# Properties

 Label 576.4.d.b.289.3 Level $576$ Weight $4$ Character 576.289 Analytic conductor $33.985$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(289,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$33.9851001633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 289.3 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 576.289 Dual form 576.4.d.b.289.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+13.8564i q^{5} -3.46410 q^{7} +O(q^{10})$$ $$q+13.8564i q^{5} -3.46410 q^{7} +48.0000i q^{11} -20.7846i q^{13} -96.0000 q^{17} +40.0000i q^{19} +110.851 q^{23} -67.0000 q^{25} +13.8564i q^{29} -204.382 q^{31} -48.0000i q^{35} -297.913i q^{37} +288.000 q^{41} +152.000i q^{43} -554.256 q^{47} -331.000 q^{49} -180.133i q^{53} -665.108 q^{55} -480.000i q^{59} -755.174i q^{61} +288.000 q^{65} +848.000i q^{67} +886.810 q^{71} -538.000 q^{73} -166.277i q^{77} -1008.05 q^{79} +432.000i q^{83} -1330.22i q^{85} -1344.00 q^{89} +72.0000i q^{91} -554.256 q^{95} -590.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 384 q^{17} - 268 q^{25} + 1152 q^{41} - 1324 q^{49} + 1152 q^{65} - 2152 q^{73} - 5376 q^{89} - 2360 q^{97}+O(q^{100})$$ 4 * q - 384 * q^17 - 268 * q^25 + 1152 * q^41 - 1324 * q^49 + 1152 * q^65 - 2152 * q^73 - 5376 * q^89 - 2360 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 13.8564i 1.23935i 0.784857 + 0.619677i $$0.212737\pi$$
−0.784857 + 0.619677i $$0.787263\pi$$
$$6$$ 0 0
$$7$$ −3.46410 −0.187044 −0.0935220 0.995617i $$-0.529813\pi$$
−0.0935220 + 0.995617i $$0.529813\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 48.0000i 1.31569i 0.753155 + 0.657843i $$0.228531\pi$$
−0.753155 + 0.657843i $$0.771469\pi$$
$$12$$ 0 0
$$13$$ − 20.7846i − 0.443432i −0.975111 0.221716i $$-0.928834\pi$$
0.975111 0.221716i $$-0.0711658\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −96.0000 −1.36961 −0.684806 0.728725i $$-0.740113\pi$$
−0.684806 + 0.728725i $$0.740113\pi$$
$$18$$ 0 0
$$19$$ 40.0000i 0.482980i 0.970403 + 0.241490i $$0.0776362\pi$$
−0.970403 + 0.241490i $$0.922364\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 110.851 1.00496 0.502480 0.864589i $$-0.332421\pi$$
0.502480 + 0.864589i $$0.332421\pi$$
$$24$$ 0 0
$$25$$ −67.0000 −0.536000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 13.8564i 0.0887266i 0.999015 + 0.0443633i $$0.0141259\pi$$
−0.999015 + 0.0443633i $$0.985874\pi$$
$$30$$ 0 0
$$31$$ −204.382 −1.18413 −0.592066 0.805890i $$-0.701688\pi$$
−0.592066 + 0.805890i $$0.701688\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 48.0000i − 0.231814i
$$36$$ 0 0
$$37$$ − 297.913i − 1.32369i −0.749640 0.661845i $$-0.769774\pi$$
0.749640 0.661845i $$-0.230226\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 288.000 1.09703 0.548513 0.836142i $$-0.315194\pi$$
0.548513 + 0.836142i $$0.315194\pi$$
$$42$$ 0 0
$$43$$ 152.000i 0.539065i 0.962991 + 0.269532i $$0.0868691\pi$$
−0.962991 + 0.269532i $$0.913131\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −554.256 −1.72014 −0.860070 0.510176i $$-0.829580\pi$$
−0.860070 + 0.510176i $$0.829580\pi$$
$$48$$ 0 0
$$49$$ −331.000 −0.965015
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 180.133i − 0.466853i −0.972374 0.233427i $$-0.925006\pi$$
0.972374 0.233427i $$-0.0749938\pi$$
$$54$$ 0 0
$$55$$ −665.108 −1.63060
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 480.000i − 1.05916i −0.848259 0.529582i $$-0.822349\pi$$
0.848259 0.529582i $$-0.177651\pi$$
$$60$$ 0 0
$$61$$ − 755.174i − 1.58508i −0.609817 0.792542i $$-0.708757\pi$$
0.609817 0.792542i $$-0.291243\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 288.000 0.549569
$$66$$ 0 0
$$67$$ 848.000i 1.54626i 0.634245 + 0.773132i $$0.281311\pi$$
−0.634245 + 0.773132i $$0.718689\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 886.810 1.48232 0.741162 0.671326i $$-0.234275\pi$$
0.741162 + 0.671326i $$0.234275\pi$$
$$72$$ 0 0
$$73$$ −538.000 −0.862577 −0.431289 0.902214i $$-0.641941\pi$$
−0.431289 + 0.902214i $$0.641941\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 166.277i − 0.246091i
$$78$$ 0 0
$$79$$ −1008.05 −1.43563 −0.717816 0.696233i $$-0.754858\pi$$
−0.717816 + 0.696233i $$0.754858\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 432.000i 0.571303i 0.958334 + 0.285652i $$0.0922100\pi$$
−0.958334 + 0.285652i $$0.907790\pi$$
$$84$$ 0 0
$$85$$ − 1330.22i − 1.69744i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1344.00 −1.60072 −0.800358 0.599522i $$-0.795357\pi$$
−0.800358 + 0.599522i $$0.795357\pi$$
$$90$$ 0 0
$$91$$ 72.0000i 0.0829412i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −554.256 −0.598584
$$96$$ 0 0
$$97$$ −590.000 −0.617582 −0.308791 0.951130i $$-0.599924\pi$$
−0.308791 + 0.951130i $$0.599924\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 983.805i 0.969230i 0.874728 + 0.484615i $$0.161040\pi$$
−0.874728 + 0.484615i $$0.838960\pi$$
$$102$$ 0 0
$$103$$ −1437.60 −1.37525 −0.687627 0.726064i $$-0.741348\pi$$
−0.687627 + 0.726064i $$0.741348\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1824.00i 1.64797i 0.566612 + 0.823985i $$0.308254\pi$$
−0.566612 + 0.823985i $$0.691746\pi$$
$$108$$ 0 0
$$109$$ − 1046.16i − 0.919301i −0.888100 0.459651i $$-0.847975\pi$$
0.888100 0.459651i $$-0.152025\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 192.000 0.159839 0.0799196 0.996801i $$-0.474534\pi$$
0.0799196 + 0.996801i $$0.474534\pi$$
$$114$$ 0 0
$$115$$ 1536.00i 1.24550i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 332.554 0.256178
$$120$$ 0 0
$$121$$ −973.000 −0.731029
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 803.672i 0.575061i
$$126$$ 0 0
$$127$$ −793.279 −0.554269 −0.277134 0.960831i $$-0.589385\pi$$
−0.277134 + 0.960831i $$0.589385\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 1344.00i − 0.896380i −0.893938 0.448190i $$-0.852069\pi$$
0.893938 0.448190i $$-0.147931\pi$$
$$132$$ 0 0
$$133$$ − 138.564i − 0.0903386i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 864.000 0.538807 0.269403 0.963027i $$-0.413174\pi$$
0.269403 + 0.963027i $$0.413174\pi$$
$$138$$ 0 0
$$139$$ − 1232.00i − 0.751776i −0.926665 0.375888i $$-0.877338\pi$$
0.926665 0.375888i $$-0.122662\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 997.661 0.583417
$$144$$ 0 0
$$145$$ −192.000 −0.109964
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 2868.28i − 1.57704i −0.615012 0.788518i $$-0.710849\pi$$
0.615012 0.788518i $$-0.289151\pi$$
$$150$$ 0 0
$$151$$ 744.782 0.401387 0.200694 0.979654i $$-0.435680\pi$$
0.200694 + 0.979654i $$0.435680\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 2832.00i − 1.46756i
$$156$$ 0 0
$$157$$ − 616.610i − 0.313445i −0.987643 0.156722i $$-0.949907\pi$$
0.987643 0.156722i $$-0.0500928\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −384.000 −0.187972
$$162$$ 0 0
$$163$$ 1960.00i 0.941835i 0.882177 + 0.470917i $$0.156077\pi$$
−0.882177 + 0.470917i $$0.843923\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3436.39 −1.59231 −0.796155 0.605093i $$-0.793136\pi$$
−0.796155 + 0.605093i $$0.793136\pi$$
$$168$$ 0 0
$$169$$ 1765.00 0.803368
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 1898.33i − 0.834261i −0.908847 0.417131i $$-0.863036\pi$$
0.908847 0.417131i $$-0.136964\pi$$
$$174$$ 0 0
$$175$$ 232.095 0.100256
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 1056.00i 0.440945i 0.975393 + 0.220472i $$0.0707599\pi$$
−0.975393 + 0.220472i $$0.929240\pi$$
$$180$$ 0 0
$$181$$ 1960.68i 0.805173i 0.915382 + 0.402586i $$0.131889\pi$$
−0.915382 + 0.402586i $$0.868111\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4128.00 1.64052
$$186$$ 0 0
$$187$$ − 4608.00i − 1.80198i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1884.47 0.713903 0.356952 0.934123i $$-0.383816\pi$$
0.356952 + 0.934123i $$0.383816\pi$$
$$192$$ 0 0
$$193$$ 962.000 0.358789 0.179394 0.983777i $$-0.442586\pi$$
0.179394 + 0.983777i $$0.442586\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4697.32i 1.69883i 0.527722 + 0.849417i $$0.323046\pi$$
−0.527722 + 0.849417i $$0.676954\pi$$
$$198$$ 0 0
$$199$$ 3107.30 1.10689 0.553444 0.832887i $$-0.313313\pi$$
0.553444 + 0.832887i $$0.313313\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 48.0000i − 0.0165958i
$$204$$ 0 0
$$205$$ 3990.65i 1.35960i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1920.00 −0.635451
$$210$$ 0 0
$$211$$ 3152.00i 1.02840i 0.857670 + 0.514201i $$0.171911\pi$$
−0.857670 + 0.514201i $$0.828089\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −2106.17 −0.668092
$$216$$ 0 0
$$217$$ 708.000 0.221485
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1995.32i 0.607330i
$$222$$ 0 0
$$223$$ 5525.24 1.65918 0.829591 0.558372i $$-0.188574\pi$$
0.829591 + 0.558372i $$0.188574\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2448.00i 0.715769i 0.933766 + 0.357884i $$0.116502\pi$$
−0.933766 + 0.357884i $$0.883498\pi$$
$$228$$ 0 0
$$229$$ 2210.10i 0.637761i 0.947795 + 0.318881i $$0.103307\pi$$
−0.947795 + 0.318881i $$0.896693\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −5568.00 −1.56554 −0.782772 0.622308i $$-0.786195\pi$$
−0.782772 + 0.622308i $$0.786195\pi$$
$$234$$ 0 0
$$235$$ − 7680.00i − 2.13186i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 5210.01 1.41007 0.705037 0.709171i $$-0.250931\pi$$
0.705037 + 0.709171i $$0.250931\pi$$
$$240$$ 0 0
$$241$$ 2798.00 0.747863 0.373932 0.927456i $$-0.378009\pi$$
0.373932 + 0.927456i $$0.378009\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 4586.47i − 1.19600i
$$246$$ 0 0
$$247$$ 831.384 0.214169
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4656.00i 1.17085i 0.810725 + 0.585427i $$0.199073\pi$$
−0.810725 + 0.585427i $$0.800927\pi$$
$$252$$ 0 0
$$253$$ 5320.86i 1.32221i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1536.00 −0.372813 −0.186407 0.982473i $$-0.559684\pi$$
−0.186407 + 0.982473i $$0.559684\pi$$
$$258$$ 0 0
$$259$$ 1032.00i 0.247588i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1330.22 0.311881 0.155940 0.987766i $$-0.450159\pi$$
0.155940 + 0.987766i $$0.450159\pi$$
$$264$$ 0 0
$$265$$ 2496.00 0.578596
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 706.677i − 0.160174i −0.996788 0.0800871i $$-0.974480\pi$$
0.996788 0.0800871i $$-0.0255198\pi$$
$$270$$ 0 0
$$271$$ 4777.00 1.07078 0.535391 0.844604i $$-0.320164\pi$$
0.535391 + 0.844604i $$0.320164\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 3216.00i − 0.705208i
$$276$$ 0 0
$$277$$ 5868.19i 1.27287i 0.771330 + 0.636435i $$0.219592\pi$$
−0.771330 + 0.636435i $$0.780408\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1536.00 −0.326086 −0.163043 0.986619i $$-0.552131\pi$$
−0.163043 + 0.986619i $$0.552131\pi$$
$$282$$ 0 0
$$283$$ − 6752.00i − 1.41825i −0.705083 0.709125i $$-0.749090\pi$$
0.705083 0.709125i $$-0.250910\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −997.661 −0.205192
$$288$$ 0 0
$$289$$ 4303.00 0.875840
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 5140.73i 1.02500i 0.858688 + 0.512499i $$0.171280\pi$$
−0.858688 + 0.512499i $$0.828720\pi$$
$$294$$ 0 0
$$295$$ 6651.08 1.31268
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 2304.00i − 0.445631i
$$300$$ 0 0
$$301$$ − 526.543i − 0.100829i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 10464.0 1.96448
$$306$$ 0 0
$$307$$ 6224.00i 1.15708i 0.815655 + 0.578538i $$0.196377\pi$$
−0.815655 + 0.578538i $$0.803623\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2438.73 −0.444655 −0.222327 0.974972i $$-0.571365\pi$$
−0.222327 + 0.974972i $$0.571365\pi$$
$$312$$ 0 0
$$313$$ 934.000 0.168667 0.0843335 0.996438i $$-0.473124\pi$$
0.0843335 + 0.996438i $$0.473124\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2618.86i 0.464006i 0.972715 + 0.232003i $$0.0745279\pi$$
−0.972715 + 0.232003i $$0.925472\pi$$
$$318$$ 0 0
$$319$$ −665.108 −0.116736
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 3840.00i − 0.661496i
$$324$$ 0 0
$$325$$ 1392.57i 0.237679i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1920.00 0.321742
$$330$$ 0 0
$$331$$ − 9824.00i − 1.63135i −0.578512 0.815674i $$-0.696367\pi$$
0.578512 0.815674i $$-0.303633\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −11750.2 −1.91637
$$336$$ 0 0
$$337$$ −4862.00 −0.785905 −0.392953 0.919559i $$-0.628546\pi$$
−0.392953 + 0.919559i $$0.628546\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 9810.34i − 1.55795i
$$342$$ 0 0
$$343$$ 2334.80 0.367544
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 528.000i 0.0816845i 0.999166 + 0.0408423i $$0.0130041\pi$$
−0.999166 + 0.0408423i $$0.986996\pi$$
$$348$$ 0 0
$$349$$ 6644.15i 1.01906i 0.860452 + 0.509532i $$0.170181\pi$$
−0.860452 + 0.509532i $$0.829819\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2112.00 0.318443 0.159222 0.987243i $$-0.449102\pi$$
0.159222 + 0.987243i $$0.449102\pi$$
$$354$$ 0 0
$$355$$ 12288.0i 1.83712i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 6207.67 0.912614 0.456307 0.889823i $$-0.349172\pi$$
0.456307 + 0.889823i $$0.349172\pi$$
$$360$$ 0 0
$$361$$ 5259.00 0.766730
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 7454.75i − 1.06904i
$$366$$ 0 0
$$367$$ −65.8179 −0.00936149 −0.00468075 0.999989i $$-0.501490\pi$$
−0.00468075 + 0.999989i $$0.501490\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 624.000i 0.0873220i
$$372$$ 0 0
$$373$$ − 8168.35i − 1.13389i −0.823755 0.566945i $$-0.808125\pi$$
0.823755 0.566945i $$-0.191875\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 288.000 0.0393442
$$378$$ 0 0
$$379$$ − 9448.00i − 1.28050i −0.768165 0.640252i $$-0.778830\pi$$
0.768165 0.640252i $$-0.221170\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 1884.47 0.251415 0.125708 0.992067i $$-0.459880\pi$$
0.125708 + 0.992067i $$0.459880\pi$$
$$384$$ 0 0
$$385$$ 2304.00 0.304994
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 9048.23i 1.17934i 0.807644 + 0.589670i $$0.200742\pi$$
−0.807644 + 0.589670i $$0.799258\pi$$
$$390$$ 0 0
$$391$$ −10641.7 −1.37641
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 13968.0i − 1.77926i
$$396$$ 0 0
$$397$$ − 6270.02i − 0.792654i −0.918110 0.396327i $$-0.870285\pi$$
0.918110 0.396327i $$-0.129715\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15648.0 −1.94869 −0.974344 0.225064i $$-0.927741\pi$$
−0.974344 + 0.225064i $$0.927741\pi$$
$$402$$ 0 0
$$403$$ 4248.00i 0.525082i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 14299.8 1.74156
$$408$$ 0 0
$$409$$ −13642.0 −1.64928 −0.824638 0.565662i $$-0.808621\pi$$
−0.824638 + 0.565662i $$0.808621\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1662.77i 0.198110i
$$414$$ 0 0
$$415$$ −5985.97 −0.708047
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 16080.0i 1.87484i 0.348196 + 0.937422i $$0.386794\pi$$
−0.348196 + 0.937422i $$0.613206\pi$$
$$420$$ 0 0
$$421$$ 1489.56i 0.172439i 0.996276 + 0.0862196i $$0.0274787\pi$$
−0.996276 + 0.0862196i $$0.972521\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6432.00 0.734113
$$426$$ 0 0
$$427$$ 2616.00i 0.296480i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6761.93 −0.755709 −0.377854 0.925865i $$-0.623338\pi$$
−0.377854 + 0.925865i $$0.623338\pi$$
$$432$$ 0 0
$$433$$ −5474.00 −0.607537 −0.303769 0.952746i $$-0.598245\pi$$
−0.303769 + 0.952746i $$0.598245\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4434.05i 0.485376i
$$438$$ 0 0
$$439$$ 1860.22 0.202240 0.101120 0.994874i $$-0.467757\pi$$
0.101120 + 0.994874i $$0.467757\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 17424.0i 1.86871i 0.356342 + 0.934356i $$0.384024\pi$$
−0.356342 + 0.934356i $$0.615976\pi$$
$$444$$ 0 0
$$445$$ − 18623.0i − 1.98385i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3168.00 −0.332978 −0.166489 0.986043i $$-0.553243\pi$$
−0.166489 + 0.986043i $$0.553243\pi$$
$$450$$ 0 0
$$451$$ 13824.0i 1.44334i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −997.661 −0.102794
$$456$$ 0 0
$$457$$ 3878.00 0.396948 0.198474 0.980106i $$-0.436401\pi$$
0.198474 + 0.980106i $$0.436401\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9851.90i 0.995334i 0.867368 + 0.497667i $$0.165810\pi$$
−0.867368 + 0.497667i $$0.834190\pi$$
$$462$$ 0 0
$$463$$ −16451.0 −1.65128 −0.825641 0.564196i $$-0.809186\pi$$
−0.825641 + 0.564196i $$0.809186\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 3888.00i − 0.385257i −0.981272 0.192629i $$-0.938299\pi$$
0.981272 0.192629i $$-0.0617013\pi$$
$$468$$ 0 0
$$469$$ − 2937.56i − 0.289219i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −7296.00 −0.709240
$$474$$ 0 0
$$475$$ − 2680.00i − 0.258878i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −9311.51 −0.888212 −0.444106 0.895974i $$-0.646479\pi$$
−0.444106 + 0.895974i $$0.646479\pi$$
$$480$$ 0 0
$$481$$ −6192.00 −0.586967
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 8175.28i − 0.765403i
$$486$$ 0 0
$$487$$ −12086.3 −1.12460 −0.562300 0.826933i $$-0.690083\pi$$
−0.562300 + 0.826933i $$0.690083\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7872.00i 0.723541i 0.932267 + 0.361770i $$0.117828\pi$$
−0.932267 + 0.361770i $$0.882172\pi$$
$$492$$ 0 0
$$493$$ − 1330.22i − 0.121521i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −3072.00 −0.277260
$$498$$ 0 0
$$499$$ 16736.0i 1.50142i 0.660635 + 0.750708i $$0.270287\pi$$
−0.660635 + 0.750708i $$0.729713\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1551.92 −0.137568 −0.0687839 0.997632i $$-0.521912\pi$$
−0.0687839 + 0.997632i $$0.521912\pi$$
$$504$$ 0 0
$$505$$ −13632.0 −1.20122
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1565.77i 0.136349i 0.997673 + 0.0681746i $$0.0217175\pi$$
−0.997673 + 0.0681746i $$0.978283\pi$$
$$510$$ 0 0
$$511$$ 1863.69 0.161340
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 19920.0i − 1.70443i
$$516$$ 0 0
$$517$$ − 26604.3i − 2.26316i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12192.0 1.02522 0.512612 0.858621i $$-0.328678\pi$$
0.512612 + 0.858621i $$0.328678\pi$$
$$522$$ 0 0
$$523$$ − 3688.00i − 0.308346i −0.988044 0.154173i $$-0.950729\pi$$
0.988044 0.154173i $$-0.0492713\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 19620.7 1.62180
$$528$$ 0 0
$$529$$ 121.000 0.00994493
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 5985.97i − 0.486456i
$$534$$ 0 0
$$535$$ −25274.1 −2.04242
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 15888.0i − 1.26966i
$$540$$ 0 0
$$541$$ − 7447.82i − 0.591879i −0.955207 0.295940i $$-0.904367\pi$$
0.955207 0.295940i $$-0.0956327\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 14496.0 1.13934
$$546$$ 0 0
$$547$$ 952.000i 0.0744142i 0.999308 + 0.0372071i $$0.0118461\pi$$
−0.999308 + 0.0372071i $$0.988154\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −554.256 −0.0428532
$$552$$ 0 0
$$553$$ 3492.00 0.268526
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 13925.7i − 1.05934i −0.848205 0.529668i $$-0.822316\pi$$
0.848205 0.529668i $$-0.177684\pi$$
$$558$$ 0 0
$$559$$ 3159.26 0.239038
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 3696.00i − 0.276675i −0.990385 0.138337i $$-0.955824\pi$$
0.990385 0.138337i $$-0.0441758\pi$$
$$564$$ 0 0
$$565$$ 2660.43i 0.198098i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6240.00 −0.459744 −0.229872 0.973221i $$-0.573831\pi$$
−0.229872 + 0.973221i $$0.573831\pi$$
$$570$$ 0 0
$$571$$ 7216.00i 0.528862i 0.964405 + 0.264431i $$0.0851841\pi$$
−0.964405 + 0.264431i $$0.914816\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −7427.03 −0.538659
$$576$$ 0 0
$$577$$ −4754.00 −0.343001 −0.171501 0.985184i $$-0.554862\pi$$
−0.171501 + 0.985184i $$0.554862\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 1496.49i − 0.106859i
$$582$$ 0 0
$$583$$ 8646.40 0.614232
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1152.00i 0.0810019i 0.999179 + 0.0405010i $$0.0128954\pi$$
−0.999179 + 0.0405010i $$0.987105\pi$$
$$588$$ 0 0
$$589$$ − 8175.28i − 0.571913i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 17472.0 1.20993 0.604965 0.796252i $$-0.293187\pi$$
0.604965 + 0.796252i $$0.293187\pi$$
$$594$$ 0 0
$$595$$ 4608.00i 0.317495i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −27158.6 −1.85254 −0.926268 0.376867i $$-0.877002\pi$$
−0.926268 + 0.376867i $$0.877002\pi$$
$$600$$ 0 0
$$601$$ −9142.00 −0.620482 −0.310241 0.950658i $$-0.600410\pi$$
−0.310241 + 0.950658i $$0.600410\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 13482.3i − 0.906005i
$$606$$ 0 0
$$607$$ −28894.1 −1.93208 −0.966041 0.258388i $$-0.916809\pi$$
−0.966041 + 0.258388i $$0.916809\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 11520.0i 0.762765i
$$612$$ 0 0
$$613$$ 9179.87i 0.604847i 0.953174 + 0.302424i $$0.0977957\pi$$
−0.953174 + 0.302424i $$0.902204\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −14208.0 −0.927054 −0.463527 0.886083i $$-0.653416\pi$$
−0.463527 + 0.886083i $$0.653416\pi$$
$$618$$ 0 0
$$619$$ − 3680.00i − 0.238953i −0.992837 0.119476i $$-0.961878\pi$$
0.992837 0.119476i $$-0.0381216\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4655.75 0.299404
$$624$$ 0 0
$$625$$ −19511.0 −1.24870
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 28599.6i 1.81294i
$$630$$ 0 0
$$631$$ 8836.92 0.557516 0.278758 0.960361i $$-0.410077\pi$$
0.278758 + 0.960361i $$0.410077\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 10992.0i − 0.686936i
$$636$$ 0 0
$$637$$ 6879.71i 0.427918i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2400.00 −0.147885 −0.0739425 0.997263i $$-0.523558\pi$$
−0.0739425 + 0.997263i $$0.523558\pi$$
$$642$$ 0 0
$$643$$ 184.000i 0.0112850i 0.999984 + 0.00564250i $$0.00179607\pi$$
−0.999984 + 0.00564250i $$0.998204\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12082.8 −0.734194 −0.367097 0.930183i $$-0.619648\pi$$
−0.367097 + 0.930183i $$0.619648\pi$$
$$648$$ 0 0
$$649$$ 23040.0 1.39353
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 26147.0i 1.56694i 0.621429 + 0.783471i $$0.286553\pi$$
−0.621429 + 0.783471i $$0.713447\pi$$
$$654$$ 0 0
$$655$$ 18623.0 1.11093
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 18816.0i 1.11224i 0.831101 + 0.556121i $$0.187711\pi$$
−0.831101 + 0.556121i $$0.812289\pi$$
$$660$$ 0 0
$$661$$ − 16676.2i − 0.981284i −0.871361 0.490642i $$-0.836762\pi$$
0.871361 0.490642i $$-0.163238\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1920.00 0.111962
$$666$$ 0 0
$$667$$ 1536.00i 0.0891667i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36248.4 2.08547
$$672$$ 0 0
$$673$$ 29326.0 1.67969 0.839847 0.542823i $$-0.182645\pi$$
0.839847 + 0.542823i $$0.182645\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 21934.7i − 1.24523i −0.782530 0.622613i $$-0.786071\pi$$
0.782530 0.622613i $$-0.213929\pi$$
$$678$$ 0 0
$$679$$ 2043.82 0.115515
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 21552.0i 1.20741i 0.797206 + 0.603707i $$0.206310\pi$$
−0.797206 + 0.603707i $$0.793690\pi$$
$$684$$ 0 0
$$685$$ 11971.9i 0.667772i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −3744.00 −0.207017
$$690$$ 0 0
$$691$$ − 26840.0i − 1.47763i −0.673909 0.738815i $$-0.735386\pi$$
0.673909 0.738815i $$-0.264614\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17071.1 0.931717
$$696$$ 0 0
$$697$$ −27648.0 −1.50250
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5611.84i 0.302363i 0.988506 + 0.151181i $$0.0483078\pi$$
−0.988506 + 0.151181i $$0.951692\pi$$
$$702$$ 0 0
$$703$$ 11916.5 0.639317
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 3408.00i − 0.181289i
$$708$$ 0 0
$$709$$ 32278.5i 1.70979i 0.518797 + 0.854897i $$0.326380\pi$$
−0.518797 + 0.854897i $$0.673620\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −22656.0 −1.19001
$$714$$ 0 0
$$715$$ 13824.0i 0.723061i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 12637.0 0.655469 0.327734 0.944770i $$-0.393715\pi$$
0.327734 + 0.944770i $$0.393715\pi$$
$$720$$ 0 0
$$721$$ 4980.00 0.257233
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 928.379i − 0.0475574i
$$726$$ 0 0
$$727$$ 20420.9 1.04177 0.520886 0.853626i $$-0.325602\pi$$
0.520886 + 0.853626i $$0.325602\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 14592.0i − 0.738310i
$$732$$ 0 0
$$733$$ 616.610i 0.0310710i 0.999879 + 0.0155355i $$0.00494530\pi$$
−0.999879 + 0.0155355i $$0.995055\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −40704.0 −2.03440
$$738$$ 0 0
$$739$$ − 25264.0i − 1.25758i −0.777575 0.628790i $$-0.783551\pi$$
0.777575 0.628790i $$-0.216449\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −18068.8 −0.892165 −0.446082 0.894992i $$-0.647181\pi$$
−0.446082 + 0.894992i $$0.647181\pi$$
$$744$$ 0 0
$$745$$ 39744.0 1.95451
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 6318.52i − 0.308243i
$$750$$ 0 0
$$751$$ −1562.31 −0.0759114 −0.0379557 0.999279i $$-0.512085\pi$$
−0.0379557 + 0.999279i $$0.512085\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 10320.0i 0.497461i
$$756$$ 0 0
$$757$$ 7115.26i 0.341623i 0.985304 + 0.170812i $$0.0546389\pi$$
−0.985304 + 0.170812i $$0.945361\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 26208.0 1.24841 0.624205 0.781261i $$-0.285423\pi$$
0.624205 + 0.781261i $$0.285423\pi$$
$$762$$ 0 0
$$763$$ 3624.00i 0.171950i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −9976.61 −0.469667
$$768$$ 0 0
$$769$$ 12866.0 0.603329 0.301664 0.953414i $$-0.402458\pi$$
0.301664 + 0.953414i $$0.402458\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 12318.3i − 0.573170i −0.958055 0.286585i $$-0.907480\pi$$
0.958055 0.286585i $$-0.0925200\pi$$
$$774$$ 0 0
$$775$$ 13693.6 0.634695
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 11520.0i 0.529842i
$$780$$ 0 0
$$781$$ 42566.9i 1.95027i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 8544.00 0.388469
$$786$$ 0 0
$$787$$ 1816.00i 0.0822534i 0.999154 + 0.0411267i $$0.0130947\pi$$
−0.999154 + 0.0411267i $$0.986905\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −665.108 −0.0298970
$$792$$ 0 0
$$793$$ −15696.0 −0.702877
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 16697.0i − 0.742079i −0.928617 0.371040i $$-0.879001\pi$$
0.928617 0.371040i $$-0.120999\pi$$
$$798$$ 0 0
$$799$$ 53208.6 2.35593
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 25824.0i − 1.13488i
$$804$$ 0 0
$$805$$ − 5320.86i − 0.232964i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 24288.0 1.05553 0.527763 0.849392i $$-0.323031\pi$$
0.527763 + 0.849392i $$0.323031\pi$$
$$810$$ 0 0
$$811$$ − 8152.00i − 0.352966i −0.984304 0.176483i $$-0.943528\pi$$
0.984304 0.176483i $$-0.0564721\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −27158.6 −1.16727
$$816$$ 0 0
$$817$$ −6080.00 −0.260358
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 44991.8i 1.91257i 0.292430 + 0.956287i $$0.405536\pi$$
−0.292430 + 0.956287i $$0.594464\pi$$
$$822$$ 0 0
$$823$$ 7285.01 0.308553 0.154277 0.988028i $$-0.450695\pi$$
0.154277 + 0.988028i $$0.450695\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 3936.00i 0.165500i 0.996570 + 0.0827498i $$0.0263702\pi$$
−0.996570 + 0.0827498i $$0.973630\pi$$
$$828$$ 0 0
$$829$$ 28163.1i 1.17991i 0.807436 + 0.589956i $$0.200855\pi$$
−0.807436 + 0.589956i $$0.799145\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 31776.0 1.32170
$$834$$ 0 0
$$835$$ − 47616.0i − 1.97344i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 11085.1 0.456139 0.228070 0.973645i $$-0.426759\pi$$
0.228070 + 0.973645i $$0.426759\pi$$
$$840$$ 0 0
$$841$$ 24197.0 0.992128
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 24456.6i 0.995658i
$$846$$ 0 0
$$847$$ 3370.57 0.136735
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 33024.0i − 1.33026i
$$852$$ 0 0
$$853$$ 44818.5i 1.79901i 0.436908 + 0.899506i $$0.356074\pi$$
−0.436908 + 0.899506i $$0.643926\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 39840.0 1.58799 0.793996 0.607923i $$-0.207997\pi$$
0.793996 + 0.607923i $$0.207997\pi$$
$$858$$ 0 0
$$859$$ 11432.0i 0.454080i 0.973885 + 0.227040i $$0.0729048\pi$$
−0.973885 + 0.227040i $$0.927095\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −4101.50 −0.161780 −0.0808902 0.996723i $$-0.525776\pi$$
−0.0808902 + 0.996723i $$0.525776\pi$$
$$864$$ 0 0
$$865$$ 26304.0 1.03395
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 48386.6i − 1.88884i
$$870$$ 0 0
$$871$$ 17625.3 0.685663
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 2784.00i − 0.107562i
$$876$$ 0 0
$$877$$ − 35839.6i − 1.37995i −0.723833 0.689976i $$-0.757621\pi$$
0.723833 0.689976i $$-0.242379\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −33024.0 −1.26289 −0.631445 0.775420i $$-0.717538\pi$$
−0.631445 + 0.775420i $$0.717538\pi$$
$$882$$ 0 0
$$883$$ − 36056.0i − 1.37416i −0.726583 0.687079i $$-0.758893\pi$$
0.726583 0.687079i $$-0.241107\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −11306.8 −0.428011 −0.214006 0.976832i $$-0.568651\pi$$
−0.214006 + 0.976832i $$0.568651\pi$$
$$888$$ 0 0
$$889$$ 2748.00 0.103673
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 22170.3i − 0.830794i
$$894$$ 0 0
$$895$$ −14632.4 −0.546487
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 2832.00i − 0.105064i
$$900$$ 0 0
$$901$$ 17292.8i 0.639408i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −27168.0 −0.997895
$$906$$ 0 0
$$907$$ 33800.0i 1.23739i 0.785632 + 0.618694i $$0.212338\pi$$
−0.785632 + 0.618694i $$0.787662\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −31149.2 −1.13284 −0.566421 0.824116i $$-0.691672\pi$$
−0.566421 + 0.824116i $$0.691672\pi$$
$$912$$ 0 0
$$913$$ −20736.0 −0.751655
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4655.75i 0.167662i
$$918$$ 0 0
$$919$$ 21751.1 0.780743 0.390371 0.920658i $$-0.372347\pi$$
0.390371 + 0.920658i $$0.372347\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 18432.0i − 0.657310i
$$924$$ 0 0
$$925$$ 19960.2i 0.709498i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 9312.00 0.328866 0.164433 0.986388i $$-0.447421\pi$$
0.164433 + 0.986388i $$0.447421\pi$$
$$930$$ 0 0
$$931$$ − 13240.0i − 0.466083i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 63850.3 2.23329
$$936$$ 0 0
$$937$$ 37846.0 1.31950 0.659752 0.751484i $$-0.270661\pi$$
0.659752 + 0.751484i $$0.270661\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 28835.2i − 0.998937i −0.866332 0.499469i $$-0.833529\pi$$
0.866332 0.499469i $$-0.166471\pi$$
$$942$$ 0 0
$$943$$ 31925.2 1.10247
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 34656.0i 1.18920i 0.804023 + 0.594598i $$0.202689\pi$$
−0.804023 + 0.594598i $$0.797311\pi$$
$$948$$ 0 0
$$949$$ 11182.1i 0.382494i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −52704.0 −1.79145 −0.895724 0.444610i $$-0.853342\pi$$
−0.895724 + 0.444610i $$0.853342\pi$$
$$954$$ 0 0
$$955$$ 26112.0i 0.884780i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2992.98 −0.100780
$$960$$ 0 0
$$961$$ 11981.0 0.402168
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 13329.9i 0.444667i
$$966$$ 0 0
$$967$$ −33418.2 −1.11133 −0.555665 0.831406i $$-0.687536\pi$$
−0.555665 + 0.831406i $$0.687536\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6384.00i 0.210991i 0.994420 + 0.105496i $$0.0336429\pi$$
−0.994420 + 0.105496i $$0.966357\pi$$
$$972$$ 0 0
$$973$$ 4267.77i 0.140615i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 31200.0 1.02167 0.510837 0.859677i $$-0.329335\pi$$
0.510837 + 0.859677i $$0.329335\pi$$
$$978$$ 0 0
$$979$$ − 64512.0i − 2.10604i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 24609.0 0.798479 0.399239 0.916847i $$-0.369274\pi$$
0.399239 + 0.916847i $$0.369274\pi$$
$$984$$ 0 0
$$985$$ −65088.0 −2.10546
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16849.4i 0.541739i
$$990$$ 0 0
$$991$$ 13153.2 0.421620 0.210810 0.977527i $$-0.432390\pi$$
0.210810 + 0.977527i $$0.432390\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 43056.0i 1.37183i
$$996$$ 0 0
$$997$$ − 30615.7i − 0.972527i −0.873812 0.486264i $$-0.838359\pi$$
0.873812 0.486264i $$-0.161641\pi$$
$$998$$ 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 576.4.d.b.289.3 yes 4
3.2 odd 2 576.4.d.h.289.1 yes 4
4.3 odd 2 inner 576.4.d.b.289.4 yes 4
8.3 odd 2 inner 576.4.d.b.289.2 yes 4
8.5 even 2 inner 576.4.d.b.289.1 4
12.11 even 2 576.4.d.h.289.2 yes 4
16.3 odd 4 2304.4.a.w.1.2 2
16.5 even 4 2304.4.a.w.1.1 2
16.11 odd 4 2304.4.a.bj.1.1 2
16.13 even 4 2304.4.a.bj.1.2 2
24.5 odd 2 576.4.d.h.289.3 yes 4
24.11 even 2 576.4.d.h.289.4 yes 4
48.5 odd 4 2304.4.a.bm.1.2 2
48.11 even 4 2304.4.a.z.1.2 2
48.29 odd 4 2304.4.a.z.1.1 2
48.35 even 4 2304.4.a.bm.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
576.4.d.b.289.1 4 8.5 even 2 inner
576.4.d.b.289.2 yes 4 8.3 odd 2 inner
576.4.d.b.289.3 yes 4 1.1 even 1 trivial
576.4.d.b.289.4 yes 4 4.3 odd 2 inner
576.4.d.h.289.1 yes 4 3.2 odd 2
576.4.d.h.289.2 yes 4 12.11 even 2
576.4.d.h.289.3 yes 4 24.5 odd 2
576.4.d.h.289.4 yes 4 24.11 even 2
2304.4.a.w.1.1 2 16.5 even 4
2304.4.a.w.1.2 2 16.3 odd 4
2304.4.a.z.1.1 2 48.29 odd 4
2304.4.a.z.1.2 2 48.11 even 4
2304.4.a.bj.1.1 2 16.11 odd 4
2304.4.a.bj.1.2 2 16.13 even 4
2304.4.a.bm.1.1 2 48.35 even 4
2304.4.a.bm.1.2 2 48.5 odd 4