# Properties

 Label 720.4.f.j.289.2 Level $720$ Weight $4$ Character 720.289 Analytic conductor $42.481$ 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] = [720,4,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not 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: $$42.4813752041$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 289.2 Root $$-2.70156i$$ of defining polynomial Character $$\chi$$ $$=$$ 720.289 Dual form 720.4.f.j.289.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+(-11.1047 + 1.29844i) q^{5} -16.2094i q^{7} +O(q^{10})$$ $$q+(-11.1047 + 1.29844i) q^{5} -16.2094i q^{7} -40.2094 q^{11} +19.7906i q^{13} +83.0156i q^{17} -48.8375 q^{19} +1.61250i q^{23} +(121.628 - 28.8375i) q^{25} -24.5344 q^{29} +12.4187 q^{31} +(21.0469 + 180.000i) q^{35} -325.884i q^{37} +242.419 q^{41} -367.350i q^{43} +204.544i q^{47} +80.2562 q^{49} +61.5281i q^{53} +(446.512 - 52.2094i) q^{55} +112.209 q^{59} +477.350 q^{61} +(-25.6969 - 219.769i) q^{65} +558.094i q^{67} +558.281 q^{71} +1011.77i q^{73} +651.769i q^{77} +1150.47 q^{79} -1157.92i q^{83} +(-107.791 - 921.862i) q^{85} +96.9751 q^{89} +320.794 q^{91} +(542.325 - 63.4124i) q^{95} +1152.37i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5}+O(q^{10})$$ 4 * q - 6 * q^5 $$4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85} + 2232 q^{89} - 1944 q^{91} + 2784 q^{95}+O(q^{100})$$ 4 * q - 6 * q^5 - 84 * q^11 + 112 * q^19 + 256 * q^25 - 636 * q^29 - 104 * q^31 - 300 * q^35 + 816 * q^41 - 140 * q^49 + 864 * q^55 + 372 * q^59 + 680 * q^61 - 948 * q^65 - 72 * q^71 + 760 * q^79 - 508 * q^85 + 2232 * q^89 - 1944 * q^91 + 2784 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −11.1047 + 1.29844i −0.993233 + 0.116136i
$$6$$ 0 0
$$7$$ 16.2094i 0.875224i −0.899164 0.437612i $$-0.855824\pi$$
0.899164 0.437612i $$-0.144176\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −40.2094 −1.10214 −0.551072 0.834458i $$-0.685781\pi$$
−0.551072 + 0.834458i $$0.685781\pi$$
$$12$$ 0 0
$$13$$ 19.7906i 0.422226i 0.977462 + 0.211113i $$0.0677087\pi$$
−0.977462 + 0.211113i $$0.932291\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 83.0156i 1.18437i 0.805803 + 0.592184i $$0.201734\pi$$
−0.805803 + 0.592184i $$0.798266\pi$$
$$18$$ 0 0
$$19$$ −48.8375 −0.589689 −0.294844 0.955545i $$-0.595268\pi$$
−0.294844 + 0.955545i $$0.595268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.61250i 0.0146186i 0.999973 + 0.00730932i $$0.00232665\pi$$
−0.999973 + 0.00730932i $$0.997673\pi$$
$$24$$ 0 0
$$25$$ 121.628 28.8375i 0.973025 0.230700i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −24.5344 −0.157101 −0.0785504 0.996910i $$-0.525029\pi$$
−0.0785504 + 0.996910i $$0.525029\pi$$
$$30$$ 0 0
$$31$$ 12.4187 0.0719507 0.0359754 0.999353i $$-0.488546\pi$$
0.0359754 + 0.999353i $$0.488546\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 21.0469 + 180.000i 0.101645 + 0.869302i
$$36$$ 0 0
$$37$$ 325.884i 1.44797i −0.689813 0.723987i $$-0.742307\pi$$
0.689813 0.723987i $$-0.257693\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 242.419 0.923401 0.461701 0.887036i $$-0.347239\pi$$
0.461701 + 0.887036i $$0.347239\pi$$
$$42$$ 0 0
$$43$$ 367.350i 1.30280i −0.758735 0.651399i $$-0.774182\pi$$
0.758735 0.651399i $$-0.225818\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 204.544i 0.634804i 0.948291 + 0.317402i $$0.102810\pi$$
−0.948291 + 0.317402i $$0.897190\pi$$
$$48$$ 0 0
$$49$$ 80.2562 0.233983
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 61.5281i 0.159463i 0.996816 + 0.0797314i $$0.0254063\pi$$
−0.996816 + 0.0797314i $$0.974594\pi$$
$$54$$ 0 0
$$55$$ 446.512 52.2094i 1.09469 0.127998i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 112.209 0.247600 0.123800 0.992307i $$-0.460492\pi$$
0.123800 + 0.992307i $$0.460492\pi$$
$$60$$ 0 0
$$61$$ 477.350 1.00194 0.500970 0.865464i $$-0.332977\pi$$
0.500970 + 0.865464i $$0.332977\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −25.6969 219.769i −0.0490355 0.419369i
$$66$$ 0 0
$$67$$ 558.094i 1.01764i 0.860872 + 0.508821i $$0.169918\pi$$
−0.860872 + 0.508821i $$0.830082\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 558.281 0.933180 0.466590 0.884474i $$-0.345482\pi$$
0.466590 + 0.884474i $$0.345482\pi$$
$$72$$ 0 0
$$73$$ 1011.77i 1.62217i 0.584927 + 0.811086i $$0.301123\pi$$
−0.584927 + 0.811086i $$0.698877\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 651.769i 0.964623i
$$78$$ 0 0
$$79$$ 1150.47 1.63845 0.819227 0.573470i $$-0.194403\pi$$
0.819227 + 0.573470i $$0.194403\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1157.92i 1.53131i −0.643251 0.765655i $$-0.722415\pi$$
0.643251 0.765655i $$-0.277585\pi$$
$$84$$ 0 0
$$85$$ −107.791 921.862i −0.137547 1.17635i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 96.9751 0.115498 0.0577491 0.998331i $$-0.481608\pi$$
0.0577491 + 0.998331i $$0.481608\pi$$
$$90$$ 0 0
$$91$$ 320.794 0.369542
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 542.325 63.4124i 0.585699 0.0684840i
$$96$$ 0 0
$$97$$ 1152.37i 1.20625i 0.797648 + 0.603123i $$0.206077\pi$$
−0.797648 + 0.603123i $$0.793923\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1156.49 1.13936 0.569679 0.821867i $$-0.307068\pi$$
0.569679 + 0.821867i $$0.307068\pi$$
$$102$$ 0 0
$$103$$ 1333.70i 1.27585i 0.770096 + 0.637927i $$0.220208\pi$$
−0.770096 + 0.637927i $$0.779792\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 798.263i 0.721224i 0.932716 + 0.360612i $$0.117432\pi$$
−0.932716 + 0.360612i $$0.882568\pi$$
$$108$$ 0 0
$$109$$ 985.119 0.865663 0.432831 0.901475i $$-0.357515\pi$$
0.432831 + 0.901475i $$0.357515\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1888.25i 1.57196i −0.618253 0.785979i $$-0.712159\pi$$
0.618253 0.785979i $$-0.287841\pi$$
$$114$$ 0 0
$$115$$ −2.09373 17.9063i −0.00169775 0.0145197i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1345.63 1.03659
$$120$$ 0 0
$$121$$ 285.794 0.214721
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1313.20 + 478.158i −0.939648 + 0.342142i
$$126$$ 0 0
$$127$$ 620.859i 0.433798i −0.976194 0.216899i $$-0.930406\pi$$
0.976194 0.216899i $$-0.0695943\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2588.35 −1.72630 −0.863151 0.504947i $$-0.831512\pi$$
−0.863151 + 0.504947i $$0.831512\pi$$
$$132$$ 0 0
$$133$$ 791.625i 0.516110i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1656.29i 1.03289i −0.856319 0.516447i $$-0.827254\pi$$
0.856319 0.516447i $$-0.172746\pi$$
$$138$$ 0 0
$$139$$ −153.256 −0.0935182 −0.0467591 0.998906i $$-0.514889\pi$$
−0.0467591 + 0.998906i $$0.514889\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 795.769i 0.465353i
$$144$$ 0 0
$$145$$ 272.447 31.8564i 0.156038 0.0182450i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1483.38 0.815591 0.407795 0.913073i $$-0.366298\pi$$
0.407795 + 0.913073i $$0.366298\pi$$
$$150$$ 0 0
$$151$$ 394.281 0.212491 0.106246 0.994340i $$-0.466117\pi$$
0.106246 + 0.994340i $$0.466117\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −137.906 + 16.1250i −0.0714639 + 0.00835606i
$$156$$ 0 0
$$157$$ 1727.05i 0.877922i −0.898506 0.438961i $$-0.855347\pi$$
0.898506 0.438961i $$-0.144653\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 26.1376 0.0127946
$$162$$ 0 0
$$163$$ 2034.28i 0.977529i 0.872416 + 0.488764i $$0.162552\pi$$
−0.872416 + 0.488764i $$0.837448\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 192.900i 0.0893835i −0.999001 0.0446918i $$-0.985769\pi$$
0.999001 0.0446918i $$-0.0142306\pi$$
$$168$$ 0 0
$$169$$ 1805.33 0.821726
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1239.91i 0.544905i 0.962169 + 0.272452i $$0.0878347\pi$$
−0.962169 + 0.272452i $$0.912165\pi$$
$$174$$ 0 0
$$175$$ −467.438 1971.52i −0.201914 0.851615i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2636.86 1.10105 0.550525 0.834818i $$-0.314427\pi$$
0.550525 + 0.834818i $$0.314427\pi$$
$$180$$ 0 0
$$181$$ 3317.58 1.36240 0.681199 0.732099i $$-0.261459\pi$$
0.681199 + 0.732099i $$0.261459\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 423.141 + 3618.84i 0.168162 + 1.43818i
$$186$$ 0 0
$$187$$ 3338.01i 1.30534i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −624.506 −0.236585 −0.118292 0.992979i $$-0.537742\pi$$
−0.118292 + 0.992979i $$0.537742\pi$$
$$192$$ 0 0
$$193$$ 436.144i 0.162665i −0.996687 0.0813324i $$-0.974082\pi$$
0.996687 0.0813324i $$-0.0259175\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3355.81i 1.21366i 0.794831 + 0.606831i $$0.207560\pi$$
−0.794831 + 0.606831i $$0.792440\pi$$
$$198$$ 0 0
$$199$$ 3799.77 1.35356 0.676780 0.736185i $$-0.263375\pi$$
0.676780 + 0.736185i $$0.263375\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 397.687i 0.137498i
$$204$$ 0 0
$$205$$ −2691.98 + 314.766i −0.917153 + 0.107240i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1963.72 0.649922
$$210$$ 0 0
$$211$$ −2365.27 −0.771715 −0.385857 0.922558i $$-0.626094\pi$$
−0.385857 + 0.922558i $$0.626094\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 476.981 + 4079.31i 0.151302 + 1.29398i
$$216$$ 0 0
$$217$$ 201.300i 0.0629730i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1642.93 −0.500070
$$222$$ 0 0
$$223$$ 3328.58i 0.999545i 0.866157 + 0.499772i $$0.166583\pi$$
−0.866157 + 0.499772i $$0.833417\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 527.100i 0.154118i 0.997027 + 0.0770592i $$0.0245530\pi$$
−0.997027 + 0.0770592i $$0.975447\pi$$
$$228$$ 0 0
$$229$$ −2566.06 −0.740479 −0.370240 0.928936i $$-0.620724\pi$$
−0.370240 + 0.928936i $$0.620724\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5534.99i 1.55626i −0.628101 0.778132i $$-0.716168\pi$$
0.628101 0.778132i $$-0.283832\pi$$
$$234$$ 0 0
$$235$$ −265.587 2271.39i −0.0737234 0.630508i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1010.01 −0.273355 −0.136678 0.990616i $$-0.543642\pi$$
−0.136678 + 0.990616i $$0.543642\pi$$
$$240$$ 0 0
$$241$$ −4074.29 −1.08900 −0.544498 0.838762i $$-0.683280\pi$$
−0.544498 + 0.838762i $$0.683280\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −891.220 + 104.208i −0.232400 + 0.0271738i
$$246$$ 0 0
$$247$$ 966.525i 0.248982i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1773.98 0.446107 0.223054 0.974806i $$-0.428398\pi$$
0.223054 + 0.974806i $$0.428398\pi$$
$$252$$ 0 0
$$253$$ 64.8375i 0.0161119i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 662.784i 0.160869i 0.996760 + 0.0804345i $$0.0256308\pi$$
−0.996760 + 0.0804345i $$0.974369\pi$$
$$258$$ 0 0
$$259$$ −5282.38 −1.26730
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 712.312i 0.167008i 0.996507 + 0.0835039i $$0.0266111\pi$$
−0.996507 + 0.0835039i $$0.973389\pi$$
$$264$$ 0 0
$$265$$ −79.8904 683.250i −0.0185194 0.158384i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3136.41 −0.710894 −0.355447 0.934696i $$-0.615671\pi$$
−0.355447 + 0.934696i $$0.615671\pi$$
$$270$$ 0 0
$$271$$ 2275.69 0.510105 0.255053 0.966927i $$-0.417907\pi$$
0.255053 + 0.966927i $$0.417907\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −4890.59 + 1159.54i −1.07241 + 0.254264i
$$276$$ 0 0
$$277$$ 5171.00i 1.12164i −0.827937 0.560821i $$-0.810485\pi$$
0.827937 0.560821i $$-0.189515\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2240.14 −0.475571 −0.237785 0.971318i $$-0.576422\pi$$
−0.237785 + 0.971318i $$0.576422\pi$$
$$282$$ 0 0
$$283$$ 225.244i 0.0473123i −0.999720 0.0236561i $$-0.992469\pi$$
0.999720 0.0236561i $$-0.00753068\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3929.46i 0.808183i
$$288$$ 0 0
$$289$$ −1978.59 −0.402726
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1139.86i 0.227274i −0.993522 0.113637i $$-0.963750\pi$$
0.993522 0.113637i $$-0.0362501\pi$$
$$294$$ 0 0
$$295$$ −1246.05 + 145.697i −0.245925 + 0.0287553i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −31.9123 −0.00617237
$$300$$ 0 0
$$301$$ −5954.51 −1.14024
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −5300.82 + 619.809i −0.995161 + 0.116361i
$$306$$ 0 0
$$307$$ 5244.86i 0.975049i 0.873109 + 0.487525i $$0.162100\pi$$
−0.873109 + 0.487525i $$0.837900\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5188.26 −0.945977 −0.472989 0.881068i $$-0.656825\pi$$
−0.472989 + 0.881068i $$0.656825\pi$$
$$312$$ 0 0
$$313$$ 486.656i 0.0878832i 0.999034 + 0.0439416i $$0.0139915\pi$$
−0.999034 + 0.0439416i $$0.986008\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4218.87i 0.747493i −0.927531 0.373747i $$-0.878073\pi$$
0.927531 0.373747i $$-0.121927\pi$$
$$318$$ 0 0
$$319$$ 986.512 0.173148
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4054.27i 0.698408i
$$324$$ 0 0
$$325$$ 570.712 + 2407.10i 0.0974074 + 0.410836i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3315.53 0.555595
$$330$$ 0 0
$$331$$ −7439.94 −1.23546 −0.617728 0.786392i $$-0.711947\pi$$
−0.617728 + 0.786392i $$0.711947\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −724.650 6197.46i −0.118185 1.01076i
$$336$$ 0 0
$$337$$ 6555.39i 1.05963i 0.848113 + 0.529815i $$0.177739\pi$$
−0.848113 + 0.529815i $$0.822261\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −499.350 −0.0793000
$$342$$ 0 0
$$343$$ 6860.72i 1.08001i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1950.56i 0.301763i −0.988552 0.150881i $$-0.951789\pi$$
0.988552 0.150881i $$-0.0482112\pi$$
$$348$$ 0 0
$$349$$ 1426.74 0.218830 0.109415 0.993996i $$-0.465102\pi$$
0.109415 + 0.993996i $$0.465102\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7078.96i 1.06735i 0.845689 + 0.533676i $$0.179190\pi$$
−0.845689 + 0.533676i $$0.820810\pi$$
$$354$$ 0 0
$$355$$ −6199.54 + 724.893i −0.926866 + 0.108376i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5409.79 0.795314 0.397657 0.917534i $$-0.369823\pi$$
0.397657 + 0.917534i $$0.369823\pi$$
$$360$$ 0 0
$$361$$ −4473.90 −0.652267
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1313.72 11235.4i −0.188392 1.61120i
$$366$$ 0 0
$$367$$ 4940.09i 0.702645i 0.936255 + 0.351322i $$0.114268\pi$$
−0.936255 + 0.351322i $$0.885732\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 997.332 0.139566
$$372$$ 0 0
$$373$$ 12891.9i 1.78959i 0.446473 + 0.894797i $$0.352680\pi$$
−0.446473 + 0.894797i $$0.647320\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 485.551i 0.0663320i
$$378$$ 0 0
$$379$$ −9475.15 −1.28418 −0.642092 0.766627i $$-0.721933\pi$$
−0.642092 + 0.766627i $$0.721933\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5800.97i 0.773931i −0.922094 0.386966i $$-0.873523\pi$$
0.922094 0.386966i $$-0.126477\pi$$
$$384$$ 0 0
$$385$$ −846.281 7237.69i −0.112027 0.958095i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 13779.7 1.79603 0.898016 0.439962i $$-0.145008\pi$$
0.898016 + 0.439962i $$0.145008\pi$$
$$390$$ 0 0
$$391$$ −133.862 −0.0173138
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −12775.6 + 1493.81i −1.62737 + 0.190283i
$$396$$ 0 0
$$397$$ 2816.46i 0.356056i 0.984025 + 0.178028i $$0.0569718\pi$$
−0.984025 + 0.178028i $$0.943028\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −11986.4 −1.49270 −0.746352 0.665551i $$-0.768196\pi$$
−0.746352 + 0.665551i $$0.768196\pi$$
$$402$$ 0 0
$$403$$ 245.775i 0.0303794i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 13103.6i 1.59588i
$$408$$ 0 0
$$409$$ 3339.07 0.403683 0.201841 0.979418i $$-0.435307\pi$$
0.201841 + 0.979418i $$0.435307\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1818.84i 0.216706i
$$414$$ 0 0
$$415$$ 1503.49 + 12858.4i 0.177840 + 1.52095i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1688.52 −0.196873 −0.0984363 0.995143i $$-0.531384\pi$$
−0.0984363 + 0.995143i $$0.531384\pi$$
$$420$$ 0 0
$$421$$ −2664.27 −0.308429 −0.154214 0.988037i $$-0.549285\pi$$
−0.154214 + 0.988037i $$0.549285\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2393.96 + 10097.0i 0.273233 + 1.15242i
$$426$$ 0 0
$$427$$ 7737.54i 0.876923i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12266.0 1.37084 0.685420 0.728148i $$-0.259619\pi$$
0.685420 + 0.728148i $$0.259619\pi$$
$$432$$ 0 0
$$433$$ 15647.3i 1.73664i −0.496008 0.868318i $$-0.665201\pi$$
0.496008 0.868318i $$-0.334799\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 78.7503i 0.00862045i
$$438$$ 0 0
$$439$$ 16131.0 1.75373 0.876867 0.480733i $$-0.159629\pi$$
0.876867 + 0.480733i $$0.159629\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 10053.7i 1.07825i 0.842225 + 0.539127i $$0.181246\pi$$
−0.842225 + 0.539127i $$0.818754\pi$$
$$444$$ 0 0
$$445$$ −1076.88 + 125.916i −0.114717 + 0.0134135i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 7477.71 0.785957 0.392979 0.919548i $$-0.371445\pi$$
0.392979 + 0.919548i $$0.371445\pi$$
$$450$$ 0 0
$$451$$ −9747.51 −1.01772
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3562.31 + 416.531i −0.367041 + 0.0429170i
$$456$$ 0 0
$$457$$ 1363.46i 0.139562i −0.997562 0.0697812i $$-0.977770\pi$$
0.997562 0.0697812i $$-0.0222301\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −5276.77 −0.533109 −0.266555 0.963820i $$-0.585885\pi$$
−0.266555 + 0.963820i $$0.585885\pi$$
$$462$$ 0 0
$$463$$ 5740.02i 0.576159i −0.957607 0.288079i $$-0.906983\pi$$
0.957607 0.288079i $$-0.0930167\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6233.36i 0.617657i −0.951118 0.308828i $$-0.900063\pi$$
0.951118 0.308828i $$-0.0999368\pi$$
$$468$$ 0 0
$$469$$ 9046.35 0.890664
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 14770.9i 1.43587i
$$474$$ 0 0
$$475$$ −5940.01 + 1408.35i −0.573782 + 0.136041i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 19688.2 1.87803 0.939013 0.343881i $$-0.111742\pi$$
0.939013 + 0.343881i $$0.111742\pi$$
$$480$$ 0 0
$$481$$ 6449.46 0.611372
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1496.29 12796.8i −0.140088 1.19808i
$$486$$ 0 0
$$487$$ 3955.08i 0.368012i −0.982925 0.184006i $$-0.941093\pi$$
0.982925 0.184006i $$-0.0589065\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13893.5 −1.27699 −0.638497 0.769624i $$-0.720443\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$492$$ 0 0
$$493$$ 2036.74i 0.186065i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 9049.39i 0.816741i
$$498$$ 0 0
$$499$$ 13523.7 1.21324 0.606618 0.794993i $$-0.292526\pi$$
0.606618 + 0.794993i $$0.292526\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 13135.4i 1.16437i 0.813057 + 0.582184i $$0.197802\pi$$
−0.813057 + 0.582184i $$0.802198\pi$$
$$504$$ 0 0
$$505$$ −12842.5 + 1501.63i −1.13165 + 0.132320i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −2222.71 −0.193556 −0.0967778 0.995306i $$-0.530854\pi$$
−0.0967778 + 0.995306i $$0.530854\pi$$
$$510$$ 0 0
$$511$$ 16400.1 1.41976
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1731.72 14810.3i −0.148172 1.26722i
$$516$$ 0 0
$$517$$ 8224.57i 0.699645i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4916.42 0.413421 0.206710 0.978402i $$-0.433724\pi$$
0.206710 + 0.978402i $$0.433724\pi$$
$$522$$ 0 0
$$523$$ 17743.4i 1.48349i −0.670681 0.741746i $$-0.733998\pi$$
0.670681 0.741746i $$-0.266002\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1030.95i 0.0852161i
$$528$$ 0 0
$$529$$ 12164.4 0.999786
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4797.62i 0.389884i
$$534$$ 0 0
$$535$$ −1036.49 8864.46i −0.0837599 0.716344i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −3227.05 −0.257883
$$540$$ 0 0
$$541$$ −12671.3 −1.00699 −0.503495 0.863998i $$-0.667953\pi$$
−0.503495 + 0.863998i $$0.667953\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −10939.4 + 1279.12i −0.859805 + 0.100534i
$$546$$ 0 0
$$547$$ 5250.90i 0.410443i 0.978716 + 0.205221i $$0.0657915\pi$$
−0.978716 + 0.205221i $$0.934209\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1198.20 0.0926406
$$552$$ 0 0
$$553$$ 18648.4i 1.43401i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 25830.2i 1.96492i 0.186465 + 0.982462i $$0.440297\pi$$
−0.186465 + 0.982462i $$0.559703\pi$$
$$558$$ 0 0
$$559$$ 7270.09 0.550075
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 2021.14i 0.151298i 0.997135 + 0.0756490i $$0.0241029\pi$$
−0.997135 + 0.0756490i $$0.975897\pi$$
$$564$$ 0 0
$$565$$ 2451.77 + 20968.4i 0.182561 + 1.56132i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8706.51 0.641469 0.320734 0.947169i $$-0.396070\pi$$
0.320734 + 0.947169i $$0.396070\pi$$
$$570$$ 0 0
$$571$$ 12194.5 0.893740 0.446870 0.894599i $$-0.352539\pi$$
0.446870 + 0.894599i $$0.352539\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 46.5004 + 196.125i 0.00337252 + 0.0142243i
$$576$$ 0 0
$$577$$ 15264.0i 1.10130i 0.834737 + 0.550649i $$0.185620\pi$$
−0.834737 + 0.550649i $$0.814380\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −18769.2 −1.34024
$$582$$ 0 0
$$583$$ 2474.01i 0.175751i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8456.89i 0.594639i 0.954778 + 0.297319i $$0.0960926\pi$$
−0.954778 + 0.297319i $$0.903907\pi$$
$$588$$ 0 0
$$589$$ −606.500 −0.0424285
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1225.23i 0.0848467i 0.999100 + 0.0424234i $$0.0135078\pi$$
−0.999100 + 0.0424234i $$0.986492\pi$$
$$594$$ 0 0
$$595$$ −14942.8 + 1747.22i −1.02957 + 0.120385i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 16060.0 1.09548 0.547741 0.836648i $$-0.315488\pi$$
0.547741 + 0.836648i $$0.315488\pi$$
$$600$$ 0 0
$$601$$ 9699.93 0.658350 0.329175 0.944269i $$-0.393229\pi$$
0.329175 + 0.944269i $$0.393229\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −3173.65 + 371.085i −0.213268 + 0.0249368i
$$606$$ 0 0
$$607$$ 23661.2i 1.58217i 0.611703 + 0.791087i $$0.290485\pi$$
−0.611703 + 0.791087i $$0.709515\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4048.05 −0.268030
$$612$$ 0 0
$$613$$ 8085.63i 0.532749i −0.963870 0.266375i $$-0.914174\pi$$
0.963870 0.266375i $$-0.0858258\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11035.1i 0.720029i −0.932947 0.360014i $$-0.882772\pi$$
0.932947 0.360014i $$-0.117228\pi$$
$$618$$ 0 0
$$619$$ −16826.3 −1.09258 −0.546290 0.837596i $$-0.683960\pi$$
−0.546290 + 0.837596i $$0.683960\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 1571.90i 0.101087i
$$624$$ 0 0
$$625$$ 13961.8 7014.90i 0.893555 0.448954i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 27053.5 1.71493
$$630$$ 0 0
$$631$$ 3705.91 0.233803 0.116902 0.993144i $$-0.462704\pi$$
0.116902 + 0.993144i $$0.462704\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 806.147 + 6894.45i 0.0503795 + 0.430863i
$$636$$ 0 0
$$637$$ 1588.32i 0.0987937i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 24597.4 1.51566 0.757829 0.652453i $$-0.226260\pi$$
0.757829 + 0.652453i $$0.226260\pi$$
$$642$$ 0 0
$$643$$ 21479.5i 1.31737i 0.752419 + 0.658685i $$0.228887\pi$$
−0.752419 + 0.658685i $$0.771113\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27119.7i 1.64789i −0.566668 0.823946i $$-0.691768\pi$$
0.566668 0.823946i $$-0.308232\pi$$
$$648$$ 0 0
$$649$$ −4511.87 −0.272891
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18476.4i 1.10725i −0.832765 0.553627i $$-0.813243\pi$$
0.832765 0.553627i $$-0.186757\pi$$
$$654$$ 0 0
$$655$$ 28742.8 3360.82i 1.71462 0.200485i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 19273.5 1.13928 0.569641 0.821894i $$-0.307082\pi$$
0.569641 + 0.821894i $$0.307082\pi$$
$$660$$ 0 0
$$661$$ 25605.3 1.50670 0.753352 0.657618i $$-0.228436\pi$$
0.753352 + 0.657618i $$0.228436\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1027.88 8790.75i −0.0599388 0.512617i
$$666$$ 0 0
$$667$$ 39.5616i 0.00229660i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −19193.9 −1.10428
$$672$$ 0 0
$$673$$ 7855.52i 0.449938i 0.974366 + 0.224969i $$0.0722280\pi$$
−0.974366 + 0.224969i $$0.927772\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6763.09i 0.383939i −0.981401 0.191970i $$-0.938512\pi$$
0.981401 0.191970i $$-0.0614875\pi$$
$$678$$ 0 0
$$679$$ 18679.3 1.05574
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 15608.6i 0.874447i 0.899353 + 0.437224i $$0.144038\pi$$
−0.899353 + 0.437224i $$0.855962\pi$$
$$684$$ 0 0
$$685$$ 2150.59 + 18392.6i 0.119956 + 1.02590i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1217.68 −0.0673293
$$690$$ 0 0
$$691$$ −6203.15 −0.341504 −0.170752 0.985314i $$-0.554620\pi$$
−0.170752 + 0.985314i $$0.554620\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1701.86 198.994i 0.0928854 0.0108608i
$$696$$ 0 0
$$697$$ 20124.5i 1.09365i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16507.9 0.889435 0.444718 0.895671i $$-0.353304\pi$$
0.444718 + 0.895671i $$0.353304\pi$$
$$702$$ 0 0
$$703$$ 15915.4i 0.853854i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18746.0i 0.997193i
$$708$$ 0 0
$$709$$ 25539.6 1.35283 0.676416 0.736520i $$-0.263532\pi$$
0.676416 + 0.736520i $$0.263532\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 20.0252i 0.00105182i
$$714$$ 0 0
$$715$$ 1033.26 + 8836.76i 0.0540442 + 0.462204i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 7353.45 0.381415 0.190708 0.981647i $$-0.438922\pi$$
0.190708 + 0.981647i $$0.438922\pi$$
$$720$$ 0 0
$$721$$ 21618.4 1.11666
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −2984.07 + 707.510i −0.152863 + 0.0362431i
$$726$$ 0 0
$$727$$ 21696.5i 1.10685i −0.832900 0.553424i $$-0.813321\pi$$
0.832900 0.553424i $$-0.186679\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 30495.8 1.54299
$$732$$ 0 0
$$733$$ 90.2714i 0.00454877i 0.999997 + 0.00227439i $$0.000723960\pi$$
−0.999997 + 0.00227439i $$0.999276\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 22440.6i 1.12159i
$$738$$ 0 0
$$739$$ 14273.1 0.710479 0.355239 0.934775i $$-0.384399\pi$$
0.355239 + 0.934775i $$0.384399\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 15866.6i 0.783429i 0.920087 + 0.391715i $$0.128118\pi$$
−0.920087 + 0.391715i $$0.871882\pi$$
$$744$$ 0 0
$$745$$ −16472.4 + 1926.07i −0.810072 + 0.0947193i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 12939.3 0.631232
$$750$$ 0 0
$$751$$ −26776.9 −1.30107 −0.650534 0.759477i $$-0.725455\pi$$
−0.650534 + 0.759477i $$0.725455\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −4378.37 + 511.950i −0.211053 + 0.0246778i
$$756$$ 0 0
$$757$$ 30478.0i 1.46333i −0.681663 0.731666i $$-0.738743\pi$$
0.681663 0.731666i $$-0.261257\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −29104.7 −1.38639 −0.693195 0.720750i $$-0.743798\pi$$
−0.693195 + 0.720750i $$0.743798\pi$$
$$762$$ 0 0
$$763$$ 15968.2i 0.757649i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2220.69i 0.104543i
$$768$$ 0 0
$$769$$ −4170.65 −0.195575 −0.0977876 0.995207i $$-0.531177\pi$$
−0.0977876 + 0.995207i $$0.531177\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 17738.5i 0.825367i 0.910874 + 0.412684i $$0.135409\pi$$
−0.910874 + 0.412684i $$0.864591\pi$$
$$774$$ 0 0
$$775$$ 1510.47 358.125i 0.0700099 0.0165990i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −11839.1 −0.544519
$$780$$ 0 0
$$781$$ −22448.1 −1.02850
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2242.47 + 19178.4i 0.101958 + 0.871982i
$$786$$ 0 0
$$787$$ 3807.92i 0.172475i −0.996275 0.0862374i $$-0.972516\pi$$
0.996275 0.0862374i $$-0.0274844\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −30607.3 −1.37582
$$792$$ 0 0
$$793$$ 9447.06i 0.423045i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 23840.3i 1.05956i 0.848136 +