416$$ 0 0
$$417$$ −4874.11 −0.572389
$$418$$ 0 0
$$419$$ 10576.7 1.23318 0.616592 0.787283i $$-0.288513\pi$$
0.616592 + 0.787283i $$0.288513\pi$$
$$420$$ 0 0
$$421$$ −4871.09 −0.563901 −0.281951 0.959429i $$-0.590981\pi$$
−0.281951 + 0.959429i $$0.590981\pi$$
$$422$$ 0 0
$$423$$ 3926.11 0.451286
$$424$$ 0 0
$$425$$ −6772.00 −0.772918
$$426$$ 0 0
$$427$$ 2166.72 0.245562
$$428$$ 0 0
$$429$$ 1381.00 0.155420
$$430$$ 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
$$435$$ −7263.52 −0.800596
$$436$$ 0 0
$$437$$ 15273.8 1.67196
$$438$$ 0 0
$$439$$ 1856.28 0.201812 0.100906 0.994896i $$-0.467826\pi$$
0.100906 + 0.994896i $$0.467826\pi$$
$$440$$ 0 0
$$441$$ −3006.52 −0.324643
$$442$$ 0 0
$$443$$ −1472.21 −0.157893 −0.0789465 0.996879i $$-0.525156\pi$$
−0.0789465 + 0.996879i $$0.525156\pi$$
$$444$$ 0 0
$$445$$ −8901.02 −0.948200
$$446$$ 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
$$451$$ 4691.06 0.489786
$$452$$ 0 0
$$453$$ 5458.03 0.566094
$$454$$ 0 0
$$455$$ 2257.51 0.232602
$$456$$ 0 0
$$457$$ −3613.53 −0.369877 −0.184938 0.982750i $$-0.559209\pi$$
−0.184938 + 0.982750i $$0.559209\pi$$
$$458$$ 0 0
$$459$$ 1022.82 0.104011
$$460$$ 0 0
$$461$$ −17710.7 −1.78931 −0.894654 0.446759i $$-0.852578\pi$$
−0.894654 + 0.446759i $$0.852578\pi$$
$$462$$ 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
$$467$$ −15208.3 −1.50697 −0.753484 0.657466i $$-0.771628\pi$$
−0.753484 + 0.657466i $$0.771628\pi$$
$$468$$ 0 0
$$469$$ 2570.26 0.253057
$$470$$ 0 0
$$471$$ 4969.51 0.486163
$$472$$ 0 0
$$473$$ −681.406 −0.0662391
$$474$$ 0 0
$$475$$ −14280.1 −1.37940
$$476$$ 0 0
$$477$$ 2505.14 0.240466
$$478$$ 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
$$483$$ −1715.28 −0.161590
$$484$$ 0 0
$$485$$ −4082.45 −0.382216
$$486$$ 0 0
$$487$$ 11337.7 1.05495 0.527474 0.849571i $$-0.323139\pi$$
0.527474 + 0.849571i $$0.323139\pi$$
$$488$$ 0 0
$$489$$ 6686.46 0.618348
$$490$$ 0 0
$$491$$ 14946.7 1.37380 0.686898 0.726754i $$-0.258972\pi$$
0.686898 + 0.726754i $$0.258972\pi$$
$$492$$ 0 0
$$493$$ −5262.51 −0.480754
$$494$$ 0 0
$$495$$ 1667.01 0.151367
$$496$$ 0 0
$$497$$ −2037.19 −0.183865
$$498$$ 0 0
$$499$$ −2631.77 −0.236101 −0.118051 0.993008i $$-0.537664\pi$$
−0.118051 + 0.993008i $$0.537664\pi$$
$$500$$ 0 0
$$501$$ 5001.04 0.445968
$$502$$ 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
$$507$$ 962.283 0.0842929
$$508$$ 0 0
$$509$$ 12020.3 1.04674 0.523368 0.852107i $$-0.324675\pi$$
0.523368 + 0.852107i $$0.324675\pi$$
$$510$$ 0 0
$$511$$ −2347.57 −0.203230
$$512$$ 0 0
$$513$$ 2156.82 0.185626
$$514$$ 0 0
$$515$$ −6821.29 −0.583654
$$516$$ 0 0
$$517$$ 4636.04 0.394377
$$518$$ 0 0
$$519$$ 7501.60 0.634458
$$520$$ 0 0
$$521$$ −15846.1 −1.33249 −0.666247 0.745731i $$-0.732101\pi$$
−0.666247 + 0.745731i $$0.732101\pi$$
$$522$$ 0 0
$$523$$ −8891.64 −0.743411 −0.371706 0.928351i $$-0.621227\pi$$
−0.371706 + 0.928351i $$0.621227\pi$$
$$524$$ 0 0
$$525$$ 1603.69 0.133316
$$526$$ 0 0
$$527$$ 8036.20 0.664255
$$528$$ 0 0
$$529$$ 24392.0 2.00477
$$530$$ 0 0
$$531$$ −7470.35 −0.610519
$$532$$ 0 0
$$533$$ −19120.0 −1.55381
$$534$$ 0 0
$$535$$ −16283.3 −1.31587
$$536$$ 0 0
$$537$$ −1136.23 −0.0913068
$$538$$ 0 0
$$539$$ −3550.17 −0.283705
$$540$$ 0 0
$$541$$ 12833.5 1.01988 0.509940 0.860210i $$-0.329667\pi$$
0.509940 + 0.860210i $$0.329667\pi$$
$$542$$ 0 0
$$543$$ −9281.63 −0.733541
$$544$$ 0 0
$$545$$ 10180.6 0.800162
$$546$$ 0 0
$$547$$ 16257.0 1.27075 0.635375 0.772204i $$-0.280845\pi$$
0.635375 + 0.772204i $$0.280845\pi$$
$$548$$ 0 0
$$549$$ −6521.22 −0.506956
$$550$$ 0 0
$$551$$ −11097.1 −0.857986
$$552$$ 0 0
$$553$$ −3046.59 −0.234275
$$554$$ 0 0
$$555$$ 14138.5 1.08134
$$556$$ 0 0
$$557$$ 1558.32 0.118542 0.0592712 0.998242i $$-0.481122\pi$$
0.0592712 + 0.998242i $$0.481122\pi$$
$$558$$ 0 0
$$559$$ 2777.30 0.210138
$$560$$ 0 0
$$561$$ 1207.77 0.0908951
$$562$$ 0 0
$$563$$ −9782.16 −0.732272 −0.366136 0.930561i $$-0.619319\pi$$
−0.366136 + 0.930561i $$0.619319\pi$$
$$564$$ 0 0
$$565$$ −10155.9 −0.756216
$$566$$ 0 0
$$567$$ −242.216 −0.0179402
$$568$$ 0 0
$$569$$ −7887.05 −0.581094 −0.290547 0.956861i $$-0.593837\pi$$
−0.290547 + 0.956861i $$0.593837\pi$$
$$570$$ 0 0
$$571$$ −21819.3 −1.59914 −0.799570 0.600573i $$-0.794939\pi$$
−0.799570 + 0.600573i $$0.794939\pi$$
$$572$$ 0 0
$$573$$ −10970.9 −0.799856
$$574$$ 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
$$579$$ −8364.35 −0.600363
$$580$$ 0 0
$$581$$ 1397.48 0.0997884
$$582$$ 0 0
$$583$$ 2958.12 0.210142
$$584$$ 0 0
$$585$$ −6794.46 −0.480199
$$586$$ 0 0
$$587$$ −13305.6 −0.935569 −0.467785 0.883843i $$-0.654948\pi$$
−0.467785 + 0.883843i $$0.654948\pi$$
$$588$$ 0 0
$$589$$ 16945.9 1.18548
$$590$$ 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
$$595$$ 1974.34 0.136034
$$596$$ 0 0
$$597$$ −14528.2 −0.995978
$$598$$ 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 $$-0.461643\pi$$
0.120209 + 0.992749i $$0.461643\pi$$
$$602$$ 0 0
$$603$$ −7735.76 −0.522429
$$604$$ 0 0
$$605$$ −21229.3 −1.42660
$$606$$ 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
$$611$$ −18895.7 −1.25113
$$612$$ 0 0
$$613$$ 22514.2 1.48343 0.741713 0.670717i $$-0.234014\pi$$
0.741713 + 0.670717i $$0.234014\pi$$
$$614$$ 0 0
$$615$$ −23079.9 −1.51328
$$616$$ 0 0
$$617$$ −4255.88 −0.277691 −0.138845 0.990314i $$-0.544339\pi$$
−0.138845 + 0.990314i $$0.544339\pi$$
$$618$$ 0 0
$$619$$ 228.949 0.0148663 0.00743315 0.999972i $$-0.497634\pi$$
0.00743315 + 0.999972i $$0.497634\pi$$
$$620$$ 0 0
$$621$$ 5162.51 0.333598
$$622$$ 0 0
$$623$$ 1527.17 0.0982102
$$624$$ 0 0
$$625$$ −6013.82 −0.384884
$$626$$ 0 0
$$627$$ 2546.83 0.162218
$$628$$ 0 0
$$629$$ 10243.5 0.649340
$$630$$ 0 0
$$631$$ −11429.2 −0.721058 −0.360529 0.932748i $$-0.617404\pi$$
−0.360529 + 0.932748i $$0.617404\pi$$
$$632$$ 0 0
$$633$$ 9667.04 0.606999
$$634$$ 0 0
$$635$$ −25464.1 −1.59136
$$636$$ 0 0
$$637$$ 14469.9 0.900030
$$638$$ 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
$$643$$ −249.316 −0.0152909 −0.00764546 0.999971i $$-0.502434\pi$$
−0.00764546 + 0.999971i $$0.502434\pi$$
$$644$$ 0 0
$$645$$ 3352.49 0.204658
$$646$$ 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
$$651$$ −1903.06 −0.114573
$$652$$ 0 0
$$653$$ 14008.6 0.839511 0.419755 0.907637i $$-0.362116\pi$$
0.419755 + 0.907637i $$0.362116\pi$$
$$654$$ 0 0
$$655$$ −1715.56 −0.102340
$$656$$ 0 0
$$657$$ 7065.52 0.419562
$$658$$ 0 0
$$659$$ 1011.91 0.0598152 0.0299076 0.999553i $$-0.490479\pi$$
0.0299076 + 0.999553i $$0.490479\pi$$
$$660$$ 0 0
$$661$$ 23619.4 1.38985 0.694923 0.719084i $$-0.255439\pi$$
0.694923 + 0.719084i $$0.255439\pi$$
$$662$$ 0 0
$$663$$ −4922.67 −0.288357
$$664$$ 0 0
$$665$$ 4163.28 0.242775
$$666$$ 0 0
$$667$$ −26561.6 −1.54194
$$668$$ 0 0
$$669$$ −14797.8 −0.855183
$$670$$ 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
$$675$$ −4826.64 −0.275226
$$676$$ 0 0
$$677$$ −18255.2 −1.03634 −0.518172 0.855277i $$-0.673387\pi$$
−0.518172 + 0.855277i $$0.673387\pi$$
$$678$$ 0 0
$$679$$ 700.438 0.0395881
$$680$$ 0 0
$$681$$ −10858.5 −0.611009
$$682$$ 0 0
$$683$$ 20090.4 1.12553 0.562765 0.826617i $$-0.309738\pi$$
0.562765 + 0.826617i $$0.309738\pi$$
$$684$$ 0 0
$$685$$ −37839.0 −2.11059
$$686$$ 0 0
$$687$$ 917.278 0.0509408
$$688$$ 0 0
$$689$$ −12056.8 −0.666659
$$690$$ 0 0
$$691$$ −16521.5 −0.909563 −0.454782 0.890603i $$-0.650283\pi$$
−0.454782 + 0.890603i $$0.650283\pi$$
$$692$$ 0 0
$$693$$ −286.014 −0.0156779
$$694$$ 0 0
$$695$$ 28316.7 1.54549
$$696$$ 0 0
$$697$$ −16721.7 −0.908720
$$698$$ 0 0
$$699$$ 1918.95 0.103836
$$700$$ 0 0
$$701$$ −12431.4 −0.669795 −0.334897 0.942255i $$-0.608702\pi$$
−0.334897 + 0.942255i $$0.608702\pi$$
$$702$$ 0 0
$$703$$ 21600.4 1.15886
$$704$$ 0 0
$$705$$ −22809.2 −1.21850
$$706$$ 0 0
$$707$$ 614.674 0.0326976
$$708$$ 0 0
$$709$$ 980.957 0.0519614 0.0259807 0.999662i $$-0.491729\pi$$
0.0259807 + 0.999662i $$0.491729\pi$$
$$710$$ 0 0
$$711$$ 9169.36 0.483654
$$712$$ 0 0
$$713$$ 40561.4 2.13048
$$714$$ 0 0
$$715$$ −8023.06 −0.419644
$$716$$ 0 0
$$717$$ −5234.81 −0.272660
$$718$$ 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
$$723$$ −10071.9 −0.518088
$$724$$ 0 0
$$725$$ 24833.5 1.27213
$$726$$ 0 0
$$727$$ 20850.1 1.06367 0.531833 0.846849i $$-0.321503\pi$$
0.531833 + 0.846849i $$0.321503\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 2428.92 0.122896
$$732$$ 0 0
$$733$$ 31517.2 1.58815 0.794074 0.607821i $$-0.207956\pi$$
0.794074 + 0.607821i $$0.207956\pi$$
$$734$$ 0 0
$$735$$ 17466.7 0.876558
$$736$$ 0 0
$$737$$ −9134.57 −0.456549
$$738$$ 0 0
$$739$$ −11415.0 −0.568213 −0.284106 0.958793i $$-0.591697\pi$$
−0.284106 + 0.958793i $$0.591697\pi$$
$$740$$ 0 0
$$741$$ −10380.4 −0.514622
$$742$$ 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
$$747$$ −4206.01 −0.206010
$$748$$ 0 0
$$749$$ 2793.78 0.136291
$$750$$ 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
$$755$$ −31709.0 −1.52849
$$756$$ 0 0
$$757$$ −29125.9 −1.39841 −0.699206 0.714920i $$-0.746463\pi$$
−0.699206 + 0.714920i $$0.746463\pi$$
$$758$$ 0 0
$$759$$ 6096.02 0.291530
$$760$$ 0 0
$$761$$ −14228.5 −0.677768 −0.338884 0.940828i $$-0.610049\pi$$
−0.338884 + 0.940828i $$0.610049\pi$$
$$762$$ 0 0
$$763$$ −1746.71 −0.0828771
$$764$$ 0 0
$$765$$ −5942.19 −0.280837
$$766$$ 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
$$771$$ 9108.35 0.425459
$$772$$ 0 0
$$773$$ −14686.9 −0.683377 −0.341689 0.939813i $$-0.610999\pi$$
−0.341689 + 0.939813i $$0.610999\pi$$
$$774$$ 0 0
$$775$$ −37922.5 −1.75770
$$776$$ 0 0
$$777$$ −2425.78 −0.112000
$$778$$ 0 0
$$779$$ −35260.9 −1.62176
$$780$$ 0 0
$$781$$ 7240.08 0.331716
$$782$$ 0 0
$$783$$ −3750.78 −0.171190
$$784$$ 0 0
$$785$$ −28870.9 −1.31267
$$786$$ 0 0
$$787$$ −21001.0 −0.951214 −0.475607 0.879658i $$-0.657772\pi$$
−0.475607 + 0.879658i $$0.657772\pi$$
$$788$$ 0 0
$$789$$ −4967.27 −0.224131
$$790$$ 0 0
$$791$$ 1742.48 0.0783253
$$792$$ 0 0
$$793$$ 31385.6 1.40547
$$794$$ 0 0
$$795$$ −14553.9 −0.649274
$$796$$ 0 0
$$797$$ 13362.7 0.593892 0.296946 0.954894i $$-0.404032\pi$$
0.296946 + 0.954894i $$0.404032\pi$$
$$798$$ 0 0
$$799$$ −16525.5 −0.731704
$$800$$ 0 0
$$801$$ −4596.36 −0.202752
$$802$$ 0 0
$$803$$ 8343.14 0.366654
$$804$$ 0 0
$$805$$ 9965.13 0.436304
$$806$$ 0 0
$$807$$ 15878.2 0.692612
$$808$$ 0 0
$$809$$ 39481.6 1.71582 0.857911 0.513798i $$-0.171762\pi$$
0.857911 + 0.513798i $$0.171762\pi$$
$$810$$ 0 0
$$811$$ 31157.5 1.34906 0.674531 0.738247i $$-0.264346\pi$$
0.674531 + 0.738247i $$0.264346\pi$$
$$812$$ 0 0
$$813$$ 24031.6 1.03668
$$814$$ 0 0
$$815$$ −38845.8 −1.66958
$$816$$ 0 0
$$817$$ 5121.87 0.219329
$$818$$ 0 0
$$819$$ 1165.75 0.0497368
$$820$$ 0 0
$$821$$ −21229.9 −0.902470 −0.451235 0.892405i $$-0.649016\pi$$
−0.451235 + 0.892405i $$0.649016\pi$$
$$822$$ 0 0
$$823$$ 24603.8 1.04208 0.521041 0.853532i $$-0.325544\pi$$
0.521041 + 0.853532i $$0.325544\pi$$
$$824$$ 0 0
$$825$$ −5699.41 −0.240519
$$826$$ 0 0
$$827$$ 13668.1 0.574710 0.287355 0.957824i $$-0.407224\pi$$
0.287355 + 0.957824i $$0.407224\pi$$
$$828$$ 0 0
$$829$$ 27518.8 1.15291 0.576457 0.817127i $$-0.304435\pi$$
0.576457 + 0.817127i $$0.304435\pi$$
$$830$$ 0 0
$$831$$ −17078.4 −0.712929
$$832$$ 0 0
$$833$$ 12654.9 0.526369
$$834$$ 0 0
$$835$$ −29054.1 −1.20414
$$836$$ 0 0
$$837$$ 5727.68 0.236532
$$838$$ 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
$$843$$ 6074.48 0.248180
$$844$$ 0 0
$$845$$ −5590.49 −0.227596
$$846$$ 0 0
$$847$$ 3642.38 0.147761
$$848$$ 0 0
$$849$$ −743.283 −0.0300464
$$850$$ 0 0
$$851$$ 51702.3 2.08265
$$852$$ 0 0
$$853$$ 5174.61 0.207708 0.103854 0.994593i $$-0.466882\pi$$
0.103854 + 0.994593i $$0.466882\pi$$
$$854$$ 0 0
$$855$$ −12530.3 −0.501201
$$856$$ 0 0
$$857$$ −9258.34 −0.369030 −0.184515 0.982830i $$-0.559071\pi$$
−0.184515 + 0.982830i $$0.559071\pi$$
$$858$$ 0 0
$$859$$ 24353.0 0.967304 0.483652 0.875260i $$-0.339310\pi$$
0.483652 + 0.875260i $$0.339310\pi$$
$$860$$ 0 0
$$861$$ 3959.88 0.156739
$$862$$ 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
$$867$$ 10433.8 0.408709
$$868$$ 0 0
$$869$$ 10827.4 0.422663
$$870$$ 0 0
$$871$$ 37231.0 1.44836
$$872$$ 0 0
$$873$$ −2108.12 −0.0817286
$$874$$ 0 0
$$875$$ −2802.08 −0.108260
$$876$$ 0 0
$$877$$ −49843.1 −1.91914 −0.959568 0.281476i $$-0.909176\pi$$
−0.959568 + 0.281476i $$0.909176\pi$$
$$878$$ 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
$$883$$ 3693.99 0.140784 0.0703922 0.997519i $$-0.477575\pi$$
0.0703922 + 0.997519i $$0.477575\pi$$
$$884$$ 0 0
$$885$$ 43399.8 1.64844
$$886$$ 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
$$891$$ 860.821 0.0323665
$$892$$ 0 0
$$893$$ −34847.4 −1.30585
$$894$$ 0 0
$$895$$ 6601.03 0.246534
$$896$$ 0 0
$$897$$ −24846.4 −0.924856
$$898$$ 0 0
$$899$$ −29469.5 −1.09328
$$900$$ 0 0
$$901$$ −10544.5 −0.389886
$$902$$ 0 0
$$903$$ −575.197 −0.0211975
$$904$$ 0 0
$$905$$ 53922.7 1.98061
$$906$$ 0 0
$$907$$ −17016.8 −0.622971 −0.311486 0.950251i $$-0.600827\pi$$
−0.311486 + 0.950251i $$0.600827\pi$$
$$908$$ 0 0
$$909$$ −1849.99 −0.0675032
$$910$$ 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
$$915$$ 37885.7 1.36881
$$916$$ 0 0
$$917$$ 294.344 0.0105999
$$918$$ 0 0
$$919$$ −5174.82 −0.185747 −0.0928736 0.995678i $$-0.529605\pi$$
−0.0928736 + 0.995678i $$0.529605\pi$$
$$920$$ 0 0
$$921$$ −8924.46 −0.319295
$$922$$ 0 0
$$923$$ −29509.3 −1.05234
$$924$$ 0 0
$$925$$ −48338.6 −1.71823
$$926$$ 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
$$931$$ 26685.3 0.939394
$$932$$ 0 0
$$933$$ −13354.6 −0.468605
$$934$$ 0 0
$$935$$ −7016.69 −0.245423
$$936$$ 0 0
$$937$$ −31566.5 −1.10057 −0.550283 0.834978i $$-0.685480\pi$$
−0.550283 + 0.834978i $$0.685480\pi$$
$$938$$ 0 0
$$939$$ −24821.3 −0.862632
$$940$$ 0 0
$$941$$ 37575.6 1.30173 0.650866 0.759193i $$-0.274406\pi$$
0.650866 + 0.759193i $$0.274406\pi$$
$$942$$ 0 0
$$943$$ −84399.7 −2.91456
$$944$$ 0 0
$$945$$ 1407.18 0.0484397
$$946$$ 0 0
$$947$$ 10289.3 0.353070 0.176535 0.984294i $$-0.443511\pi$$
0.176535 + 0.984294i $$0.443511\pi$$
$$948$$ 0 0
$$949$$ −34005.2 −1.16318
$$950$$ 0 0
$$951$$ −1287.11 −0.0438879
$$952$$ 0 0
$$953$$ 36779.7 1.25017 0.625085 0.780557i $$-0.285064\pi$$
0.625085 + 0.780557i $$0.285064\pi$$
$$954$$ 0 0
$$955$$ 63736.9 2.15966
$$956$$ 0 0
$$957$$ −4429.01 −0.149602
$$958$$ 0 0
$$959$$ 6492.15 0.218605
$$960$$ 0 0
$$961$$ 15210.9 0.510586
$$962$$ 0 0
$$963$$ −8408.47 −0.281370
$$964$$ 0 0
$$965$$ 48593.6 1.62102
$$966$$ 0 0
$$967$$ −35228.9 −1.17155 −0.585773 0.810475i $$-0.699209\pi$$
−0.585773 + 0.810475i $$0.699209\pi$$
$$968$$ 0 0
$$969$$ −9078.36 −0.300969
$$970$$ 0 0
$$971$$ 15314.8 0.506155 0.253077 0.967446i $$-0.418557\pi$$
0.253077 + 0.967446i $$0.418557\pi$$
$$972$$ 0 0
$$973$$ −4858.38 −0.160074
$$974$$ 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
$$979$$ −5427.49 −0.177184
$$980$$ 0 0
$$981$$ 5257.10 0.171097
$$982$$ 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
$$987$$ 3913.43 0.126207
$$988$$ 0 0
$$989$$ 12259.6 0.394168
$$990$$ 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
$$995$$ 84403.1 2.68920
$$996$$ 0 0
$$997$$ −1717.01 −0.0545420 −0.0272710 0.999628i $$-0.508682\pi$$
−0.0272710 + 0.999628i $$0.508682\pi$$
$$998$$ 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 768.4.a.u.1.4 4
3.2 odd 2 2304.4.a.cb.1.1 4
4.3 odd 2 768.4.a.v.1.4 4
8.3 odd 2 inner 768.4.a.u.1.1 4
8.5 even 2 768.4.a.v.1.1 4
12.11 even 2 2304.4.a.by.1.1 4
16.3 odd 4 384.4.d.f.193.5 yes 8
16.5 even 4 384.4.d.f.193.8 yes 8
16.11 odd 4 384.4.d.f.193.4 yes 8
16.13 even 4 384.4.d.f.193.1 8
24.5 odd 2 2304.4.a.by.1.4 4
24.11 even 2 2304.4.a.cb.1.4 4
48.5 odd 4 1152.4.d.p.577.2 8
48.11 even 4 1152.4.d.p.577.1 8
48.29 odd 4 1152.4.d.p.577.8 8
48.35 even 4 1152.4.d.p.577.7 8

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