# Properties

 Label 2304.4.a.by.1.1 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, 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("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$135.940400653$$ Analytic rank: $$0$$ 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 1.1 Root $$1.06909$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-17.4288 q^{5} +2.99032 q^{7} +O(q^{10})$$ $$q-17.4288 q^{5} +2.99032 q^{7} +10.6274 q^{11} -43.3156 q^{13} +37.8823 q^{17} +79.8823 q^{19} -191.204 q^{23} +178.765 q^{25} -138.918 q^{29} +212.136 q^{31} -52.1177 q^{35} -270.404 q^{37} -441.411 q^{41} +64.1177 q^{43} +436.234 q^{47} -334.058 q^{49} -278.348 q^{53} -185.224 q^{55} -830.039 q^{59} -724.580 q^{61} +754.940 q^{65} +859.529 q^{67} +681.264 q^{71} +785.058 q^{73} +31.7793 q^{77} -1018.82 q^{79} -467.334 q^{83} -660.244 q^{85} +510.706 q^{89} -129.527 q^{91} -1392.26 q^{95} -234.235 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 48 q^{11} - 120 q^{17} + 48 q^{19} + 172 q^{25} - 480 q^{35} - 408 q^{41} + 528 q^{43} + 836 q^{49} - 1872 q^{59} + 576 q^{65} + 2352 q^{67} + 968 q^{73} - 3408 q^{83} + 3672 q^{89} + 5184 q^{91} - 1480 q^{97}+O(q^{100})$$ 4 * q - 48 * q^11 - 120 * q^17 + 48 * q^19 + 172 * q^25 - 480 * q^35 - 408 * q^41 + 528 * q^43 + 836 * q^49 - 1872 * q^59 + 576 * q^65 + 2352 * q^67 + 968 * q^73 - 3408 * q^83 + 3672 * q^89 + 5184 * q^91 - 1480 * q^97

## 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
$$5$$ −17.4288 −1.55888 −0.779441 0.626475i $$-0.784497\pi$$
−0.779441 + 0.626475i $$0.784497\pi$$
$$6$$ 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
$$11$$ 10.6274 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
$$15$$ 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
$$19$$ 79.8823 0.964539 0.482270 0.876023i $$-0.339813\pi$$
0.482270 + 0.876023i $$0.339813\pi$$
$$20$$ 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
$$29$$ −138.918 −0.889530 −0.444765 0.895647i $$-0.646713\pi$$
−0.444765 + 0.895647i $$0.646713\pi$$
$$30$$ 0 0
$$31$$ 212.136 1.22906 0.614529 0.788894i $$-0.289346\pi$$
0.614529 + 0.788894i $$0.289346\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 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
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −441.411 −1.68139 −0.840693 0.541511i $$-0.817852\pi$$
−0.840693 + 0.541511i $$0.817852\pi$$
$$42$$ 0 0
$$43$$ 64.1177 0.227392 0.113696 0.993516i $$-0.463731\pi$$
0.113696 + 0.993516i $$0.463731\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 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
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −278.348 −0.721398 −0.360699 0.932682i $$-0.617462\pi$$
−0.360699 + 0.932682i $$0.617462\pi$$
$$54$$ 0 0
$$55$$ −185.224 −0.454101
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 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
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 754.940 1.44060
$$66$$ 0 0
$$67$$ 859.529 1.56729 0.783643 0.621211i $$-0.213359\pi$$
0.783643 + 0.621211i $$0.213359\pi$$
$$68$$ 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 $$-0.283324\pi$$
0.629343 + 0.777128i $$0.283324\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 31.7793 0.0470337
$$78$$ 0 0
$$79$$ −1018.82 −1.45096 −0.725481 0.688243i $$-0.758382\pi$$
−0.725481 + 0.688243i $$0.758382\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 510.706 0.608256 0.304128 0.952631i $$-0.401635\pi$$
0.304128 + 0.952631i $$0.401635\pi$$
$$90$$ 0 0
$$91$$ −129.527 −0.149210
$$92$$ 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
$$101$$ 205.555 0.202509 0.101255 0.994861i $$-0.467714\pi$$
0.101255 + 0.994861i $$0.467714\pi$$
$$102$$ 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
$$107$$ −934.274 −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
$$111$$ 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
$$115$$ 3332.47 2.70221
$$116$$ 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
$$125$$ −937.053 −0.670501
$$126$$ 0 0
$$127$$ 1461.03 1.02083 0.510416 0.859928i $$-0.329491\pi$$
0.510416 + 0.859928i $$0.329491\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 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
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2171.06 1.35391 0.676956 0.736024i $$-0.263299\pi$$
0.676956 + 0.736024i $$0.263299\pi$$
$$138$$ 0 0
$$139$$ −1624.70 −0.991407 −0.495703 0.868492i $$-0.665090\pi$$
−0.495703 + 0.868492i $$0.665090\pi$$
$$140$$ 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
$$149$$ −636.658 −0.350047 −0.175024 0.984564i $$-0.556000\pi$$
−0.175024 + 0.984564i $$0.556000\pi$$
$$150$$ 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
$$155$$ −3697.29 −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
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −571.761 −0.279882
$$162$$ 0 0
$$163$$ 2228.82 1.07101 0.535505 0.844532i $$-0.320121\pi$$
0.535505 + 0.844532i $$0.320121\pi$$
$$164$$ 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
$$173$$ 2500.53 1.09891 0.549457 0.835522i $$-0.314835\pi$$
0.549457 + 0.835522i $$0.314835\pi$$
$$174$$ 0 0
$$175$$ 534.562 0.230909
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 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
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4712.82 1.87294
$$186$$ 0 0
$$187$$ 402.590 0.157435
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1147.74 0.415091 0.207546 0.978225i $$-0.433452\pi$$
0.207546 + 0.978225i $$0.433452\pi$$
$$198$$ 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
$$203$$ −415.408 −0.143625
$$204$$ 0 0
$$205$$ 7693.29 2.62109
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 848.942 0.280969
$$210$$ 0 0
$$211$$ 3222.35 1.05135 0.525677 0.850684i $$-0.323812\pi$$
0.525677 + 0.850684i $$0.323812\pi$$
$$212$$ 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
$$221$$ −1640.89 −0.499449
$$222$$ 0 0
$$223$$ −4932.61 −1.48122 −0.740610 0.671935i $$-0.765463\pi$$
−0.740610 + 0.671935i $$0.765463\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 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
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 639.648 0.179849 0.0899244 0.995949i $$-0.471337\pi$$
0.0899244 + 0.995949i $$0.471337\pi$$
$$234$$ 0 0
$$235$$ −7603.05 −2.11050
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 5822.24 1.51824
$$246$$ 0 0
$$247$$ −3460.15 −0.891351
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$255$$ 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
$$259$$ −808.592 −0.193990
$$260$$ 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
$$269$$ 5292.72 1.19964 0.599820 0.800135i $$-0.295239\pi$$
0.599820 + 0.800135i $$0.295239\pi$$
$$270$$ 0 0
$$271$$ 8010.52 1.79559 0.897795 0.440414i $$-0.145168\pi$$
0.897795 + 0.440414i $$0.145168\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2024.83 0.429861 0.214931 0.976629i $$-0.431047\pi$$
0.214931 + 0.976629i $$0.431047\pi$$
$$282$$ 0 0
$$283$$ −247.761 −0.0520419 −0.0260210 0.999661i $$-0.508284\pi$$
−0.0260210 + 0.999661i $$0.508284\pi$$
$$284$$ 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
$$293$$ −8133.61 −1.62174 −0.810871 0.585225i $$-0.801006\pi$$
−0.810871 + 0.585225i $$0.801006\pi$$
$$294$$ 0 0
$$295$$ 14466.6 2.85518
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 8282.12 1.60190
$$300$$ 0 0
$$301$$ 191.732 0.0367152
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12628.6 2.37085
$$306$$ 0 0
$$307$$ −2974.82 −0.553035 −0.276518 0.961009i $$-0.589180\pi$$
−0.276518 + 0.961009i $$0.589180\pi$$
$$308$$ 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 $$-0.231464\pi$$
0.747061 + 0.664755i $$0.231464\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −429.036 −0.0760160 −0.0380080 0.999277i $$-0.512101\pi$$
−0.0380080 + 0.999277i $$0.512101\pi$$
$$318$$ 0 0
$$319$$ −1476.34 −0.259119
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3026.12 0.521293
$$324$$ 0 0
$$325$$ −7743.29 −1.32160
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1304.48 0.218596
$$330$$ 0 0
$$331$$ −8196.71 −1.36112 −0.680562 0.732691i $$-0.738264\pi$$
−0.680562 + 0.732691i $$0.738264\pi$$
$$332$$ 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
$$341$$ 2254.46 0.358023
$$342$$ 0 0
$$343$$ −2024.62 −0.318715
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 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
$$351$$ 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
$$355$$ −11873.6 −1.77518
$$356$$ 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
$$365$$ −13682.7 −1.96214
$$366$$ 0 0
$$367$$ −11272.4 −1.60331 −0.801657 0.597785i $$-0.796048\pi$$
−0.801657 + 0.597785i $$0.796048\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6017.30 0.822034
$$378$$ 0 0
$$379$$ 10201.3 1.38260 0.691299 0.722569i $$-0.257039\pi$$
0.691299 + 0.722569i $$0.257039\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 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
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 546.451 0.0712240 0.0356120 0.999366i $$-0.488662\pi$$
0.0356120 + 0.999366i $$0.488662\pi$$
$$390$$ 0 0
$$391$$ −7243.25 −0.936846
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 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
$$399$$ 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
$$403$$ −9188.81 −1.13580
$$404$$ 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 $$-0.368114\pi$$
0.402580 + 0.915385i $$0.368114\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2482.08 −0.295727
$$414$$ 0 0
$$415$$ 8145.09 0.963438
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6772.00 0.772918
$$426$$ 0 0
$$427$$ −2166.72 −0.245562
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −15273.8 −1.67196
$$438$$ 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
$$443$$ −1472.21 −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
$$447$$ 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
$$451$$ −4691.06 −0.489786
$$452$$ 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 $$-0.559209\pi$$
−0.184938 + 0.982750i $$0.559209\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 17710.7 1.78931 0.894654 0.446759i $$-0.147422\pi$$
0.894654 + 0.446759i $$0.147422\pi$$
$$462$$ 0 0
$$463$$ 1674.57 0.168087 0.0840433 0.996462i $$-0.473217\pi$$
0.0840433 + 0.996462i $$0.473217\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 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
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 681.406 0.0662391
$$474$$ 0 0
$$475$$ 14280.1 1.37940
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 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
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4082.45 0.382216
$$486$$ 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
$$491$$ 14946.7 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
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2037.19 0.183865
$$498$$ 0 0
$$499$$ 2631.77 0.236101 0.118051 0.993008i $$-0.462336\pi$$
0.118051 + 0.993008i $$0.462336\pi$$
$$500$$ 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
$$509$$ −12020.3 −1.04674 −0.523368 0.852107i $$-0.675325\pi$$
−0.523368 + 0.852107i $$0.675325\pi$$
$$510$$ 0 0
$$511$$ 2347.57 0.203230
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −6821.29 −0.583654
$$516$$ 0 0
$$517$$ 4636.04 0.394377
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 15846.1 1.33249 0.666247 0.745731i $$-0.267899\pi$$
0.666247 + 0.745731i $$0.267899\pi$$
$$522$$ 0 0
$$523$$ 8891.64 0.743411 0.371706 0.928351i $$-0.378773\pi$$
0.371706 + 0.928351i $$0.378773\pi$$
$$524$$ 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
$$533$$ 19120.0 1.55381
$$534$$ 0 0
$$535$$ 16283.3 1.31587
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −10180.6 −0.800162
$$546$$ 0 0
$$547$$ −16257.0 −1.27075 −0.635375 0.772204i $$-0.719155\pi$$
−0.635375 + 0.772204i $$0.719155\pi$$
$$548$$ 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
$$557$$ −1558.32 −0.118542 −0.0592712 0.998242i $$-0.518878\pi$$
−0.0592712 + 0.998242i $$0.518878\pi$$
$$558$$ 0 0
$$559$$ −2777.30 −0.210138
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 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
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 7887.05 0.581094 0.290547 0.956861i $$-0.406163\pi$$
0.290547 + 0.956861i $$0.406163\pi$$
$$570$$ 0 0
$$571$$ 21819.3 1.59914 0.799570 0.600573i $$-0.205061\pi$$
0.799570 + 0.600573i $$0.205061\pi$$
$$572$$ 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
$$581$$ −1397.48 −0.0997884
$$582$$ 0 0
$$583$$ −2958.12 −0.210142
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 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
$$591$$ 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
$$595$$ −1974.34 −0.136034
$$596$$ 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 $$-0.461643\pi$$
0.120209 + 0.992749i $$0.461643\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 21229.3 1.42660
$$606$$ 0 0
$$607$$ 6050.64 0.404593 0.202296 0.979324i $$-0.435160\pi$$
0.202296 + 0.979324i $$0.435160\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4255.88 0.277691 0.138845 0.990314i $$-0.455661\pi$$
0.138845 + 0.990314i $$0.455661\pi$$
$$618$$ 0 0
$$619$$ −228.949 −0.0148663 −0.00743315 0.999972i $$-0.502366\pi$$
−0.00743315 + 0.999972i $$0.502366\pi$$
$$620$$ 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
$$629$$ −10243.5 −0.649340
$$630$$ 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
$$635$$ −25464.1 −1.59136
$$636$$ 0 0
$$637$$ 14469.9 0.900030
$$638$$ 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
$$643$$ 249.316 0.0152909 0.00764546 0.999971i $$-0.497566\pi$$
0.00764546 + 0.999971i $$0.497566\pi$$
$$644$$ 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
$$653$$ −14008.6 −0.839511 −0.419755 0.907637i $$-0.637884\pi$$
−0.419755 + 0.907637i $$0.637884\pi$$
$$654$$ 0 0
$$655$$ 1715.56 0.102340
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 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
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4163.28 −0.242775
$$666$$ 0 0
$$667$$ 26561.6 1.54194
$$668$$ 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
$$677$$ 18255.2 1.03634 0.518172 0.855277i $$-0.326613\pi$$
0.518172 + 0.855277i $$0.326613\pi$$
$$678$$ 0 0
$$679$$ −700.438 −0.0395881
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 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
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 12056.8 0.666659
$$690$$ 0 0
$$691$$ 16521.5 0.909563 0.454782 0.890603i $$-0.349717\pi$$
0.454782 + 0.890603i $$0.349717\pi$$
$$692$$ 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
$$701$$ 12431.4 0.669795 0.334897 0.942255i $$-0.391298\pi$$
0.334897 + 0.942255i $$0.391298\pi$$
$$702$$ 0 0
$$703$$ −21600.4 −1.15886
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −40561.4 −2.13048
$$714$$ 0 0
$$715$$ 8023.06 0.419644
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 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
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −24833.5 −1.27213
$$726$$ 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
$$731$$ 2428.92 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
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 9134.57 0.456549
$$738$$ 0 0
$$739$$ 11415.0 0.568213 0.284106 0.958793i $$-0.408303\pi$$
0.284106 + 0.958793i $$0.408303\pi$$
$$740$$ 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
$$749$$ −2793.78 −0.136291
$$750$$ 0 0
$$751$$ −7843.07 −0.381089 −0.190544 0.981679i $$-0.561025\pi$$
−0.190544 + 0.981679i $$0.561025\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 14228.5 0.677768 0.338884 0.940828i $$-0.389951\pi$$
0.338884 + 0.940828i $$0.389951\pi$$
$$762$$ 0 0
$$763$$ 1746.71 0.0828771
$$764$$ 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
$$773$$ 14686.9 0.683377 0.341689 0.939813i $$-0.389001\pi$$
0.341689 + 0.939813i $$0.389001\pi$$
$$774$$ 0 0
$$775$$ 37922.5 1.75770
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −35260.9 −1.62176
$$780$$ 0 0
$$781$$ 7240.08 0.331716
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 28870.9 1.31267
$$786$$ 0 0
$$787$$ 21001.0 0.951214 0.475607 0.879658i $$-0.342228\pi$$
0.475607 + 0.879658i $$0.342228\pi$$
$$788$$ 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
$$797$$ −13362.7 −0.593892 −0.296946 0.954894i $$-0.595968\pi$$
−0.296946 + 0.954894i $$0.595968\pi$$
$$798$$ 0 0
$$799$$ 16525.5 0.731704
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8343.14 0.366654
$$804$$ 0 0
$$805$$ 9965.13 0.436304
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −39481.6 −1.71582 −0.857911 0.513798i $$-0.828238\pi$$
−0.857911 + 0.513798i $$0.828238\pi$$
$$810$$ 0 0
$$811$$ −31157.5 −1.34906 −0.674531 0.738247i $$-0.735654\pi$$
−0.674531 + 0.738247i $$0.735654\pi$$
$$812$$ 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
$$821$$ 21229.9 0.902470 0.451235 0.892405i $$-0.350984\pi$$
0.451235 + 0.892405i $$0.350984\pi$$
$$822$$ 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
$$827$$ 13668.1 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
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −12654.9 −0.526369
$$834$$ 0 0
$$835$$ 29054.1 1.20414
$$836$$ 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
$$845$$ 5590.49 0.227596
$$846$$ 0 0
$$847$$ −3642.38 −0.147761
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 9258.34 0.369030 0.184515 0.982830i $$-0.440929\pi$$
0.184515 + 0.982830i $$0.440929\pi$$
$$858$$ 0 0
$$859$$ −24353.0 −0.967304 −0.483652 0.875260i $$-0.660690\pi$$
−0.483652 + 0.875260i $$0.660690\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −10827.4 −0.422663
$$870$$ 0 0
$$871$$ −37231.0 −1.44836
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 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
$$879$$ 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
$$883$$ −3693.99 −0.140784 −0.0703922 0.997519i $$-0.522425\pi$$
−0.0703922 + 0.997519i $$0.522425\pi$$
$$884$$ 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
$$893$$ 34847.4 1.30585
$$894$$ 0 0
$$895$$ −6601.03 −0.246534
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −29469.5 −1.09328
$$900$$ 0 0
$$901$$ −10544.5 −0.389886
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −53922.7 −1.98061
$$906$$ 0 0
$$907$$ 17016.8 0.622971 0.311486 0.950251i $$-0.399173\pi$$
0.311486 + 0.950251i $$0.399173\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 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
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −294.344 −0.0105999
$$918$$ 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
$$923$$ −29509.3 −1.05234
$$924$$ 0 0
$$925$$ −48338.6 −1.71823
$$926$$ 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
$$931$$ −26685.3 −0.939394
$$932$$ 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 $$-0.685480\pi$$
−0.550283 + 0.834978i $$0.685480\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −37575.6 −1.30173 −0.650866 0.759193i $$-0.725594\pi$$
−0.650866 + 0.759193i $$0.725594\pi$$
$$942$$ 0 0
$$943$$ 84399.7 2.91456
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 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
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −36779.7 −1.25017 −0.625085 0.780557i $$-0.714936\pi$$
−0.625085 + 0.780557i $$0.714936\pi$$
$$954$$ 0 0
$$955$$ −63736.9 −2.15966
$$956$$ 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
$$965$$ −48593.6 −1.62102
$$966$$ 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
$$971$$ 15314.8 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
$$975$$ 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
$$979$$ 5427.49 0.177184
$$980$$ 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
$$989$$ −12259.6 −0.394168
$$990$$ 0 0
$$991$$ −37518.6 −1.20264 −0.601321 0.799008i $$-0.705359\pi$$
−0.601321 + 0.799008i $$0.705359\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 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
$$999$$ 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 2304.4.a.by.1.1 4
3.2 odd 2 768.4.a.v.1.4 4
4.3 odd 2 2304.4.a.cb.1.1 4
8.3 odd 2 inner 2304.4.a.by.1.4 4
8.5 even 2 2304.4.a.cb.1.4 4
12.11 even 2 768.4.a.u.1.4 4
16.3 odd 4 1152.4.d.p.577.8 8
16.5 even 4 1152.4.d.p.577.1 8
16.11 odd 4 1152.4.d.p.577.2 8
16.13 even 4 1152.4.d.p.577.7 8
24.5 odd 2 768.4.a.u.1.1 4
24.11 even 2 768.4.a.v.1.1 4
48.5 odd 4 384.4.d.f.193.4 yes 8
48.11 even 4 384.4.d.f.193.8 yes 8
48.29 odd 4 384.4.d.f.193.5 yes 8
48.35 even 4 384.4.d.f.193.1 8

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