# Properties

 Label 480.4.a.j.1.1 Level $480$ Weight $4$ Character 480.1 Self dual yes Analytic conductor $28.321$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(1,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 480.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$28.3209168028$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 480.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +5.00000 q^{5} -12.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +5.00000 q^{5} -12.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} +38.0000 q^{13} +15.0000 q^{15} -6.00000 q^{17} +104.000 q^{19} -36.0000 q^{21} +100.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +230.000 q^{29} -56.0000 q^{31} -72.0000 q^{33} -60.0000 q^{35} +190.000 q^{37} +114.000 q^{39} +202.000 q^{41} -148.000 q^{43} +45.0000 q^{45} +124.000 q^{47} -199.000 q^{49} -18.0000 q^{51} +206.000 q^{53} -120.000 q^{55} +312.000 q^{57} -128.000 q^{59} +190.000 q^{61} -108.000 q^{63} +190.000 q^{65} -204.000 q^{67} +300.000 q^{69} -440.000 q^{71} +1210.00 q^{73} +75.0000 q^{75} +288.000 q^{77} +816.000 q^{79} +81.0000 q^{81} -1412.00 q^{83} -30.0000 q^{85} +690.000 q^{87} -214.000 q^{89} -456.000 q^{91} -168.000 q^{93} +520.000 q^{95} +1202.00 q^{97} -216.000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −12.0000 −0.647939 −0.323970 0.946068i $$-0.605018\pi$$
−0.323970 + 0.946068i $$0.605018\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 0 0
$$13$$ 38.0000 0.810716 0.405358 0.914158i $$-0.367147\pi$$
0.405358 + 0.914158i $$0.367147\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ −6.00000 −0.0856008 −0.0428004 0.999084i $$-0.513628\pi$$
−0.0428004 + 0.999084i $$0.513628\pi$$
$$18$$ 0 0
$$19$$ 104.000 1.25575 0.627875 0.778314i $$-0.283925\pi$$
0.627875 + 0.778314i $$0.283925\pi$$
$$20$$ 0 0
$$21$$ −36.0000 −0.374088
$$22$$ 0 0
$$23$$ 100.000 0.906584 0.453292 0.891362i $$-0.350249\pi$$
0.453292 + 0.891362i $$0.350249\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 230.000 1.47276 0.736378 0.676570i $$-0.236535\pi$$
0.736378 + 0.676570i $$0.236535\pi$$
$$30$$ 0 0
$$31$$ −56.0000 −0.324448 −0.162224 0.986754i $$-0.551867\pi$$
−0.162224 + 0.986754i $$0.551867\pi$$
$$32$$ 0 0
$$33$$ −72.0000 −0.379806
$$34$$ 0 0
$$35$$ −60.0000 −0.289767
$$36$$ 0 0
$$37$$ 190.000 0.844211 0.422106 0.906547i $$-0.361291\pi$$
0.422106 + 0.906547i $$0.361291\pi$$
$$38$$ 0 0
$$39$$ 114.000 0.468067
$$40$$ 0 0
$$41$$ 202.000 0.769441 0.384721 0.923033i $$-0.374298\pi$$
0.384721 + 0.923033i $$0.374298\pi$$
$$42$$ 0 0
$$43$$ −148.000 −0.524879 −0.262439 0.964948i $$-0.584527\pi$$
−0.262439 + 0.964948i $$0.584527\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ 124.000 0.384835 0.192418 0.981313i $$-0.438367\pi$$
0.192418 + 0.981313i $$0.438367\pi$$
$$48$$ 0 0
$$49$$ −199.000 −0.580175
$$50$$ 0 0
$$51$$ −18.0000 −0.0494217
$$52$$ 0 0
$$53$$ 206.000 0.533892 0.266946 0.963711i $$-0.413985\pi$$
0.266946 + 0.963711i $$0.413985\pi$$
$$54$$ 0 0
$$55$$ −120.000 −0.294196
$$56$$ 0 0
$$57$$ 312.000 0.725007
$$58$$ 0 0
$$59$$ −128.000 −0.282444 −0.141222 0.989978i $$-0.545103\pi$$
−0.141222 + 0.989978i $$0.545103\pi$$
$$60$$ 0 0
$$61$$ 190.000 0.398803 0.199402 0.979918i $$-0.436100\pi$$
0.199402 + 0.979918i $$0.436100\pi$$
$$62$$ 0 0
$$63$$ −108.000 −0.215980
$$64$$ 0 0
$$65$$ 190.000 0.362563
$$66$$ 0 0
$$67$$ −204.000 −0.371979 −0.185989 0.982552i $$-0.559549\pi$$
−0.185989 + 0.982552i $$0.559549\pi$$
$$68$$ 0 0
$$69$$ 300.000 0.523417
$$70$$ 0 0
$$71$$ −440.000 −0.735470 −0.367735 0.929931i $$-0.619867\pi$$
−0.367735 + 0.929931i $$0.619867\pi$$
$$72$$ 0 0
$$73$$ 1210.00 1.94000 0.969999 0.243111i $$-0.0781678\pi$$
0.969999 + 0.243111i $$0.0781678\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ 288.000 0.426242
$$78$$ 0 0
$$79$$ 816.000 1.16212 0.581058 0.813862i $$-0.302639\pi$$
0.581058 + 0.813862i $$0.302639\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1412.00 −1.86731 −0.933657 0.358167i $$-0.883402\pi$$
−0.933657 + 0.358167i $$0.883402\pi$$
$$84$$ 0 0
$$85$$ −30.0000 −0.0382818
$$86$$ 0 0
$$87$$ 690.000 0.850296
$$88$$ 0 0
$$89$$ −214.000 −0.254876 −0.127438 0.991847i $$-0.540675\pi$$
−0.127438 + 0.991847i $$0.540675\pi$$
$$90$$ 0 0
$$91$$ −456.000 −0.525294
$$92$$ 0 0
$$93$$ −168.000 −0.187320
$$94$$ 0 0
$$95$$ 520.000 0.561588
$$96$$ 0 0
$$97$$ 1202.00 1.25819 0.629096 0.777328i $$-0.283425\pi$$
0.629096 + 0.777328i $$0.283425\pi$$
$$98$$ 0 0
$$99$$ −216.000 −0.219281
$$100$$ 0 0
$$101$$ 1342.00 1.32212 0.661059 0.750334i $$-0.270107\pi$$
0.661059 + 0.750334i $$0.270107\pi$$
$$102$$ 0 0
$$103$$ 908.000 0.868620 0.434310 0.900763i $$-0.356992\pi$$
0.434310 + 0.900763i $$0.356992\pi$$
$$104$$ 0 0
$$105$$ −180.000 −0.167297
$$106$$ 0 0
$$107$$ −876.000 −0.791459 −0.395730 0.918367i $$-0.629508\pi$$
−0.395730 + 0.918367i $$0.629508\pi$$
$$108$$ 0 0
$$109$$ 302.000 0.265379 0.132690 0.991158i $$-0.457639\pi$$
0.132690 + 0.991158i $$0.457639\pi$$
$$110$$ 0 0
$$111$$ 570.000 0.487405
$$112$$ 0 0
$$113$$ −998.000 −0.830831 −0.415416 0.909632i $$-0.636364\pi$$
−0.415416 + 0.909632i $$0.636364\pi$$
$$114$$ 0 0
$$115$$ 500.000 0.405437
$$116$$ 0 0
$$117$$ 342.000 0.270239
$$118$$ 0 0
$$119$$ 72.0000 0.0554641
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ 606.000 0.444237
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 836.000 0.584118 0.292059 0.956400i $$-0.405660\pi$$
0.292059 + 0.956400i $$0.405660\pi$$
$$128$$ 0 0
$$129$$ −444.000 −0.303039
$$130$$ 0 0
$$131$$ −1480.00 −0.987085 −0.493543 0.869722i $$-0.664298\pi$$
−0.493543 + 0.869722i $$0.664298\pi$$
$$132$$ 0 0
$$133$$ −1248.00 −0.813649
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 1346.00 0.839391 0.419695 0.907665i $$-0.362137\pi$$
0.419695 + 0.907665i $$0.362137\pi$$
$$138$$ 0 0
$$139$$ 824.000 0.502811 0.251406 0.967882i $$-0.419107\pi$$
0.251406 + 0.967882i $$0.419107\pi$$
$$140$$ 0 0
$$141$$ 372.000 0.222185
$$142$$ 0 0
$$143$$ −912.000 −0.533324
$$144$$ 0 0
$$145$$ 1150.00 0.658637
$$146$$ 0 0
$$147$$ −597.000 −0.334964
$$148$$ 0 0
$$149$$ −2450.00 −1.34706 −0.673530 0.739160i $$-0.735223\pi$$
−0.673530 + 0.739160i $$0.735223\pi$$
$$150$$ 0 0
$$151$$ 2696.00 1.45296 0.726481 0.687186i $$-0.241154\pi$$
0.726481 + 0.687186i $$0.241154\pi$$
$$152$$ 0 0
$$153$$ −54.0000 −0.0285336
$$154$$ 0 0
$$155$$ −280.000 −0.145098
$$156$$ 0 0
$$157$$ −554.000 −0.281618 −0.140809 0.990037i $$-0.544970\pi$$
−0.140809 + 0.990037i $$0.544970\pi$$
$$158$$ 0 0
$$159$$ 618.000 0.308243
$$160$$ 0 0
$$161$$ −1200.00 −0.587411
$$162$$ 0 0
$$163$$ −1364.00 −0.655440 −0.327720 0.944775i $$-0.606280\pi$$
−0.327720 + 0.944775i $$0.606280\pi$$
$$164$$ 0 0
$$165$$ −360.000 −0.169854
$$166$$ 0 0
$$167$$ −2004.00 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −753.000 −0.342740
$$170$$ 0 0
$$171$$ 936.000 0.418583
$$172$$ 0 0
$$173$$ −1546.00 −0.679423 −0.339712 0.940530i $$-0.610329\pi$$
−0.339712 + 0.940530i $$0.610329\pi$$
$$174$$ 0 0
$$175$$ −300.000 −0.129588
$$176$$ 0 0
$$177$$ −384.000 −0.163069
$$178$$ 0 0
$$179$$ −1072.00 −0.447626 −0.223813 0.974632i $$-0.571850\pi$$
−0.223813 + 0.974632i $$0.571850\pi$$
$$180$$ 0 0
$$181$$ −3754.00 −1.54162 −0.770808 0.637067i $$-0.780147\pi$$
−0.770808 + 0.637067i $$0.780147\pi$$
$$182$$ 0 0
$$183$$ 570.000 0.230249
$$184$$ 0 0
$$185$$ 950.000 0.377543
$$186$$ 0 0
$$187$$ 144.000 0.0563119
$$188$$ 0 0
$$189$$ −324.000 −0.124696
$$190$$ 0 0
$$191$$ 1224.00 0.463694 0.231847 0.972752i $$-0.425523\pi$$
0.231847 + 0.972752i $$0.425523\pi$$
$$192$$ 0 0
$$193$$ −1694.00 −0.631797 −0.315898 0.948793i $$-0.602306\pi$$
−0.315898 + 0.948793i $$0.602306\pi$$
$$194$$ 0 0
$$195$$ 570.000 0.209326
$$196$$ 0 0
$$197$$ 3134.00 1.13344 0.566721 0.823909i $$-0.308212\pi$$
0.566721 + 0.823909i $$0.308212\pi$$
$$198$$ 0 0
$$199$$ 2560.00 0.911928 0.455964 0.889998i $$-0.349295\pi$$
0.455964 + 0.889998i $$0.349295\pi$$
$$200$$ 0 0
$$201$$ −612.000 −0.214762
$$202$$ 0 0
$$203$$ −2760.00 −0.954256
$$204$$ 0 0
$$205$$ 1010.00 0.344105
$$206$$ 0 0
$$207$$ 900.000 0.302195
$$208$$ 0 0
$$209$$ −2496.00 −0.826086
$$210$$ 0 0
$$211$$ −1856.00 −0.605556 −0.302778 0.953061i $$-0.597914\pi$$
−0.302778 + 0.953061i $$0.597914\pi$$
$$212$$ 0 0
$$213$$ −1320.00 −0.424624
$$214$$ 0 0
$$215$$ −740.000 −0.234733
$$216$$ 0 0
$$217$$ 672.000 0.210223
$$218$$ 0 0
$$219$$ 3630.00 1.12006
$$220$$ 0 0
$$221$$ −228.000 −0.0693979
$$222$$ 0 0
$$223$$ 1596.00 0.479265 0.239632 0.970864i $$-0.422973\pi$$
0.239632 + 0.970864i $$0.422973\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −3996.00 −1.16839 −0.584193 0.811614i $$-0.698589\pi$$
−0.584193 + 0.811614i $$0.698589\pi$$
$$228$$ 0 0
$$229$$ −6090.00 −1.75737 −0.878687 0.477399i $$-0.841580\pi$$
−0.878687 + 0.477399i $$0.841580\pi$$
$$230$$ 0 0
$$231$$ 864.000 0.246091
$$232$$ 0 0
$$233$$ −894.000 −0.251364 −0.125682 0.992071i $$-0.540112\pi$$
−0.125682 + 0.992071i $$0.540112\pi$$
$$234$$ 0 0
$$235$$ 620.000 0.172104
$$236$$ 0 0
$$237$$ 2448.00 0.670948
$$238$$ 0 0
$$239$$ −6336.00 −1.71482 −0.857410 0.514635i $$-0.827928\pi$$
−0.857410 + 0.514635i $$0.827928\pi$$
$$240$$ 0 0
$$241$$ 338.000 0.0903423 0.0451711 0.998979i $$-0.485617\pi$$
0.0451711 + 0.998979i $$0.485617\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −995.000 −0.259462
$$246$$ 0 0
$$247$$ 3952.00 1.01806
$$248$$ 0 0
$$249$$ −4236.00 −1.07809
$$250$$ 0 0
$$251$$ −4872.00 −1.22517 −0.612585 0.790404i $$-0.709870\pi$$
−0.612585 + 0.790404i $$0.709870\pi$$
$$252$$ 0 0
$$253$$ −2400.00 −0.596390
$$254$$ 0 0
$$255$$ −90.0000 −0.0221020
$$256$$ 0 0
$$257$$ −1878.00 −0.455823 −0.227911 0.973682i $$-0.573190\pi$$
−0.227911 + 0.973682i $$0.573190\pi$$
$$258$$ 0 0
$$259$$ −2280.00 −0.546997
$$260$$ 0 0
$$261$$ 2070.00 0.490919
$$262$$ 0 0
$$263$$ 2244.00 0.526125 0.263063 0.964779i $$-0.415267\pi$$
0.263063 + 0.964779i $$0.415267\pi$$
$$264$$ 0 0
$$265$$ 1030.00 0.238764
$$266$$ 0 0
$$267$$ −642.000 −0.147153
$$268$$ 0 0
$$269$$ −4314.00 −0.977804 −0.488902 0.872339i $$-0.662602\pi$$
−0.488902 + 0.872339i $$0.662602\pi$$
$$270$$ 0 0
$$271$$ −6392.00 −1.43279 −0.716395 0.697694i $$-0.754209\pi$$
−0.716395 + 0.697694i $$0.754209\pi$$
$$272$$ 0 0
$$273$$ −1368.00 −0.303279
$$274$$ 0 0
$$275$$ −600.000 −0.131569
$$276$$ 0 0
$$277$$ 4398.00 0.953972 0.476986 0.878911i $$-0.341729\pi$$
0.476986 + 0.878911i $$0.341729\pi$$
$$278$$ 0 0
$$279$$ −504.000 −0.108149
$$280$$ 0 0
$$281$$ −7622.00 −1.61812 −0.809058 0.587729i $$-0.800022\pi$$
−0.809058 + 0.587729i $$0.800022\pi$$
$$282$$ 0 0
$$283$$ 1020.00 0.214250 0.107125 0.994246i $$-0.465836\pi$$
0.107125 + 0.994246i $$0.465836\pi$$
$$284$$ 0 0
$$285$$ 1560.00 0.324233
$$286$$ 0 0
$$287$$ −2424.00 −0.498551
$$288$$ 0 0
$$289$$ −4877.00 −0.992673
$$290$$ 0 0
$$291$$ 3606.00 0.726417
$$292$$ 0 0
$$293$$ −3746.00 −0.746907 −0.373453 0.927649i $$-0.621826\pi$$
−0.373453 + 0.927649i $$0.621826\pi$$
$$294$$ 0 0
$$295$$ −640.000 −0.126313
$$296$$ 0 0
$$297$$ −648.000 −0.126602
$$298$$ 0 0
$$299$$ 3800.00 0.734982
$$300$$ 0 0
$$301$$ 1776.00 0.340089
$$302$$ 0 0
$$303$$ 4026.00 0.763326
$$304$$ 0 0
$$305$$ 950.000 0.178350
$$306$$ 0 0
$$307$$ 9700.00 1.80328 0.901642 0.432483i $$-0.142362\pi$$
0.901642 + 0.432483i $$0.142362\pi$$
$$308$$ 0 0
$$309$$ 2724.00 0.501498
$$310$$ 0 0
$$311$$ 4152.00 0.757036 0.378518 0.925594i $$-0.376434\pi$$
0.378518 + 0.925594i $$0.376434\pi$$
$$312$$ 0 0
$$313$$ 6362.00 1.14889 0.574443 0.818544i $$-0.305219\pi$$
0.574443 + 0.818544i $$0.305219\pi$$
$$314$$ 0 0
$$315$$ −540.000 −0.0965891
$$316$$ 0 0
$$317$$ 10886.0 1.92877 0.964383 0.264511i $$-0.0852104\pi$$
0.964383 + 0.264511i $$0.0852104\pi$$
$$318$$ 0 0
$$319$$ −5520.00 −0.968842
$$320$$ 0 0
$$321$$ −2628.00 −0.456949
$$322$$ 0 0
$$323$$ −624.000 −0.107493
$$324$$ 0 0
$$325$$ 950.000 0.162143
$$326$$ 0 0
$$327$$ 906.000 0.153217
$$328$$ 0 0
$$329$$ −1488.00 −0.249350
$$330$$ 0 0
$$331$$ 4128.00 0.685485 0.342742 0.939429i $$-0.388644\pi$$
0.342742 + 0.939429i $$0.388644\pi$$
$$332$$ 0 0
$$333$$ 1710.00 0.281404
$$334$$ 0 0
$$335$$ −1020.00 −0.166354
$$336$$ 0 0
$$337$$ 12002.0 1.94003 0.970016 0.243042i $$-0.0781453\pi$$
0.970016 + 0.243042i $$0.0781453\pi$$
$$338$$ 0 0
$$339$$ −2994.00 −0.479681
$$340$$ 0 0
$$341$$ 1344.00 0.213436
$$342$$ 0 0
$$343$$ 6504.00 1.02386
$$344$$ 0 0
$$345$$ 1500.00 0.234079
$$346$$ 0 0
$$347$$ 6276.00 0.970932 0.485466 0.874256i $$-0.338650\pi$$
0.485466 + 0.874256i $$0.338650\pi$$
$$348$$ 0 0
$$349$$ −9362.00 −1.43592 −0.717960 0.696084i $$-0.754924\pi$$
−0.717960 + 0.696084i $$0.754924\pi$$
$$350$$ 0 0
$$351$$ 1026.00 0.156022
$$352$$ 0 0
$$353$$ −838.000 −0.126352 −0.0631760 0.998002i $$-0.520123\pi$$
−0.0631760 + 0.998002i $$0.520123\pi$$
$$354$$ 0 0
$$355$$ −2200.00 −0.328912
$$356$$ 0 0
$$357$$ 216.000 0.0320222
$$358$$ 0 0
$$359$$ −6896.00 −1.01381 −0.506904 0.862003i $$-0.669210\pi$$
−0.506904 + 0.862003i $$0.669210\pi$$
$$360$$ 0 0
$$361$$ 3957.00 0.576906
$$362$$ 0 0
$$363$$ −2265.00 −0.327498
$$364$$ 0 0
$$365$$ 6050.00 0.867593
$$366$$ 0 0
$$367$$ −1132.00 −0.161008 −0.0805040 0.996754i $$-0.525653\pi$$
−0.0805040 + 0.996754i $$0.525653\pi$$
$$368$$ 0 0
$$369$$ 1818.00 0.256480
$$370$$ 0 0
$$371$$ −2472.00 −0.345930
$$372$$ 0 0
$$373$$ −12578.0 −1.74602 −0.873008 0.487705i $$-0.837834\pi$$
−0.873008 + 0.487705i $$0.837834\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 8740.00 1.19399
$$378$$ 0 0
$$379$$ 11752.0 1.59277 0.796385 0.604790i $$-0.206743\pi$$
0.796385 + 0.604790i $$0.206743\pi$$
$$380$$ 0 0
$$381$$ 2508.00 0.337241
$$382$$ 0 0
$$383$$ −7372.00 −0.983529 −0.491764 0.870728i $$-0.663648\pi$$
−0.491764 + 0.870728i $$0.663648\pi$$
$$384$$ 0 0
$$385$$ 1440.00 0.190621
$$386$$ 0 0
$$387$$ −1332.00 −0.174960
$$388$$ 0 0
$$389$$ 6654.00 0.867278 0.433639 0.901087i $$-0.357229\pi$$
0.433639 + 0.901087i $$0.357229\pi$$
$$390$$ 0 0
$$391$$ −600.000 −0.0776044
$$392$$ 0 0
$$393$$ −4440.00 −0.569894
$$394$$ 0 0
$$395$$ 4080.00 0.519714
$$396$$ 0 0
$$397$$ 4278.00 0.540823 0.270411 0.962745i $$-0.412840\pi$$
0.270411 + 0.962745i $$0.412840\pi$$
$$398$$ 0 0
$$399$$ −3744.00 −0.469761
$$400$$ 0 0
$$401$$ 9074.00 1.13001 0.565005 0.825088i $$-0.308874\pi$$
0.565005 + 0.825088i $$0.308874\pi$$
$$402$$ 0 0
$$403$$ −2128.00 −0.263035
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ −4560.00 −0.555358
$$408$$ 0 0
$$409$$ 6682.00 0.807833 0.403916 0.914796i $$-0.367649\pi$$
0.403916 + 0.914796i $$0.367649\pi$$
$$410$$ 0 0
$$411$$ 4038.00 0.484623
$$412$$ 0 0
$$413$$ 1536.00 0.183006
$$414$$ 0 0
$$415$$ −7060.00 −0.835089
$$416$$ 0 0
$$417$$ 2472.00 0.290298
$$418$$ 0 0
$$419$$ 4832.00 0.563386 0.281693 0.959505i $$-0.409104\pi$$
0.281693 + 0.959505i $$0.409104\pi$$
$$420$$ 0 0
$$421$$ 3974.00 0.460050 0.230025 0.973185i $$-0.426119\pi$$
0.230025 + 0.973185i $$0.426119\pi$$
$$422$$ 0 0
$$423$$ 1116.00 0.128278
$$424$$ 0 0
$$425$$ −150.000 −0.0171202
$$426$$ 0 0
$$427$$ −2280.00 −0.258400
$$428$$ 0 0
$$429$$ −2736.00 −0.307915
$$430$$ 0 0
$$431$$ 1112.00 0.124276 0.0621382 0.998068i $$-0.480208\pi$$
0.0621382 + 0.998068i $$0.480208\pi$$
$$432$$ 0 0
$$433$$ 1106.00 0.122751 0.0613753 0.998115i $$-0.480451\pi$$
0.0613753 + 0.998115i $$0.480451\pi$$
$$434$$ 0 0
$$435$$ 3450.00 0.380264
$$436$$ 0 0
$$437$$ 10400.0 1.13844
$$438$$ 0 0
$$439$$ −9280.00 −1.00891 −0.504454 0.863439i $$-0.668306\pi$$
−0.504454 + 0.863439i $$0.668306\pi$$
$$440$$ 0 0
$$441$$ −1791.00 −0.193392
$$442$$ 0 0
$$443$$ 7004.00 0.751174 0.375587 0.926787i $$-0.377441\pi$$
0.375587 + 0.926787i $$0.377441\pi$$
$$444$$ 0 0
$$445$$ −1070.00 −0.113984
$$446$$ 0 0
$$447$$ −7350.00 −0.777725
$$448$$ 0 0
$$449$$ −11502.0 −1.20894 −0.604469 0.796629i $$-0.706615\pi$$
−0.604469 + 0.796629i $$0.706615\pi$$
$$450$$ 0 0
$$451$$ −4848.00 −0.506172
$$452$$ 0 0
$$453$$ 8088.00 0.838868
$$454$$ 0 0
$$455$$ −2280.00 −0.234919
$$456$$ 0 0
$$457$$ 11578.0 1.18511 0.592556 0.805529i $$-0.298119\pi$$
0.592556 + 0.805529i $$0.298119\pi$$
$$458$$ 0 0
$$459$$ −162.000 −0.0164739
$$460$$ 0 0
$$461$$ −6362.00 −0.642750 −0.321375 0.946952i $$-0.604145\pi$$
−0.321375 + 0.946952i $$0.604145\pi$$
$$462$$ 0 0
$$463$$ 2892.00 0.290286 0.145143 0.989411i $$-0.453636\pi$$
0.145143 + 0.989411i $$0.453636\pi$$
$$464$$ 0 0
$$465$$ −840.000 −0.0837722
$$466$$ 0 0
$$467$$ −11036.0 −1.09354 −0.546772 0.837281i $$-0.684144\pi$$
−0.546772 + 0.837281i $$0.684144\pi$$
$$468$$ 0 0
$$469$$ 2448.00 0.241019
$$470$$ 0 0
$$471$$ −1662.00 −0.162592
$$472$$ 0 0
$$473$$ 3552.00 0.345288
$$474$$ 0 0
$$475$$ 2600.00 0.251150
$$476$$ 0 0
$$477$$ 1854.00 0.177964
$$478$$ 0 0
$$479$$ −13664.0 −1.30339 −0.651695 0.758481i $$-0.725942\pi$$
−0.651695 + 0.758481i $$0.725942\pi$$
$$480$$ 0 0
$$481$$ 7220.00 0.684415
$$482$$ 0 0
$$483$$ −3600.00 −0.339142
$$484$$ 0 0
$$485$$ 6010.00 0.562680
$$486$$ 0 0
$$487$$ 4820.00 0.448491 0.224245 0.974533i $$-0.428008\pi$$
0.224245 + 0.974533i $$0.428008\pi$$
$$488$$ 0 0
$$489$$ −4092.00 −0.378418
$$490$$ 0 0
$$491$$ 17464.0 1.60517 0.802586 0.596537i $$-0.203457\pi$$
0.802586 + 0.596537i $$0.203457\pi$$
$$492$$ 0 0
$$493$$ −1380.00 −0.126069
$$494$$ 0 0
$$495$$ −1080.00 −0.0980654
$$496$$ 0 0
$$497$$ 5280.00 0.476540
$$498$$ 0 0
$$499$$ 13960.0 1.25238 0.626188 0.779672i $$-0.284614\pi$$
0.626188 + 0.779672i $$0.284614\pi$$
$$500$$ 0 0
$$501$$ −6012.00 −0.536120
$$502$$ 0 0
$$503$$ 20388.0 1.80727 0.903634 0.428305i $$-0.140889\pi$$
0.903634 + 0.428305i $$0.140889\pi$$
$$504$$ 0 0
$$505$$ 6710.00 0.591269
$$506$$ 0 0
$$507$$ −2259.00 −0.197881
$$508$$ 0 0
$$509$$ −12954.0 −1.12805 −0.564024 0.825759i $$-0.690747\pi$$
−0.564024 + 0.825759i $$0.690747\pi$$
$$510$$ 0 0
$$511$$ −14520.0 −1.25700
$$512$$ 0 0
$$513$$ 2808.00 0.241669
$$514$$ 0 0
$$515$$ 4540.00 0.388459
$$516$$ 0 0
$$517$$ −2976.00 −0.253161
$$518$$ 0 0
$$519$$ −4638.00 −0.392265
$$520$$ 0 0
$$521$$ −3542.00 −0.297846 −0.148923 0.988849i $$-0.547581\pi$$
−0.148923 + 0.988849i $$0.547581\pi$$
$$522$$ 0 0
$$523$$ 1532.00 0.128087 0.0640437 0.997947i $$-0.479600\pi$$
0.0640437 + 0.997947i $$0.479600\pi$$
$$524$$ 0 0
$$525$$ −900.000 −0.0748176
$$526$$ 0 0
$$527$$ 336.000 0.0277730
$$528$$ 0 0
$$529$$ −2167.00 −0.178105
$$530$$ 0 0
$$531$$ −1152.00 −0.0941479
$$532$$ 0 0
$$533$$ 7676.00 0.623798
$$534$$ 0 0
$$535$$ −4380.00 −0.353951
$$536$$ 0 0
$$537$$ −3216.00 −0.258437
$$538$$ 0 0
$$539$$ 4776.00 0.381664
$$540$$ 0 0
$$541$$ −11826.0 −0.939814 −0.469907 0.882716i $$-0.655713\pi$$
−0.469907 + 0.882716i $$0.655713\pi$$
$$542$$ 0 0
$$543$$ −11262.0 −0.890053
$$544$$ 0 0
$$545$$ 1510.00 0.118681
$$546$$ 0 0
$$547$$ −11260.0 −0.880151 −0.440076 0.897961i $$-0.645048\pi$$
−0.440076 + 0.897961i $$0.645048\pi$$
$$548$$ 0 0
$$549$$ 1710.00 0.132934
$$550$$ 0 0
$$551$$ 23920.0 1.84941
$$552$$ 0 0
$$553$$ −9792.00 −0.752980
$$554$$ 0 0
$$555$$ 2850.00 0.217974
$$556$$ 0 0
$$557$$ 4374.00 0.332733 0.166367 0.986064i $$-0.446797\pi$$
0.166367 + 0.986064i $$0.446797\pi$$
$$558$$ 0 0
$$559$$ −5624.00 −0.425527
$$560$$ 0 0
$$561$$ 432.000 0.0325117
$$562$$ 0 0
$$563$$ −11604.0 −0.868651 −0.434325 0.900756i $$-0.643013\pi$$
−0.434325 + 0.900756i $$0.643013\pi$$
$$564$$ 0 0
$$565$$ −4990.00 −0.371559
$$566$$ 0 0
$$567$$ −972.000 −0.0719932
$$568$$ 0 0
$$569$$ −7990.00 −0.588679 −0.294339 0.955701i $$-0.595100\pi$$
−0.294339 + 0.955701i $$0.595100\pi$$
$$570$$ 0 0
$$571$$ −26080.0 −1.91141 −0.955704 0.294329i $$-0.904904\pi$$
−0.955704 + 0.294329i $$0.904904\pi$$
$$572$$ 0 0
$$573$$ 3672.00 0.267714
$$574$$ 0 0
$$575$$ 2500.00 0.181317
$$576$$ 0 0
$$577$$ 13922.0 1.00447 0.502236 0.864731i $$-0.332511\pi$$
0.502236 + 0.864731i $$0.332511\pi$$
$$578$$ 0 0
$$579$$ −5082.00 −0.364768
$$580$$ 0 0
$$581$$ 16944.0 1.20991
$$582$$ 0 0
$$583$$ −4944.00 −0.351217
$$584$$ 0 0
$$585$$ 1710.00 0.120854
$$586$$ 0 0
$$587$$ 26340.0 1.85208 0.926038 0.377431i $$-0.123193\pi$$
0.926038 + 0.377431i $$0.123193\pi$$
$$588$$ 0 0
$$589$$ −5824.00 −0.407426
$$590$$ 0 0
$$591$$ 9402.00 0.654394
$$592$$ 0 0
$$593$$ −9478.00 −0.656349 −0.328174 0.944617i $$-0.606433\pi$$
−0.328174 + 0.944617i $$0.606433\pi$$
$$594$$ 0 0
$$595$$ 360.000 0.0248043
$$596$$ 0 0
$$597$$ 7680.00 0.526502
$$598$$ 0 0
$$599$$ 6528.00 0.445287 0.222643 0.974900i $$-0.428531\pi$$
0.222643 + 0.974900i $$0.428531\pi$$
$$600$$ 0 0
$$601$$ 2090.00 0.141852 0.0709259 0.997482i $$-0.477405\pi$$
0.0709259 + 0.997482i $$0.477405\pi$$
$$602$$ 0 0
$$603$$ −1836.00 −0.123993
$$604$$ 0 0
$$605$$ −3775.00 −0.253679
$$606$$ 0 0
$$607$$ 8788.00 0.587634 0.293817 0.955862i $$-0.405074\pi$$
0.293817 + 0.955862i $$0.405074\pi$$
$$608$$ 0 0
$$609$$ −8280.00 −0.550940
$$610$$ 0 0
$$611$$ 4712.00 0.311992
$$612$$ 0 0
$$613$$ −2626.00 −0.173023 −0.0865115 0.996251i $$-0.527572\pi$$
−0.0865115 + 0.996251i $$0.527572\pi$$
$$614$$ 0 0
$$615$$ 3030.00 0.198669
$$616$$ 0 0
$$617$$ −29214.0 −1.90618 −0.953089 0.302691i $$-0.902115\pi$$
−0.953089 + 0.302691i $$0.902115\pi$$
$$618$$ 0 0
$$619$$ 22984.0 1.49242 0.746208 0.665713i $$-0.231873\pi$$
0.746208 + 0.665713i $$0.231873\pi$$
$$620$$ 0 0
$$621$$ 2700.00 0.174472
$$622$$ 0 0
$$623$$ 2568.00 0.165144
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −7488.00 −0.476941
$$628$$ 0 0
$$629$$ −1140.00 −0.0722651
$$630$$ 0 0
$$631$$ 4472.00 0.282136 0.141068 0.990000i $$-0.454946\pi$$
0.141068 + 0.990000i $$0.454946\pi$$
$$632$$ 0 0
$$633$$ −5568.00 −0.349618
$$634$$ 0 0
$$635$$ 4180.00 0.261226
$$636$$ 0 0
$$637$$ −7562.00 −0.470357
$$638$$ 0 0
$$639$$ −3960.00 −0.245157
$$640$$ 0 0
$$641$$ −8798.00 −0.542122 −0.271061 0.962562i $$-0.587374\pi$$
−0.271061 + 0.962562i $$0.587374\pi$$
$$642$$ 0 0
$$643$$ −29428.0 −1.80486 −0.902432 0.430833i $$-0.858220\pi$$
−0.902432 + 0.430833i $$0.858220\pi$$
$$644$$ 0 0
$$645$$ −2220.00 −0.135523
$$646$$ 0 0
$$647$$ 4860.00 0.295311 0.147656 0.989039i $$-0.452827\pi$$
0.147656 + 0.989039i $$0.452827\pi$$
$$648$$ 0 0
$$649$$ 3072.00 0.185804
$$650$$ 0 0
$$651$$ 2016.00 0.121372
$$652$$ 0 0
$$653$$ −6570.00 −0.393727 −0.196864 0.980431i $$-0.563076\pi$$
−0.196864 + 0.980431i $$0.563076\pi$$
$$654$$ 0 0
$$655$$ −7400.00 −0.441438
$$656$$ 0 0
$$657$$ 10890.0 0.646666
$$658$$ 0 0
$$659$$ 26496.0 1.56622 0.783109 0.621885i $$-0.213633\pi$$
0.783109 + 0.621885i $$0.213633\pi$$
$$660$$ 0 0
$$661$$ −19642.0 −1.15580 −0.577901 0.816107i $$-0.696128\pi$$
−0.577901 + 0.816107i $$0.696128\pi$$
$$662$$ 0 0
$$663$$ −684.000 −0.0400669
$$664$$ 0 0
$$665$$ −6240.00 −0.363875
$$666$$ 0 0
$$667$$ 23000.0 1.33518
$$668$$ 0 0
$$669$$ 4788.00 0.276704
$$670$$ 0 0
$$671$$ −4560.00 −0.262350
$$672$$ 0 0
$$673$$ −19582.0 −1.12159 −0.560795 0.827954i $$-0.689505\pi$$
−0.560795 + 0.827954i $$0.689505\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −22914.0 −1.30082 −0.650411 0.759582i $$-0.725403\pi$$
−0.650411 + 0.759582i $$0.725403\pi$$
$$678$$ 0 0
$$679$$ −14424.0 −0.815232
$$680$$ 0 0
$$681$$ −11988.0 −0.674569
$$682$$ 0 0
$$683$$ −13764.0 −0.771105 −0.385553 0.922686i $$-0.625989\pi$$
−0.385553 + 0.922686i $$0.625989\pi$$
$$684$$ 0 0
$$685$$ 6730.00 0.375387
$$686$$ 0 0
$$687$$ −18270.0 −1.01462
$$688$$ 0 0
$$689$$ 7828.00 0.432835
$$690$$ 0 0
$$691$$ −34688.0 −1.90969 −0.954843 0.297109i $$-0.903977\pi$$
−0.954843 + 0.297109i $$0.903977\pi$$
$$692$$ 0 0
$$693$$ 2592.00 0.142081
$$694$$ 0 0
$$695$$ 4120.00 0.224864
$$696$$ 0 0
$$697$$ −1212.00 −0.0658648
$$698$$ 0 0
$$699$$ −2682.00 −0.145125
$$700$$ 0 0
$$701$$ −17226.0 −0.928127 −0.464064 0.885802i $$-0.653609\pi$$
−0.464064 + 0.885802i $$0.653609\pi$$
$$702$$ 0 0
$$703$$ 19760.0 1.06012
$$704$$ 0 0
$$705$$ 1860.00 0.0993640
$$706$$ 0 0
$$707$$ −16104.0 −0.856652
$$708$$ 0 0
$$709$$ −12970.0 −0.687022 −0.343511 0.939149i $$-0.611616\pi$$
−0.343511 + 0.939149i $$0.611616\pi$$
$$710$$ 0 0
$$711$$ 7344.00 0.387372
$$712$$ 0 0
$$713$$ −5600.00 −0.294140
$$714$$ 0 0
$$715$$ −4560.00 −0.238510
$$716$$ 0 0
$$717$$ −19008.0 −0.990051
$$718$$ 0 0
$$719$$ 10832.0 0.561843 0.280922 0.959731i $$-0.409360\pi$$
0.280922 + 0.959731i $$0.409360\pi$$
$$720$$ 0 0
$$721$$ −10896.0 −0.562813
$$722$$ 0 0
$$723$$ 1014.00 0.0521592
$$724$$ 0 0
$$725$$ 5750.00 0.294551
$$726$$ 0 0
$$727$$ 35588.0 1.81552 0.907762 0.419486i $$-0.137790\pi$$
0.907762 + 0.419486i $$0.137790\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 888.000 0.0449300
$$732$$ 0 0
$$733$$ 19238.0 0.969402 0.484701 0.874680i $$-0.338928\pi$$
0.484701 + 0.874680i $$0.338928\pi$$
$$734$$ 0 0
$$735$$ −2985.00 −0.149801
$$736$$ 0 0
$$737$$ 4896.00 0.244703
$$738$$ 0 0
$$739$$ −4072.00 −0.202694 −0.101347 0.994851i $$-0.532315\pi$$
−0.101347 + 0.994851i $$0.532315\pi$$
$$740$$ 0 0
$$741$$ 11856.0 0.587775
$$742$$ 0 0
$$743$$ 31268.0 1.54389 0.771946 0.635688i $$-0.219284\pi$$
0.771946 + 0.635688i $$0.219284\pi$$
$$744$$ 0 0
$$745$$ −12250.0 −0.602423
$$746$$ 0 0
$$747$$ −12708.0 −0.622438
$$748$$ 0 0
$$749$$ 10512.0 0.512817
$$750$$ 0 0
$$751$$ −28216.0 −1.37099 −0.685497 0.728075i $$-0.740415\pi$$
−0.685497 + 0.728075i $$0.740415\pi$$
$$752$$ 0 0
$$753$$ −14616.0 −0.707353
$$754$$ 0 0
$$755$$ 13480.0 0.649785
$$756$$ 0 0
$$757$$ −33874.0 −1.62638 −0.813191 0.581997i $$-0.802272\pi$$
−0.813191 + 0.581997i $$0.802272\pi$$
$$758$$ 0 0
$$759$$ −7200.00 −0.344326
$$760$$ 0 0
$$761$$ 5466.00 0.260371 0.130186 0.991490i $$-0.458443\pi$$
0.130186 + 0.991490i $$0.458443\pi$$
$$762$$ 0 0
$$763$$ −3624.00 −0.171950
$$764$$ 0 0
$$765$$ −270.000 −0.0127606
$$766$$ 0 0
$$767$$ −4864.00 −0.228982
$$768$$ 0 0
$$769$$ −8878.00 −0.416318 −0.208159 0.978095i $$-0.566747\pi$$
−0.208159 + 0.978095i $$0.566747\pi$$
$$770$$ 0 0
$$771$$ −5634.00 −0.263169
$$772$$ 0 0
$$773$$ −9538.00 −0.443801 −0.221900 0.975069i $$-0.571226\pi$$
−0.221900 + 0.975069i $$0.571226\pi$$
$$774$$ 0 0
$$775$$ −1400.00 −0.0648897
$$776$$ 0 0
$$777$$ −6840.00 −0.315809
$$778$$ 0 0
$$779$$ 21008.0 0.966226
$$780$$ 0 0
$$781$$ 10560.0 0.483824
$$782$$ 0 0
$$783$$ 6210.00 0.283432
$$784$$ 0 0
$$785$$ −2770.00 −0.125943
$$786$$ 0 0
$$787$$ −5404.00 −0.244767 −0.122384 0.992483i $$-0.539054\pi$$
−0.122384 + 0.992483i $$0.539054\pi$$
$$788$$ 0 0
$$789$$ 6732.00 0.303759
$$790$$ 0 0
$$791$$ 11976.0 0.538328
$$792$$ 0 0
$$793$$ 7220.00 0.323316
$$794$$ 0 0
$$795$$ 3090.00 0.137850
$$796$$ 0 0
$$797$$ 16326.0 0.725592 0.362796 0.931869i $$-0.381822\pi$$
0.362796 + 0.931869i $$0.381822\pi$$
$$798$$ 0 0
$$799$$ −744.000 −0.0329422
$$800$$ 0 0
$$801$$ −1926.00 −0.0849586
$$802$$ 0 0
$$803$$ −29040.0 −1.27621
$$804$$ 0 0
$$805$$ −6000.00 −0.262698
$$806$$ 0 0
$$807$$ −12942.0 −0.564535
$$808$$ 0 0
$$809$$ 26202.0 1.13871 0.569353 0.822093i $$-0.307194\pi$$
0.569353 + 0.822093i $$0.307194\pi$$
$$810$$ 0 0
$$811$$ −26208.0 −1.13476 −0.567378 0.823457i $$-0.692042\pi$$
−0.567378 + 0.823457i $$0.692042\pi$$
$$812$$ 0 0
$$813$$ −19176.0 −0.827222
$$814$$ 0 0
$$815$$ −6820.00 −0.293122
$$816$$ 0 0
$$817$$ −15392.0 −0.659116
$$818$$ 0 0
$$819$$ −4104.00 −0.175098
$$820$$ 0 0
$$821$$ −1986.00 −0.0844237 −0.0422119 0.999109i $$-0.513440\pi$$
−0.0422119 + 0.999109i $$0.513440\pi$$
$$822$$ 0 0
$$823$$ −5236.00 −0.221769 −0.110884 0.993833i $$-0.535368\pi$$
−0.110884 + 0.993833i $$0.535368\pi$$
$$824$$ 0 0
$$825$$ −1800.00 −0.0759612
$$826$$ 0 0
$$827$$ −26044.0 −1.09509 −0.547545 0.836777i $$-0.684437\pi$$
−0.547545 + 0.836777i $$0.684437\pi$$
$$828$$ 0 0
$$829$$ 21246.0 0.890113 0.445057 0.895502i $$-0.353184\pi$$
0.445057 + 0.895502i $$0.353184\pi$$
$$830$$ 0 0
$$831$$ 13194.0 0.550776
$$832$$ 0 0
$$833$$ 1194.00 0.0496634
$$834$$ 0 0
$$835$$ −10020.0 −0.415277
$$836$$ 0 0
$$837$$ −1512.00 −0.0624401
$$838$$ 0 0
$$839$$ −17440.0 −0.717635 −0.358817 0.933408i $$-0.616820\pi$$
−0.358817 + 0.933408i $$0.616820\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ 0 0
$$843$$ −22866.0 −0.934219
$$844$$ 0 0
$$845$$ −3765.00 −0.153278
$$846$$ 0 0
$$847$$ 9060.00 0.367539
$$848$$ 0 0
$$849$$ 3060.00 0.123697
$$850$$ 0 0
$$851$$ 19000.0 0.765349
$$852$$ 0 0
$$853$$ 33582.0 1.34798 0.673989 0.738741i $$-0.264579\pi$$
0.673989 + 0.738741i $$0.264579\pi$$
$$854$$ 0 0
$$855$$ 4680.00 0.187196
$$856$$ 0 0
$$857$$ −18942.0 −0.755013 −0.377507 0.926007i $$-0.623218\pi$$
−0.377507 + 0.926007i $$0.623218\pi$$
$$858$$ 0 0
$$859$$ 25720.0 1.02160 0.510800 0.859699i $$-0.329349\pi$$
0.510800 + 0.859699i $$0.329349\pi$$
$$860$$ 0 0
$$861$$ −7272.00 −0.287839
$$862$$ 0 0
$$863$$ 32436.0 1.27941 0.639707 0.768619i $$-0.279056\pi$$
0.639707 + 0.768619i $$0.279056\pi$$
$$864$$ 0 0
$$865$$ −7730.00 −0.303847
$$866$$ 0 0
$$867$$ −14631.0 −0.573120
$$868$$ 0 0
$$869$$ −19584.0 −0.764490
$$870$$ 0 0
$$871$$ −7752.00 −0.301569
$$872$$ 0 0
$$873$$ 10818.0 0.419397
$$874$$ 0 0
$$875$$ −1500.00 −0.0579534
$$876$$ 0 0
$$877$$ 8646.00 0.332902 0.166451 0.986050i $$-0.446769\pi$$
0.166451 + 0.986050i $$0.446769\pi$$
$$878$$ 0 0
$$879$$ −11238.0 −0.431227
$$880$$ 0 0
$$881$$ 27442.0 1.04943 0.524713 0.851279i $$-0.324173\pi$$
0.524713 + 0.851279i $$0.324173\pi$$
$$882$$ 0 0
$$883$$ −14116.0 −0.537986 −0.268993 0.963142i $$-0.586691\pi$$
−0.268993 + 0.963142i $$0.586691\pi$$
$$884$$ 0 0
$$885$$ −1920.00 −0.0729267
$$886$$ 0 0
$$887$$ 4124.00 0.156111 0.0780554 0.996949i $$-0.475129\pi$$
0.0780554 + 0.996949i $$0.475129\pi$$
$$888$$ 0 0
$$889$$ −10032.0 −0.378473
$$890$$ 0 0
$$891$$ −1944.00 −0.0730937
$$892$$ 0 0
$$893$$ 12896.0 0.483257
$$894$$ 0 0
$$895$$ −5360.00 −0.200184
$$896$$ 0 0
$$897$$ 11400.0 0.424342
$$898$$ 0 0
$$899$$ −12880.0 −0.477833
$$900$$ 0 0
$$901$$ −1236.00 −0.0457016
$$902$$ 0 0
$$903$$ 5328.00 0.196351
$$904$$ 0 0
$$905$$ −18770.0 −0.689432
$$906$$ 0 0
$$907$$ 42100.0 1.54124 0.770622 0.637293i $$-0.219946\pi$$
0.770622 + 0.637293i $$0.219946\pi$$
$$908$$ 0 0
$$909$$ 12078.0 0.440706
$$910$$ 0 0
$$911$$ 34152.0 1.24205 0.621024 0.783791i $$-0.286717\pi$$
0.621024 + 0.783791i $$0.286717\pi$$
$$912$$ 0 0
$$913$$ 33888.0 1.22840
$$914$$ 0 0
$$915$$ 2850.00 0.102971
$$916$$ 0 0
$$917$$ 17760.0 0.639571
$$918$$ 0 0
$$919$$ 41984.0 1.50699 0.753495 0.657453i $$-0.228366\pi$$
0.753495 + 0.657453i $$0.228366\pi$$
$$920$$ 0 0
$$921$$ 29100.0 1.04113
$$922$$ 0 0
$$923$$ −16720.0 −0.596257
$$924$$ 0 0
$$925$$ 4750.00 0.168842
$$926$$ 0 0
$$927$$ 8172.00 0.289540
$$928$$ 0 0
$$929$$ 48434.0 1.71051 0.855257 0.518204i $$-0.173399\pi$$
0.855257 + 0.518204i $$0.173399\pi$$
$$930$$ 0 0
$$931$$ −20696.0 −0.728554
$$932$$ 0 0
$$933$$ 12456.0 0.437075
$$934$$ 0 0
$$935$$ 720.000 0.0251834
$$936$$ 0 0
$$937$$ −25590.0 −0.892197 −0.446099 0.894984i $$-0.647187\pi$$
−0.446099 + 0.894984i $$0.647187\pi$$
$$938$$ 0 0
$$939$$ 19086.0 0.663310
$$940$$ 0 0
$$941$$ −18906.0 −0.654961 −0.327480 0.944858i $$-0.606200\pi$$
−0.327480 + 0.944858i $$0.606200\pi$$
$$942$$ 0 0
$$943$$ 20200.0 0.697564
$$944$$ 0 0
$$945$$ −1620.00 −0.0557657
$$946$$ 0 0
$$947$$ −36316.0 −1.24616 −0.623079 0.782159i $$-0.714118\pi$$
−0.623079 + 0.782159i $$0.714118\pi$$
$$948$$ 0 0
$$949$$ 45980.0 1.57279
$$950$$ 0 0
$$951$$ 32658.0 1.11357
$$952$$ 0 0
$$953$$ 15890.0 0.540113 0.270056 0.962844i $$-0.412958\pi$$
0.270056 + 0.962844i $$0.412958\pi$$
$$954$$ 0 0
$$955$$ 6120.00 0.207370
$$956$$ 0 0
$$957$$ −16560.0 −0.559361
$$958$$ 0 0
$$959$$ −16152.0 −0.543874
$$960$$ 0 0
$$961$$ −26655.0 −0.894733
$$962$$ 0 0
$$963$$ −7884.00 −0.263820
$$964$$ 0 0
$$965$$ −8470.00 −0.282548
$$966$$ 0 0
$$967$$ 116.000 0.00385761 0.00192880 0.999998i $$-0.499386\pi$$
0.00192880 + 0.999998i $$0.499386\pi$$
$$968$$ 0 0
$$969$$ −1872.00 −0.0620612
$$970$$ 0 0
$$971$$ 4744.00 0.156789 0.0783945 0.996922i $$-0.475021\pi$$
0.0783945 + 0.996922i $$0.475021\pi$$
$$972$$ 0 0
$$973$$ −9888.00 −0.325791
$$974$$ 0 0
$$975$$ 2850.00 0.0936134
$$976$$ 0 0
$$977$$ −18374.0 −0.601675 −0.300837 0.953675i $$-0.597266\pi$$
−0.300837 + 0.953675i $$0.597266\pi$$
$$978$$ 0 0
$$979$$ 5136.00 0.167668
$$980$$ 0 0
$$981$$ 2718.00 0.0884598
$$982$$ 0 0
$$983$$ −13036.0 −0.422974 −0.211487 0.977381i $$-0.567831\pi$$
−0.211487 + 0.977381i $$0.567831\pi$$
$$984$$ 0 0
$$985$$ 15670.0 0.506891
$$986$$ 0 0
$$987$$ −4464.00 −0.143962
$$988$$ 0 0
$$989$$ −14800.0 −0.475847
$$990$$ 0 0
$$991$$ 15224.0 0.487998 0.243999 0.969775i $$-0.421541\pi$$
0.243999 + 0.969775i $$0.421541\pi$$
$$992$$ 0 0
$$993$$ 12384.0 0.395765
$$994$$ 0 0
$$995$$ 12800.0 0.407826
$$996$$ 0 0
$$997$$ −43794.0 −1.39114 −0.695572 0.718457i $$-0.744849\pi$$
−0.695572 + 0.718457i $$0.744849\pi$$
$$998$$ 0 0
$$999$$ 5130.00 0.162468
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 480.4.a.j.1.1 yes 1
3.2 odd 2 1440.4.a.d.1.1 1
4.3 odd 2 480.4.a.e.1.1 1
5.4 even 2 2400.4.a.g.1.1 1
8.3 odd 2 960.4.a.y.1.1 1
8.5 even 2 960.4.a.d.1.1 1
12.11 even 2 1440.4.a.g.1.1 1
20.19 odd 2 2400.4.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.e.1.1 1 4.3 odd 2
480.4.a.j.1.1 yes 1 1.1 even 1 trivial
960.4.a.d.1.1 1 8.5 even 2
960.4.a.y.1.1 1 8.3 odd 2
1440.4.a.d.1.1 1 3.2 odd 2
1440.4.a.g.1.1 1 12.11 even 2
2400.4.a.g.1.1 1 5.4 even 2
2400.4.a.p.1.1 1 20.19 odd 2