# Properties

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

# Related objects

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: $$2^{6}$$ Twist minimal: no (minimal twist has level 192) 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.g.289.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.3923i q^{5} -3.46410 q^{7} +O(q^{10})$$ $$q+10.3923i q^{5} -3.46410 q^{7} -55.4256i q^{13} +90.0000 q^{17} +116.000i q^{19} +103.923 q^{23} +17.0000 q^{25} +259.808i q^{29} -301.377 q^{31} -36.0000i q^{35} -34.6410i q^{37} +54.0000 q^{41} -20.0000i q^{43} -394.908 q^{47} -331.000 q^{49} +488.438i q^{53} +324.000i q^{59} +575.041i q^{61} +576.000 q^{65} -116.000i q^{67} -1101.58 q^{71} +1106.00 q^{73} -148.956 q^{79} +1152.00i q^{83} +935.307i q^{85} -918.000 q^{89} +192.000i q^{91} -1205.51 q^{95} +190.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 360 q^{17} + 68 q^{25} + 216 q^{41} - 1324 q^{49} + 2304 q^{65} + 4424 q^{73} - 3672 q^{89} + 760 q^{97}+O(q^{100})$$ 4 * q + 360 * q^17 + 68 * q^25 + 216 * q^41 - 1324 * q^49 + 2304 * q^65 + 4424 * q^73 - 3672 * q^89 + 760 * 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
$$5$$ 10.3923i 0.929516i 0.885438 + 0.464758i $$0.153859\pi$$
−0.885438 + 0.464758i $$0.846141\pi$$
$$6$$ 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
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ − 55.4256i − 1.18248i −0.806494 0.591242i $$-0.798638\pi$$
0.806494 0.591242i $$-0.201362\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 90.0000 1.28401 0.642006 0.766700i $$-0.278102\pi$$
0.642006 + 0.766700i $$0.278102\pi$$
$$18$$ 0 0
$$19$$ 116.000i 1.40064i 0.713827 + 0.700322i $$0.246960\pi$$
−0.713827 + 0.700322i $$0.753040\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 103.923 0.942150 0.471075 0.882093i $$-0.343866\pi$$
0.471075 + 0.882093i $$0.343866\pi$$
$$24$$ 0 0
$$25$$ 17.0000 0.136000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 259.808i 1.66362i 0.555058 + 0.831811i $$0.312696\pi$$
−0.555058 + 0.831811i $$0.687304\pi$$
$$30$$ 0 0
$$31$$ −301.377 −1.74609 −0.873046 0.487637i $$-0.837859\pi$$
−0.873046 + 0.487637i $$0.837859\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 36.0000i − 0.173860i
$$36$$ 0 0
$$37$$ − 34.6410i − 0.153918i −0.997034 0.0769588i $$-0.975479\pi$$
0.997034 0.0769588i $$-0.0245210\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 54.0000 0.205692 0.102846 0.994697i $$-0.467205\pi$$
0.102846 + 0.994697i $$0.467205\pi$$
$$42$$ 0 0
$$43$$ − 20.0000i − 0.0709296i −0.999371 0.0354648i $$-0.988709\pi$$
0.999371 0.0354648i $$-0.0112912\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −394.908 −1.22560 −0.612800 0.790238i $$-0.709957\pi$$
−0.612800 + 0.790238i $$0.709957\pi$$
$$48$$ 0 0
$$49$$ −331.000 −0.965015
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 488.438i 1.26589i 0.774197 + 0.632945i $$0.218154\pi$$
−0.774197 + 0.632945i $$0.781846\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 324.000i 0.714936i 0.933925 + 0.357468i $$0.116360\pi$$
−0.933925 + 0.357468i $$0.883640\pi$$
$$60$$ 0 0
$$61$$ 575.041i 1.20699i 0.797366 + 0.603495i $$0.206226\pi$$
−0.797366 + 0.603495i $$0.793774\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 576.000 1.09914
$$66$$ 0 0
$$67$$ − 116.000i − 0.211517i −0.994392 0.105759i $$-0.966273\pi$$
0.994392 0.105759i $$-0.0337271\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1101.58 −1.84132 −0.920662 0.390361i $$-0.872350\pi$$
−0.920662 + 0.390361i $$0.872350\pi$$
$$72$$ 0 0
$$73$$ 1106.00 1.77325 0.886627 0.462486i $$-0.153042\pi$$
0.886627 + 0.462486i $$0.153042\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −148.956 −0.212138 −0.106069 0.994359i $$-0.533826\pi$$
−0.106069 + 0.994359i $$0.533826\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1152.00i 1.52348i 0.647886 + 0.761738i $$0.275653\pi$$
−0.647886 + 0.761738i $$0.724347\pi$$
$$84$$ 0 0
$$85$$ 935.307i 1.19351i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −918.000 −1.09335 −0.546673 0.837346i $$-0.684106\pi$$
−0.546673 + 0.837346i $$0.684106\pi$$
$$90$$ 0 0
$$91$$ 192.000i 0.221177i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1205.51 −1.30192
$$96$$ 0 0
$$97$$ 190.000 0.198882 0.0994411 0.995043i $$-0.468295\pi$$
0.0994411 + 0.995043i $$0.468295\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 10.3923i − 0.0102383i −0.999987 0.00511917i $$-0.998371\pi$$
0.999987 0.00511917i $$-0.00162949\pi$$
$$102$$ 0 0
$$103$$ 793.279 0.758875 0.379438 0.925217i $$-0.376118\pi$$
0.379438 + 0.925217i $$0.376118\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 252.000i − 0.227680i −0.993499 0.113840i $$-0.963685\pi$$
0.993499 0.113840i $$-0.0363151\pi$$
$$108$$ 0 0
$$109$$ 457.261i 0.401814i 0.979610 + 0.200907i $$0.0643889\pi$$
−0.979610 + 0.200907i $$0.935611\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2214.00 1.84315 0.921573 0.388204i $$-0.126904\pi$$
0.921573 + 0.388204i $$0.126904\pi$$
$$114$$ 0 0
$$115$$ 1080.00i 0.875744i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −311.769 −0.240167
$$120$$ 0 0
$$121$$ 1331.00 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1475.71i 1.05593i
$$126$$ 0 0
$$127$$ −696.284 −0.486498 −0.243249 0.969964i $$-0.578213\pi$$
−0.243249 + 0.969964i $$0.578213\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2268.00i 1.51264i 0.654201 + 0.756321i $$0.273005\pi$$
−0.654201 + 0.756321i $$0.726995\pi$$
$$132$$ 0 0
$$133$$ − 401.836i − 0.261982i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −522.000 −0.325529 −0.162764 0.986665i $$-0.552041\pi$$
−0.162764 + 0.986665i $$0.552041\pi$$
$$138$$ 0 0
$$139$$ − 676.000i − 0.412501i −0.978499 0.206250i $$-0.933874\pi$$
0.978499 0.206250i $$-0.0661261\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2700.00 −1.54636
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1465.31i 0.805660i 0.915275 + 0.402830i $$0.131973\pi$$
−0.915275 + 0.402830i $$0.868027\pi$$
$$150$$ 0 0
$$151$$ −2386.77 −1.28631 −0.643153 0.765738i $$-0.722374\pi$$
−0.643153 + 0.765738i $$0.722374\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 3132.00i − 1.62302i
$$156$$ 0 0
$$157$$ − 2016.11i − 1.02486i −0.858729 0.512430i $$-0.828746\pi$$
0.858729 0.512430i $$-0.171254\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −360.000 −0.176223
$$162$$ 0 0
$$163$$ − 388.000i − 0.186445i −0.995645 0.0932224i $$-0.970283\pi$$
0.995645 0.0932224i $$-0.0297168\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2203.17 1.02088 0.510438 0.859915i $$-0.329483\pi$$
0.510438 + 0.859915i $$0.329483\pi$$
$$168$$ 0 0
$$169$$ −875.000 −0.398270
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 197.454i 0.0867753i 0.999058 + 0.0433877i $$0.0138151\pi$$
−0.999058 + 0.0433877i $$0.986185\pi$$
$$174$$ 0 0
$$175$$ −58.8897 −0.0254380
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 2844.00i − 1.18754i −0.804633 0.593772i $$-0.797638\pi$$
0.804633 0.593772i $$-0.202362\pi$$
$$180$$ 0 0
$$181$$ 96.9948i 0.0398319i 0.999802 + 0.0199159i $$0.00633986\pi$$
−0.999802 + 0.0199159i $$0.993660\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 360.000 0.143069
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3200.83 1.21259 0.606293 0.795241i $$-0.292656\pi$$
0.606293 + 0.795241i $$0.292656\pi$$
$$192$$ 0 0
$$193$$ −1342.00 −0.500514 −0.250257 0.968179i $$-0.580515\pi$$
−0.250257 + 0.968179i $$0.580515\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 966.484i − 0.349539i −0.984609 0.174769i $$-0.944082\pi$$
0.984609 0.174769i $$-0.0559180\pi$$
$$198$$ 0 0
$$199$$ −1784.01 −0.635504 −0.317752 0.948174i $$-0.602928\pi$$
−0.317752 + 0.948174i $$0.602928\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 900.000i − 0.311171i
$$204$$ 0 0
$$205$$ 561.184i 0.191194i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 2764.00i 0.901809i 0.892572 + 0.450904i $$0.148898\pi$$
−0.892572 + 0.450904i $$0.851102\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 207.846 0.0659302
$$216$$ 0 0
$$217$$ 1044.00 0.326596
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 4988.31i − 1.51832i
$$222$$ 0 0
$$223$$ 4292.02 1.28886 0.644428 0.764665i $$-0.277095\pi$$
0.644428 + 0.764665i $$0.277095\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5688.00i 1.66311i 0.555443 + 0.831555i $$0.312549\pi$$
−0.555443 + 0.831555i $$0.687451\pi$$
$$228$$ 0 0
$$229$$ − 5570.28i − 1.60740i −0.595036 0.803699i $$-0.702862\pi$$
0.595036 0.803699i $$-0.297138\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2718.00 −0.764215 −0.382108 0.924118i $$-0.624802\pi$$
−0.382108 + 0.924118i $$0.624802\pi$$
$$234$$ 0 0
$$235$$ − 4104.00i − 1.13921i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3574.95 0.967550 0.483775 0.875192i $$-0.339265\pi$$
0.483775 + 0.875192i $$0.339265\pi$$
$$240$$ 0 0
$$241$$ 4490.00 1.20011 0.600055 0.799959i $$-0.295146\pi$$
0.600055 + 0.799959i $$0.295146\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 3439.85i − 0.896996i
$$246$$ 0 0
$$247$$ 6429.37 1.65624
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 4608.00i − 1.15878i −0.815050 0.579391i $$-0.803290\pi$$
0.815050 0.579391i $$-0.196710\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4626.00 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 120.000i 0.0287893i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −1995.32 −0.467821 −0.233910 0.972258i $$-0.575152\pi$$
−0.233910 + 0.972258i $$0.575152\pi$$
$$264$$ 0 0
$$265$$ −5076.00 −1.17666
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 3148.87i − 0.713717i −0.934158 0.356859i $$-0.883848\pi$$
0.934158 0.356859i $$-0.116152\pi$$
$$270$$ 0 0
$$271$$ 5345.11 1.19813 0.599063 0.800702i $$-0.295540\pi$$
0.599063 + 0.800702i $$0.295540\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6526.37i 1.41564i 0.706394 + 0.707818i $$0.250321\pi$$
−0.706394 + 0.707818i $$0.749679\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1170.00 0.248386 0.124193 0.992258i $$-0.460366\pi$$
0.124193 + 0.992258i $$0.460366\pi$$
$$282$$ 0 0
$$283$$ − 5740.00i − 1.20568i −0.797862 0.602840i $$-0.794036\pi$$
0.797862 0.602840i $$-0.205964\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −187.061 −0.0384735
$$288$$ 0 0
$$289$$ 3187.00 0.648687
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 7991.68i − 1.59344i −0.604346 0.796722i $$-0.706566\pi$$
0.604346 0.796722i $$-0.293434\pi$$
$$294$$ 0 0
$$295$$ −3367.11 −0.664544
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 5760.00i − 1.11408i
$$300$$ 0 0
$$301$$ 69.2820i 0.0132669i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −5976.00 −1.12192
$$306$$ 0 0
$$307$$ 5452.00i 1.01356i 0.862076 + 0.506779i $$0.169164\pi$$
−0.862076 + 0.506779i $$0.830836\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2203.17 0.401705 0.200852 0.979622i $$-0.435629\pi$$
0.200852 + 0.979622i $$0.435629\pi$$
$$312$$ 0 0
$$313$$ −1034.00 −0.186726 −0.0933628 0.995632i $$-0.529762\pi$$
−0.0933628 + 0.995632i $$0.529762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2650.04i − 0.469530i −0.972052 0.234765i $$-0.924568\pi$$
0.972052 0.234765i $$-0.0754320\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 10440.0i 1.79844i
$$324$$ 0 0
$$325$$ − 942.236i − 0.160818i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1368.00 0.229241
$$330$$ 0 0
$$331$$ − 4132.00i − 0.686149i −0.939308 0.343074i $$-0.888532\pi$$
0.939308 0.343074i $$-0.111468\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 1205.51 0.196609
$$336$$ 0 0
$$337$$ −458.000 −0.0740322 −0.0370161 0.999315i $$-0.511785\pi$$
−0.0370161 + 0.999315i $$0.511785\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 2334.80 0.367544
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 11016.0i − 1.70424i −0.523350 0.852118i $$-0.675318\pi$$
0.523350 0.852118i $$-0.324682\pi$$
$$348$$ 0 0
$$349$$ − 2528.79i − 0.387860i −0.981015 0.193930i $$-0.937876\pi$$
0.981015 0.193930i $$-0.0621235\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −5562.00 −0.838627 −0.419314 0.907841i $$-0.637729\pi$$
−0.419314 + 0.907841i $$0.637729\pi$$
$$354$$ 0 0
$$355$$ − 11448.0i − 1.71154i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8875.03 1.30475 0.652376 0.757895i $$-0.273772\pi$$
0.652376 + 0.757895i $$0.273772\pi$$
$$360$$ 0 0
$$361$$ −6597.00 −0.961802
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 11493.9i 1.64827i
$$366$$ 0 0
$$367$$ −12799.9 −1.82056 −0.910282 0.413989i $$-0.864135\pi$$
−0.910282 + 0.413989i $$0.864135\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 1692.00i − 0.236777i
$$372$$ 0 0
$$373$$ − 4981.38i − 0.691491i −0.938328 0.345745i $$-0.887626\pi$$
0.938328 0.345745i $$-0.112374\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 14400.0 1.96721
$$378$$ 0 0
$$379$$ 9892.00i 1.34068i 0.742054 + 0.670340i $$0.233852\pi$$
−0.742054 + 0.670340i $$0.766148\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −8771.11 −1.17019 −0.585095 0.810965i $$-0.698943\pi$$
−0.585095 + 0.810965i $$0.698943\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 9903.87i 1.29086i 0.763818 + 0.645432i $$0.223323\pi$$
−0.763818 + 0.645432i $$0.776677\pi$$
$$390$$ 0 0
$$391$$ 9353.07 1.20973
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 1548.00i − 0.197186i
$$396$$ 0 0
$$397$$ 103.923i 0.0131379i 0.999978 + 0.00656895i $$0.00209098\pi$$
−0.999978 + 0.00656895i $$0.997909\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1062.00 −0.132254 −0.0661269 0.997811i $$-0.521064\pi$$
−0.0661269 + 0.997811i $$0.521064\pi$$
$$402$$ 0 0
$$403$$ 16704.0i 2.06473i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −8614.00 −1.04141 −0.520703 0.853738i $$-0.674330\pi$$
−0.520703 + 0.853738i $$0.674330\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 1122.37i − 0.133724i
$$414$$ 0 0
$$415$$ −11971.9 −1.41609
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 10440.0i 1.21725i 0.793458 + 0.608625i $$0.208278\pi$$
−0.793458 + 0.608625i $$0.791722\pi$$
$$420$$ 0 0
$$421$$ 900.666i 0.104266i 0.998640 + 0.0521328i $$0.0166019\pi$$
−0.998640 + 0.0521328i $$0.983398\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1530.00 0.174626
$$426$$ 0 0
$$427$$ − 1992.00i − 0.225760i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 394.908 0.0441346 0.0220673 0.999756i $$-0.492975\pi$$
0.0220673 + 0.999756i $$0.492975\pi$$
$$432$$ 0 0
$$433$$ 12958.0 1.43816 0.719078 0.694929i $$-0.244564\pi$$
0.719078 + 0.694929i $$0.244564\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 12055.1i 1.31962i
$$438$$ 0 0
$$439$$ −11441.9 −1.24395 −0.621974 0.783038i $$-0.713669\pi$$
−0.621974 + 0.783038i $$0.713669\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 1800.00i − 0.193049i −0.995331 0.0965244i $$-0.969227\pi$$
0.995331 0.0965244i $$-0.0307726\pi$$
$$444$$ 0 0
$$445$$ − 9540.14i − 1.01628i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 13626.0 1.43218 0.716092 0.698006i $$-0.245929\pi$$
0.716092 + 0.698006i $$0.245929\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1995.32 −0.205587
$$456$$ 0 0
$$457$$ 12602.0 1.28993 0.644964 0.764213i $$-0.276873\pi$$
0.644964 + 0.764213i $$0.276873\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 1839.44i − 0.185838i −0.995674 0.0929188i $$-0.970380\pi$$
0.995674 0.0929188i $$-0.0296197\pi$$
$$462$$ 0 0
$$463$$ 11012.4 1.10538 0.552688 0.833389i $$-0.313602\pi$$
0.552688 + 0.833389i $$0.313602\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9144.00i 0.906068i 0.891493 + 0.453034i $$0.149658\pi$$
−0.891493 + 0.453034i $$0.850342\pi$$
$$468$$ 0 0
$$469$$ 401.836i 0.0395630i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1972.00i 0.190488i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 6173.03 0.588837 0.294418 0.955677i $$-0.404874\pi$$
0.294418 + 0.955677i $$0.404874\pi$$
$$480$$ 0 0
$$481$$ −1920.00 −0.182005
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1974.54i 0.184864i
$$486$$ 0 0
$$487$$ −3204.29 −0.298153 −0.149076 0.988826i $$-0.547630\pi$$
−0.149076 + 0.988826i $$0.547630\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 396.000i 0.0363976i 0.999834 + 0.0181988i $$0.00579318\pi$$
−0.999834 + 0.0181988i $$0.994207\pi$$
$$492$$ 0 0
$$493$$ 23382.7i 2.13611i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3816.00 0.344408
$$498$$ 0 0
$$499$$ 12436.0i 1.11565i 0.829957 + 0.557827i $$0.188365\pi$$
−0.829957 + 0.557827i $$0.811635\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −16482.2 −1.46104 −0.730522 0.682890i $$-0.760723\pi$$
−0.730522 + 0.682890i $$0.760723\pi$$
$$504$$ 0 0
$$505$$ 108.000 0.00951671
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 9155.62i 0.797280i 0.917107 + 0.398640i $$0.130518\pi$$
−0.917107 + 0.398640i $$0.869482\pi$$
$$510$$ 0 0
$$511$$ −3831.30 −0.331676
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 8244.00i 0.705386i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −7650.00 −0.643287 −0.321644 0.946861i $$-0.604235\pi$$
−0.321644 + 0.946861i $$0.604235\pi$$
$$522$$ 0 0
$$523$$ − 18332.0i − 1.53270i −0.642423 0.766350i $$-0.722071\pi$$
0.642423 0.766350i $$-0.277929\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −27123.9 −2.24200
$$528$$ 0 0
$$529$$ −1367.00 −0.112353
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 2992.98i − 0.243228i
$$534$$ 0 0
$$535$$ 2618.86 0.211632
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 16863.2i 1.34012i 0.742305 + 0.670062i $$0.233733\pi$$
−0.742305 + 0.670062i $$0.766267\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −4752.00 −0.373492
$$546$$ 0 0
$$547$$ − 1684.00i − 0.131632i −0.997832 0.0658159i $$-0.979035\pi$$
0.997832 0.0658159i $$-0.0209650\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −30137.7 −2.33014
$$552$$ 0 0
$$553$$ 516.000 0.0396791
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 2275.91i 0.173130i 0.996246 + 0.0865652i $$0.0275891\pi$$
−0.996246 + 0.0865652i $$0.972411\pi$$
$$558$$ 0 0
$$559$$ −1108.51 −0.0838731
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 7992.00i − 0.598264i −0.954212 0.299132i $$-0.903303\pi$$
0.954212 0.299132i $$-0.0966971\pi$$
$$564$$ 0 0
$$565$$ 23008.6i 1.71323i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 5526.00 0.407139 0.203569 0.979061i $$-0.434746\pi$$
0.203569 + 0.979061i $$0.434746\pi$$
$$570$$ 0 0
$$571$$ − 13420.0i − 0.983554i −0.870721 0.491777i $$-0.836347\pi$$
0.870721 0.491777i $$-0.163653\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1766.69 0.128132
$$576$$ 0 0
$$577$$ −10178.0 −0.734343 −0.367171 0.930153i $$-0.619674\pi$$
−0.367171 + 0.930153i $$0.619674\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 3990.65i − 0.284957i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 18684.0i 1.31375i 0.754000 + 0.656875i $$0.228122\pi$$
−0.754000 + 0.656875i $$0.771878\pi$$
$$588$$ 0 0
$$589$$ − 34959.7i − 2.44565i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 5094.00 0.352758 0.176379 0.984322i $$-0.443562\pi$$
0.176379 + 0.984322i $$0.443562\pi$$
$$594$$ 0 0
$$595$$ − 3240.00i − 0.223239i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 19433.6 1.32560 0.662801 0.748795i $$-0.269367\pi$$
0.662801 + 0.748795i $$0.269367\pi$$
$$600$$ 0 0
$$601$$ 27722.0 1.88154 0.940769 0.339049i $$-0.110105\pi$$
0.940769 + 0.339049i $$0.110105\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 13832.2i 0.929516i
$$606$$ 0 0
$$607$$ 26684.0 1.78430 0.892149 0.451741i $$-0.149197\pi$$
0.892149 + 0.451741i $$0.149197\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 21888.0i 1.44925i
$$612$$ 0 0
$$613$$ 16911.7i 1.11429i 0.830416 + 0.557144i $$0.188103\pi$$
−0.830416 + 0.557144i $$0.811897\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −17694.0 −1.15451 −0.577256 0.816563i $$-0.695876\pi$$
−0.577256 + 0.816563i $$0.695876\pi$$
$$618$$ 0 0
$$619$$ 13652.0i 0.886462i 0.896407 + 0.443231i $$0.146168\pi$$
−0.896407 + 0.443231i $$0.853832\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 3180.05 0.204504
$$624$$ 0 0
$$625$$ −13211.0 −0.845504
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 3117.69i − 0.197632i
$$630$$ 0 0
$$631$$ −9162.55 −0.578059 −0.289030 0.957320i $$-0.593333\pi$$
−0.289030 + 0.957320i $$0.593333\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 7236.00i − 0.452208i
$$636$$ 0 0
$$637$$ 18345.9i 1.14112i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 5202.00 0.320541 0.160270 0.987073i $$-0.448763\pi$$
0.160270 + 0.987073i $$0.448763\pi$$
$$642$$ 0 0
$$643$$ − 15892.0i − 0.974680i −0.873212 0.487340i $$-0.837967\pi$$
0.873212 0.487340i $$-0.162033\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −478.046 −0.0290478 −0.0145239 0.999895i $$-0.504623\pi$$
−0.0145239 + 0.999895i $$0.504623\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 24660.9i − 1.47788i −0.673770 0.738941i $$-0.735326\pi$$
0.673770 0.738941i $$-0.264674\pi$$
$$654$$ 0 0
$$655$$ −23569.7 −1.40602
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 28260.0i − 1.67049i −0.549878 0.835245i $$-0.685326\pi$$
0.549878 0.835245i $$-0.314674\pi$$
$$660$$ 0 0
$$661$$ − 25863.0i − 1.52187i −0.648830 0.760933i $$-0.724742\pi$$
0.648830 0.760933i $$-0.275258\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4176.00 0.243516
$$666$$ 0 0
$$667$$ 27000.0i 1.56738i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 190.000 0.0108826 0.00544128 0.999985i $$-0.498268\pi$$
0.00544128 + 0.999985i $$0.498268\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4998.70i 0.283775i 0.989883 + 0.141887i $$0.0453171\pi$$
−0.989883 + 0.141887i $$0.954683\pi$$
$$678$$ 0 0
$$679$$ −658.179 −0.0371997
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 8064.00i − 0.451772i −0.974154 0.225886i $$-0.927472\pi$$
0.974154 0.225886i $$-0.0725277\pi$$
$$684$$ 0 0
$$685$$ − 5424.78i − 0.302584i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 27072.0 1.49690
$$690$$ 0 0
$$691$$ 19244.0i 1.05944i 0.848171 + 0.529722i $$0.177704\pi$$
−0.848171 + 0.529722i $$0.822296\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 7025.20 0.383426
$$696$$ 0 0
$$697$$ 4860.00 0.264111
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6204.21i 0.334279i 0.985933 + 0.167140i $$0.0534530\pi$$
−0.985933 + 0.167140i $$0.946547\pi$$
$$702$$ 0 0
$$703$$ 4018.36 0.215584
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 36.0000i 0.00191502i
$$708$$ 0 0
$$709$$ − 15020.3i − 0.795629i −0.917466 0.397814i $$-0.869769\pi$$
0.917466 0.397814i $$-0.130231\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −31320.0 −1.64508
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 30740.4 1.59447 0.797236 0.603668i $$-0.206295\pi$$
0.797236 + 0.603668i $$0.206295\pi$$
$$720$$ 0 0
$$721$$ −2748.00 −0.141943
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 4416.73i 0.226253i
$$726$$ 0 0
$$727$$ −12127.8 −0.618701 −0.309351 0.950948i $$-0.600112\pi$$
−0.309351 + 0.950948i $$0.600112\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 1800.00i − 0.0910744i
$$732$$ 0 0
$$733$$ − 12387.6i − 0.624212i −0.950047 0.312106i $$-0.898966\pi$$
0.950047 0.312106i $$-0.101034\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ − 11180.0i − 0.556513i −0.960507 0.278256i $$-0.910244\pi$$
0.960507 0.278256i $$-0.0897565\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 35500.1 1.75286 0.876429 0.481532i $$-0.159919\pi$$
0.876429 + 0.481532i $$0.159919\pi$$
$$744$$ 0 0
$$745$$ −15228.0 −0.748873
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 872.954i 0.0425862i
$$750$$ 0 0
$$751$$ 37970.0 1.84493 0.922467 0.386076i $$-0.126170\pi$$
0.922467 + 0.386076i $$0.126170\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 24804.0i − 1.19564i
$$756$$ 0 0
$$757$$ − 39047.4i − 1.87477i −0.348296 0.937385i $$-0.613240\pi$$
0.348296 0.937385i $$-0.386760\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 12222.0 0.582191 0.291095 0.956694i $$-0.405980\pi$$
0.291095 + 0.956694i $$0.405980\pi$$
$$762$$ 0 0
$$763$$ − 1584.00i − 0.0751568i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 17957.9 0.845401
$$768$$ 0 0
$$769$$ −34030.0 −1.59578 −0.797889 0.602804i $$-0.794050\pi$$
−0.797889 + 0.602804i $$0.794050\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 4873.99i − 0.226786i −0.993550 0.113393i $$-0.963828\pi$$
0.993550 0.113393i $$-0.0361718\pi$$
$$774$$ 0 0
$$775$$ −5123.41 −0.237469
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6264.00i 0.288102i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 20952.0 0.952623
$$786$$ 0 0
$$787$$ − 30988.0i − 1.40356i −0.712393 0.701781i $$-0.752389\pi$$
0.712393 0.701781i $$-0.247611\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7669.52 −0.344749
$$792$$ 0 0
$$793$$ 31872.0 1.42725
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 7160.30i − 0.318232i −0.987260 0.159116i $$-0.949136\pi$$
0.987260 0.159116i $$-0.0508644\pi$$
$$798$$ 0 0
$$799$$ −35541.7 −1.57369
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ − 3741.23i − 0.163803i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −37530.0 −1.63101 −0.815503 0.578752i $$-0.803540\pi$$
−0.815503 + 0.578752i $$0.803540\pi$$
$$810$$ 0 0
$$811$$ 10852.0i 0.469871i 0.972011 + 0.234935i $$0.0754879\pi$$
−0.972011 + 0.234935i $$0.924512\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 4032.21 0.173303
$$816$$ 0 0
$$817$$ 2320.00 0.0993470
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 31353.6i − 1.33282i −0.745584 0.666411i $$-0.767829\pi$$
0.745584 0.666411i $$-0.232171\pi$$
$$822$$ 0 0
$$823$$ 32947.1 1.39546 0.697729 0.716361i $$-0.254194\pi$$
0.697729 + 0.716361i $$0.254194\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 10044.0i − 0.422327i −0.977451 0.211163i $$-0.932275\pi$$
0.977451 0.211163i $$-0.0677252\pi$$
$$828$$ 0 0
$$829$$ − 9796.48i − 0.410429i −0.978717 0.205215i $$-0.934211\pi$$
0.978717 0.205215i $$-0.0657892\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −29790.0 −1.23909
$$834$$ 0 0
$$835$$ 22896.0i 0.948921i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 21054.8 0.866380 0.433190 0.901303i $$-0.357388\pi$$
0.433190 + 0.901303i $$0.357388\pi$$
$$840$$ 0 0
$$841$$ −43111.0 −1.76764
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 9093.27i − 0.370199i
$$846$$ 0 0
$$847$$ −4610.72 −0.187044
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 3600.00i − 0.145013i
$$852$$ 0 0
$$853$$ 40703.2i 1.63382i 0.576763 + 0.816911i $$0.304316\pi$$
−0.576763 + 0.816911i $$0.695684\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18342.0 0.731098 0.365549 0.930792i $$-0.380881\pi$$
0.365549 + 0.930792i $$0.380881\pi$$
$$858$$ 0 0
$$859$$ − 26324.0i − 1.04559i −0.852458 0.522796i $$-0.824889\pi$$
0.852458 0.522796i $$-0.175111\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −12761.8 −0.503378 −0.251689 0.967808i $$-0.580986\pi$$
−0.251689 + 0.967808i $$0.580986\pi$$
$$864$$ 0 0
$$865$$ −2052.00 −0.0806591
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −6429.37 −0.250116
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 5112.00i − 0.197505i
$$876$$ 0 0
$$877$$ 2459.51i 0.0946999i 0.998878 + 0.0473500i $$0.0150776\pi$$
−0.998878 + 0.0473500i $$0.984922\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −37314.0 −1.42695 −0.713474 0.700682i $$-0.752879\pi$$
−0.713474 + 0.700682i $$0.752879\pi$$
$$882$$ 0 0
$$883$$ − 18244.0i − 0.695311i −0.937622 0.347655i $$-0.886978\pi$$
0.937622 0.347655i $$-0.113022\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −17957.9 −0.679783 −0.339891 0.940465i $$-0.610390\pi$$
−0.339891 + 0.940465i $$0.610390\pi$$
$$888$$ 0 0
$$889$$ 2412.00 0.0909965
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 45809.3i − 1.71663i
$$894$$ 0 0
$$895$$ 29555.7 1.10384
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 78300.0i − 2.90484i
$$900$$ 0 0
$$901$$ 43959.4i 1.62542i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −1008.00 −0.0370244
$$906$$ 0 0
$$907$$ − 16388.0i − 0.599950i −0.953947 0.299975i $$-0.903022\pi$$
0.953947 0.299975i $$-0.0969783\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 25107.8 0.913127 0.456564 0.889691i $$-0.349080\pi$$
0.456564 + 0.889691i $$0.349080\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 7856.58i − 0.282930i
$$918$$ 0 0
$$919$$ 27155.1 0.974716 0.487358 0.873202i $$-0.337961\pi$$
0.487358 + 0.873202i $$0.337961\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 61056.0i 2.17734i
$$924$$ 0 0
$$925$$ − 588.897i − 0.0209328i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −48006.0 −1.69540 −0.847700 0.530477i $$-0.822013\pi$$
−0.847700 + 0.530477i $$0.822013\pi$$
$$930$$ 0 0
$$931$$ − 38396.0i − 1.35164i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7894.00 0.275225 0.137612 0.990486i $$-0.456057\pi$$
0.137612 + 0.990486i $$0.456057\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 2670.82i − 0.0925253i −0.998929 0.0462627i $$-0.985269\pi$$
0.998929 0.0462627i $$-0.0147311\pi$$
$$942$$ 0 0
$$943$$ 5611.84 0.193793
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 22356.0i 0.767130i 0.923514 + 0.383565i $$0.125304\pi$$
−0.923514 + 0.383565i $$0.874696\pi$$
$$948$$ 0 0
$$949$$ − 61300.7i − 2.09685i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 14958.0 0.508434 0.254217 0.967147i $$-0.418182\pi$$
0.254217 + 0.967147i $$0.418182\pi$$
$$954$$ 0 0
$$955$$ 33264.0i 1.12712i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1808.26 0.0608882
$$960$$ 0 0
$$961$$ 61037.0 2.04884
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 13946.5i − 0.465236i
$$966$$ 0 0
$$967$$ 17476.4 0.581182 0.290591 0.956847i $$-0.406148\pi$$
0.290591 + 0.956847i $$0.406148\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 48528.0i 1.60385i 0.597425 + 0.801925i $$0.296190\pi$$
−0.597425 + 0.801925i $$0.703810\pi$$
$$972$$ 0 0
$$973$$ 2341.73i 0.0771557i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39978.0 1.30912 0.654560 0.756010i $$-0.272854\pi$$
0.654560 + 0.756010i $$0.272854\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −17957.9 −0.582674 −0.291337 0.956621i $$-0.594100\pi$$
−0.291337 + 0.956621i $$0.594100\pi$$
$$984$$ 0 0
$$985$$ 10044.0 0.324902
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 2078.46i − 0.0668263i
$$990$$ 0 0
$$991$$ 1666.23 0.0534103 0.0267052 0.999643i $$-0.491498\pi$$
0.0267052 + 0.999643i $$0.491498\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 18540.0i − 0.590711i
$$996$$ 0 0
$$997$$ 41465.3i 1.31717i 0.752506 + 0.658585i $$0.228845\pi$$
−0.752506 + 0.658585i $$0.771155\pi$$
$$998$$ 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.g.289.3 4
3.2 odd 2 192.4.d.a.97.1 4
4.3 odd 2 inner 576.4.d.g.289.4 4
8.3 odd 2 inner 576.4.d.g.289.2 4
8.5 even 2 inner 576.4.d.g.289.1 4
12.11 even 2 192.4.d.a.97.3 yes 4
16.3 odd 4 2304.4.a.bg.1.2 2
16.5 even 4 2304.4.a.bg.1.1 2
16.11 odd 4 2304.4.a.be.1.1 2
16.13 even 4 2304.4.a.be.1.2 2
24.5 odd 2 192.4.d.a.97.4 yes 4
24.11 even 2 192.4.d.a.97.2 yes 4
48.5 odd 4 768.4.a.g.1.2 2
48.11 even 4 768.4.a.n.1.2 2
48.29 odd 4 768.4.a.n.1.1 2
48.35 even 4 768.4.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.a.97.1 4 3.2 odd 2
192.4.d.a.97.2 yes 4 24.11 even 2
192.4.d.a.97.3 yes 4 12.11 even 2
192.4.d.a.97.4 yes 4 24.5 odd 2
576.4.d.g.289.1 4 8.5 even 2 inner
576.4.d.g.289.2 4 8.3 odd 2 inner
576.4.d.g.289.3 4 1.1 even 1 trivial
576.4.d.g.289.4 4 4.3 odd 2 inner
768.4.a.g.1.1 2 48.35 even 4
768.4.a.g.1.2 2 48.5 odd 4
768.4.a.n.1.1 2 48.29 odd 4
768.4.a.n.1.2 2 48.11 even 4
2304.4.a.be.1.1 2 16.11 odd 4
2304.4.a.be.1.2 2 16.13 even 4
2304.4.a.bg.1.1 2 16.5 even 4
2304.4.a.bg.1.2 2 16.3 odd 4