# Properties

 Label 480.4.a.b.1.1 Level $480$ Weight $4$ Character 480.1 Self dual yes Analytic conductor $28.321$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.00000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.00000 q^{5} +8.00000 q^{7} +9.00000 q^{9} -4.00000 q^{11} -6.00000 q^{13} +15.0000 q^{15} -2.00000 q^{17} +16.0000 q^{19} -24.0000 q^{21} +60.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -142.000 q^{29} +176.000 q^{31} +12.0000 q^{33} -40.0000 q^{35} -214.000 q^{37} +18.0000 q^{39} -278.000 q^{41} +68.0000 q^{43} -45.0000 q^{45} -116.000 q^{47} -279.000 q^{49} +6.00000 q^{51} -350.000 q^{53} +20.0000 q^{55} -48.0000 q^{57} -684.000 q^{59} -394.000 q^{61} +72.0000 q^{63} +30.0000 q^{65} -108.000 q^{67} -180.000 q^{69} +96.0000 q^{71} -398.000 q^{73} -75.0000 q^{75} -32.0000 q^{77} -136.000 q^{79} +81.0000 q^{81} -436.000 q^{83} +10.0000 q^{85} +426.000 q^{87} -750.000 q^{89} -48.0000 q^{91} -528.000 q^{93} -80.0000 q^{95} +82.0000 q^{97} -36.0000 q^{99} +O(q^{100})$$

## 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
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −0.109640 −0.0548202 0.998496i $$-0.517459\pi$$
−0.0548202 + 0.998496i $$0.517459\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −0.128008 −0.0640039 0.997950i $$-0.520387\pi$$
−0.0640039 + 0.997950i $$0.520387\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ −2.00000 −0.0285336 −0.0142668 0.999898i $$-0.504541\pi$$
−0.0142668 + 0.999898i $$0.504541\pi$$
$$18$$ 0 0
$$19$$ 16.0000 0.193192 0.0965961 0.995324i $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ 0 0
$$21$$ −24.0000 −0.249392
$$22$$ 0 0
$$23$$ 60.0000 0.543951 0.271975 0.962304i $$-0.412323\pi$$
0.271975 + 0.962304i $$0.412323\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −142.000 −0.909267 −0.454633 0.890679i $$-0.650230\pi$$
−0.454633 + 0.890679i $$0.650230\pi$$
$$30$$ 0 0
$$31$$ 176.000 1.01969 0.509847 0.860265i $$-0.329702\pi$$
0.509847 + 0.860265i $$0.329702\pi$$
$$32$$ 0 0
$$33$$ 12.0000 0.0633010
$$34$$ 0 0
$$35$$ −40.0000 −0.193178
$$36$$ 0 0
$$37$$ −214.000 −0.950848 −0.475424 0.879757i $$-0.657705\pi$$
−0.475424 + 0.879757i $$0.657705\pi$$
$$38$$ 0 0
$$39$$ 18.0000 0.0739053
$$40$$ 0 0
$$41$$ −278.000 −1.05893 −0.529467 0.848330i $$-0.677608\pi$$
−0.529467 + 0.848330i $$0.677608\pi$$
$$42$$ 0 0
$$43$$ 68.0000 0.241161 0.120580 0.992704i $$-0.461524\pi$$
0.120580 + 0.992704i $$0.461524\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ −116.000 −0.360007 −0.180004 0.983666i $$-0.557611\pi$$
−0.180004 + 0.983666i $$0.557611\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ 6.00000 0.0164739
$$52$$ 0 0
$$53$$ −350.000 −0.907098 −0.453549 0.891231i $$-0.649842\pi$$
−0.453549 + 0.891231i $$0.649842\pi$$
$$54$$ 0 0
$$55$$ 20.0000 0.0490327
$$56$$ 0 0
$$57$$ −48.0000 −0.111540
$$58$$ 0 0
$$59$$ −684.000 −1.50931 −0.754654 0.656123i $$-0.772195\pi$$
−0.754654 + 0.656123i $$0.772195\pi$$
$$60$$ 0 0
$$61$$ −394.000 −0.826992 −0.413496 0.910506i $$-0.635692\pi$$
−0.413496 + 0.910506i $$0.635692\pi$$
$$62$$ 0 0
$$63$$ 72.0000 0.143986
$$64$$ 0 0
$$65$$ 30.0000 0.0572468
$$66$$ 0 0
$$67$$ −108.000 −0.196930 −0.0984649 0.995141i $$-0.531393\pi$$
−0.0984649 + 0.995141i $$0.531393\pi$$
$$68$$ 0 0
$$69$$ −180.000 −0.314050
$$70$$ 0 0
$$71$$ 96.0000 0.160466 0.0802331 0.996776i $$-0.474434\pi$$
0.0802331 + 0.996776i $$0.474434\pi$$
$$72$$ 0 0
$$73$$ −398.000 −0.638115 −0.319057 0.947735i $$-0.603366\pi$$
−0.319057 + 0.947735i $$0.603366\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ −32.0000 −0.0473602
$$78$$ 0 0
$$79$$ −136.000 −0.193686 −0.0968430 0.995300i $$-0.530874\pi$$
−0.0968430 + 0.995300i $$0.530874\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −436.000 −0.576593 −0.288296 0.957541i $$-0.593089\pi$$
−0.288296 + 0.957541i $$0.593089\pi$$
$$84$$ 0 0
$$85$$ 10.0000 0.0127606
$$86$$ 0 0
$$87$$ 426.000 0.524965
$$88$$ 0 0
$$89$$ −750.000 −0.893257 −0.446628 0.894720i $$-0.647375\pi$$
−0.446628 + 0.894720i $$0.647375\pi$$
$$90$$ 0 0
$$91$$ −48.0000 −0.0552941
$$92$$ 0 0
$$93$$ −528.000 −0.588721
$$94$$ 0 0
$$95$$ −80.0000 −0.0863982
$$96$$ 0 0
$$97$$ 82.0000 0.0858334 0.0429167 0.999079i $$-0.486335\pi$$
0.0429167 + 0.999079i $$0.486335\pi$$
$$98$$ 0 0
$$99$$ −36.0000 −0.0365468
$$100$$ 0 0
$$101$$ 154.000 0.151719 0.0758593 0.997119i $$-0.475830\pi$$
0.0758593 + 0.997119i $$0.475830\pi$$
$$102$$ 0 0
$$103$$ 1216.00 1.16326 0.581631 0.813453i $$-0.302415\pi$$
0.581631 + 0.813453i $$0.302415\pi$$
$$104$$ 0 0
$$105$$ 120.000 0.111531
$$106$$ 0 0
$$107$$ −732.000 −0.661356 −0.330678 0.943744i $$-0.607277\pi$$
−0.330678 + 0.943744i $$0.607277\pi$$
$$108$$ 0 0
$$109$$ −554.000 −0.486822 −0.243411 0.969923i $$-0.578266\pi$$
−0.243411 + 0.969923i $$0.578266\pi$$
$$110$$ 0 0
$$111$$ 642.000 0.548972
$$112$$ 0 0
$$113$$ −474.000 −0.394603 −0.197302 0.980343i $$-0.563218\pi$$
−0.197302 + 0.980343i $$0.563218\pi$$
$$114$$ 0 0
$$115$$ −300.000 −0.243262
$$116$$ 0 0
$$117$$ −54.0000 −0.0426692
$$118$$ 0 0
$$119$$ −16.0000 −0.0123254
$$120$$ 0 0
$$121$$ −1315.00 −0.987979
$$122$$ 0 0
$$123$$ 834.000 0.611376
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 840.000 0.586913 0.293456 0.955972i $$-0.405194\pi$$
0.293456 + 0.955972i $$0.405194\pi$$
$$128$$ 0 0
$$129$$ −204.000 −0.139234
$$130$$ 0 0
$$131$$ −1484.00 −0.989753 −0.494877 0.868963i $$-0.664787\pi$$
−0.494877 + 0.868963i $$0.664787\pi$$
$$132$$ 0 0
$$133$$ 128.000 0.0834512
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 1006.00 0.627360 0.313680 0.949529i $$-0.398438\pi$$
0.313680 + 0.949529i $$0.398438\pi$$
$$138$$ 0 0
$$139$$ −1400.00 −0.854291 −0.427146 0.904183i $$-0.640481\pi$$
−0.427146 + 0.904183i $$0.640481\pi$$
$$140$$ 0 0
$$141$$ 348.000 0.207850
$$142$$ 0 0
$$143$$ 24.0000 0.0140348
$$144$$ 0 0
$$145$$ 710.000 0.406636
$$146$$ 0 0
$$147$$ 837.000 0.469623
$$148$$ 0 0
$$149$$ 2.00000 0.00109964 0.000549820 1.00000i $$-0.499825\pi$$
0.000549820 1.00000i $$0.499825\pi$$
$$150$$ 0 0
$$151$$ 2248.00 1.21152 0.605760 0.795647i $$-0.292869\pi$$
0.605760 + 0.795647i $$0.292869\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −0.00951120
$$154$$ 0 0
$$155$$ −880.000 −0.456021
$$156$$ 0 0
$$157$$ 2890.00 1.46909 0.734545 0.678560i $$-0.237396\pi$$
0.734545 + 0.678560i $$0.237396\pi$$
$$158$$ 0 0
$$159$$ 1050.00 0.523713
$$160$$ 0 0
$$161$$ 480.000 0.234965
$$162$$ 0 0
$$163$$ 252.000 0.121093 0.0605465 0.998165i $$-0.480716\pi$$
0.0605465 + 0.998165i $$0.480716\pi$$
$$164$$ 0 0
$$165$$ −60.0000 −0.0283091
$$166$$ 0 0
$$167$$ 884.000 0.409617 0.204808 0.978802i $$-0.434343\pi$$
0.204808 + 0.978802i $$0.434343\pi$$
$$168$$ 0 0
$$169$$ −2161.00 −0.983614
$$170$$ 0 0
$$171$$ 144.000 0.0643974
$$172$$ 0 0
$$173$$ −1398.00 −0.614381 −0.307191 0.951648i $$-0.599389\pi$$
−0.307191 + 0.951648i $$0.599389\pi$$
$$174$$ 0 0
$$175$$ 200.000 0.0863919
$$176$$ 0 0
$$177$$ 2052.00 0.871400
$$178$$ 0 0
$$179$$ −3612.00 −1.50823 −0.754116 0.656741i $$-0.771934\pi$$
−0.754116 + 0.656741i $$0.771934\pi$$
$$180$$ 0 0
$$181$$ 3150.00 1.29358 0.646789 0.762669i $$-0.276111\pi$$
0.646789 + 0.762669i $$0.276111\pi$$
$$182$$ 0 0
$$183$$ 1182.00 0.477464
$$184$$ 0 0
$$185$$ 1070.00 0.425232
$$186$$ 0 0
$$187$$ 8.00000 0.00312844
$$188$$ 0 0
$$189$$ −216.000 −0.0831306
$$190$$ 0 0
$$191$$ −3040.00 −1.15166 −0.575829 0.817570i $$-0.695321\pi$$
−0.575829 + 0.817570i $$0.695321\pi$$
$$192$$ 0 0
$$193$$ 1994.00 0.743685 0.371843 0.928296i $$-0.378726\pi$$
0.371843 + 0.928296i $$0.378726\pi$$
$$194$$ 0 0
$$195$$ −90.0000 −0.0330515
$$196$$ 0 0
$$197$$ 4066.00 1.47051 0.735255 0.677790i $$-0.237062\pi$$
0.735255 + 0.677790i $$0.237062\pi$$
$$198$$ 0 0
$$199$$ 3904.00 1.39069 0.695345 0.718676i $$-0.255252\pi$$
0.695345 + 0.718676i $$0.255252\pi$$
$$200$$ 0 0
$$201$$ 324.000 0.113698
$$202$$ 0 0
$$203$$ −1136.00 −0.392766
$$204$$ 0 0
$$205$$ 1390.00 0.473570
$$206$$ 0 0
$$207$$ 540.000 0.181317
$$208$$ 0 0
$$209$$ −64.0000 −0.0211817
$$210$$ 0 0
$$211$$ 560.000 0.182711 0.0913554 0.995818i $$-0.470880\pi$$
0.0913554 + 0.995818i $$0.470880\pi$$
$$212$$ 0 0
$$213$$ −288.000 −0.0926452
$$214$$ 0 0
$$215$$ −340.000 −0.107850
$$216$$ 0 0
$$217$$ 1408.00 0.440467
$$218$$ 0 0
$$219$$ 1194.00 0.368416
$$220$$ 0 0
$$221$$ 12.0000 0.00365252
$$222$$ 0 0
$$223$$ −608.000 −0.182577 −0.0912885 0.995824i $$-0.529099\pi$$
−0.0912885 + 0.995824i $$0.529099\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −5124.00 −1.49820 −0.749101 0.662456i $$-0.769514\pi$$
−0.749101 + 0.662456i $$0.769514\pi$$
$$228$$ 0 0
$$229$$ 1190.00 0.343395 0.171697 0.985150i $$-0.445075\pi$$
0.171697 + 0.985150i $$0.445075\pi$$
$$230$$ 0 0
$$231$$ 96.0000 0.0273434
$$232$$ 0 0
$$233$$ 3862.00 1.08587 0.542936 0.839774i $$-0.317313\pi$$
0.542936 + 0.839774i $$0.317313\pi$$
$$234$$ 0 0
$$235$$ 580.000 0.161000
$$236$$ 0 0
$$237$$ 408.000 0.111825
$$238$$ 0 0
$$239$$ −1640.00 −0.443861 −0.221931 0.975062i $$-0.571236\pi$$
−0.221931 + 0.975062i $$0.571236\pi$$
$$240$$ 0 0
$$241$$ −2334.00 −0.623843 −0.311921 0.950108i $$-0.600973\pi$$
−0.311921 + 0.950108i $$0.600973\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1395.00 0.363768
$$246$$ 0 0
$$247$$ −96.0000 −0.0247301
$$248$$ 0 0
$$249$$ 1308.00 0.332896
$$250$$ 0 0
$$251$$ 772.000 0.194136 0.0970681 0.995278i $$-0.469054\pi$$
0.0970681 + 0.995278i $$0.469054\pi$$
$$252$$ 0 0
$$253$$ −240.000 −0.0596390
$$254$$ 0 0
$$255$$ −30.0000 −0.00736734
$$256$$ 0 0
$$257$$ 6214.00 1.50824 0.754122 0.656734i $$-0.228063\pi$$
0.754122 + 0.656734i $$0.228063\pi$$
$$258$$ 0 0
$$259$$ −1712.00 −0.410728
$$260$$ 0 0
$$261$$ −1278.00 −0.303089
$$262$$ 0 0
$$263$$ −4948.00 −1.16010 −0.580051 0.814580i $$-0.696967\pi$$
−0.580051 + 0.814580i $$0.696967\pi$$
$$264$$ 0 0
$$265$$ 1750.00 0.405667
$$266$$ 0 0
$$267$$ 2250.00 0.515722
$$268$$ 0 0
$$269$$ −7038.00 −1.59522 −0.797610 0.603173i $$-0.793903\pi$$
−0.797610 + 0.603173i $$0.793903\pi$$
$$270$$ 0 0
$$271$$ 728.000 0.163184 0.0815920 0.996666i $$-0.474000\pi$$
0.0815920 + 0.996666i $$0.474000\pi$$
$$272$$ 0 0
$$273$$ 144.000 0.0319241
$$274$$ 0 0
$$275$$ −100.000 −0.0219281
$$276$$ 0 0
$$277$$ 7274.00 1.57781 0.788903 0.614518i $$-0.210649\pi$$
0.788903 + 0.614518i $$0.210649\pi$$
$$278$$ 0 0
$$279$$ 1584.00 0.339898
$$280$$ 0 0
$$281$$ 1714.00 0.363874 0.181937 0.983310i $$-0.441763\pi$$
0.181937 + 0.983310i $$0.441763\pi$$
$$282$$ 0 0
$$283$$ 4316.00 0.906571 0.453285 0.891365i $$-0.350252\pi$$
0.453285 + 0.891365i $$0.350252\pi$$
$$284$$ 0 0
$$285$$ 240.000 0.0498820
$$286$$ 0 0
$$287$$ −2224.00 −0.457417
$$288$$ 0 0
$$289$$ −4909.00 −0.999186
$$290$$ 0 0
$$291$$ −246.000 −0.0495559
$$292$$ 0 0
$$293$$ −6078.00 −1.21188 −0.605940 0.795511i $$-0.707203\pi$$
−0.605940 + 0.795511i $$0.707203\pi$$
$$294$$ 0 0
$$295$$ 3420.00 0.674983
$$296$$ 0 0
$$297$$ 108.000 0.0211003
$$298$$ 0 0
$$299$$ −360.000 −0.0696299
$$300$$ 0 0
$$301$$ 544.000 0.104172
$$302$$ 0 0
$$303$$ −462.000 −0.0875947
$$304$$ 0 0
$$305$$ 1970.00 0.369842
$$306$$ 0 0
$$307$$ 1564.00 0.290756 0.145378 0.989376i $$-0.453560\pi$$
0.145378 + 0.989376i $$0.453560\pi$$
$$308$$ 0 0
$$309$$ −3648.00 −0.671610
$$310$$ 0 0
$$311$$ −5216.00 −0.951036 −0.475518 0.879706i $$-0.657739\pi$$
−0.475518 + 0.879706i $$0.657739\pi$$
$$312$$ 0 0
$$313$$ −5790.00 −1.04559 −0.522796 0.852458i $$-0.675111\pi$$
−0.522796 + 0.852458i $$0.675111\pi$$
$$314$$ 0 0
$$315$$ −360.000 −0.0643927
$$316$$ 0 0
$$317$$ 3946.00 0.699146 0.349573 0.936909i $$-0.386327\pi$$
0.349573 + 0.936909i $$0.386327\pi$$
$$318$$ 0 0
$$319$$ 568.000 0.0996925
$$320$$ 0 0
$$321$$ 2196.00 0.381834
$$322$$ 0 0
$$323$$ −32.0000 −0.00551247
$$324$$ 0 0
$$325$$ −150.000 −0.0256015
$$326$$ 0 0
$$327$$ 1662.00 0.281067
$$328$$ 0 0
$$329$$ −928.000 −0.155508
$$330$$ 0 0
$$331$$ 9992.00 1.65924 0.829622 0.558325i $$-0.188556\pi$$
0.829622 + 0.558325i $$0.188556\pi$$
$$332$$ 0 0
$$333$$ −1926.00 −0.316949
$$334$$ 0 0
$$335$$ 540.000 0.0880697
$$336$$ 0 0
$$337$$ 3274.00 0.529217 0.264609 0.964356i $$-0.414757\pi$$
0.264609 + 0.964356i $$0.414757\pi$$
$$338$$ 0 0
$$339$$ 1422.00 0.227824
$$340$$ 0 0
$$341$$ −704.000 −0.111800
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 900.000 0.140447
$$346$$ 0 0
$$347$$ 5580.00 0.863257 0.431628 0.902052i $$-0.357939\pi$$
0.431628 + 0.902052i $$0.357939\pi$$
$$348$$ 0 0
$$349$$ −4154.00 −0.637130 −0.318565 0.947901i $$-0.603201\pi$$
−0.318565 + 0.947901i $$0.603201\pi$$
$$350$$ 0 0
$$351$$ 162.000 0.0246351
$$352$$ 0 0
$$353$$ −2802.00 −0.422480 −0.211240 0.977434i $$-0.567750\pi$$
−0.211240 + 0.977434i $$0.567750\pi$$
$$354$$ 0 0
$$355$$ −480.000 −0.0717627
$$356$$ 0 0
$$357$$ 48.0000 0.00711605
$$358$$ 0 0
$$359$$ 11296.0 1.66067 0.830334 0.557265i $$-0.188149\pi$$
0.830334 + 0.557265i $$0.188149\pi$$
$$360$$ 0 0
$$361$$ −6603.00 −0.962677
$$362$$ 0 0
$$363$$ 3945.00 0.570410
$$364$$ 0 0
$$365$$ 1990.00 0.285374
$$366$$ 0 0
$$367$$ −10536.0 −1.49857 −0.749284 0.662248i $$-0.769602\pi$$
−0.749284 + 0.662248i $$0.769602\pi$$
$$368$$ 0 0
$$369$$ −2502.00 −0.352978
$$370$$ 0 0
$$371$$ −2800.00 −0.391830
$$372$$ 0 0
$$373$$ −2222.00 −0.308447 −0.154224 0.988036i $$-0.549288\pi$$
−0.154224 + 0.988036i $$0.549288\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 852.000 0.116393
$$378$$ 0 0
$$379$$ 1512.00 0.204924 0.102462 0.994737i $$-0.467328\pi$$
0.102462 + 0.994737i $$0.467328\pi$$
$$380$$ 0 0
$$381$$ −2520.00 −0.338854
$$382$$ 0 0
$$383$$ −7916.00 −1.05611 −0.528053 0.849211i $$-0.677078\pi$$
−0.528053 + 0.849211i $$0.677078\pi$$
$$384$$ 0 0
$$385$$ 160.000 0.0211801
$$386$$ 0 0
$$387$$ 612.000 0.0803868
$$388$$ 0 0
$$389$$ −9950.00 −1.29688 −0.648438 0.761267i $$-0.724578\pi$$
−0.648438 + 0.761267i $$0.724578\pi$$
$$390$$ 0 0
$$391$$ −120.000 −0.0155209
$$392$$ 0 0
$$393$$ 4452.00 0.571434
$$394$$ 0 0
$$395$$ 680.000 0.0866190
$$396$$ 0 0
$$397$$ 9554.00 1.20781 0.603906 0.797055i $$-0.293610\pi$$
0.603906 + 0.797055i $$0.293610\pi$$
$$398$$ 0 0
$$399$$ −384.000 −0.0481806
$$400$$ 0 0
$$401$$ −6390.00 −0.795764 −0.397882 0.917437i $$-0.630255\pi$$
−0.397882 + 0.917437i $$0.630255\pi$$
$$402$$ 0 0
$$403$$ −1056.00 −0.130529
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ 856.000 0.104251
$$408$$ 0 0
$$409$$ −13030.0 −1.57529 −0.787643 0.616132i $$-0.788699\pi$$
−0.787643 + 0.616132i $$0.788699\pi$$
$$410$$ 0 0
$$411$$ −3018.00 −0.362207
$$412$$ 0 0
$$413$$ −5472.00 −0.651960
$$414$$ 0 0
$$415$$ 2180.00 0.257860
$$416$$ 0 0
$$417$$ 4200.00 0.493225
$$418$$ 0 0
$$419$$ −2100.00 −0.244849 −0.122424 0.992478i $$-0.539067\pi$$
−0.122424 + 0.992478i $$0.539067\pi$$
$$420$$ 0 0
$$421$$ 3478.00 0.402630 0.201315 0.979527i $$-0.435478\pi$$
0.201315 + 0.979527i $$0.435478\pi$$
$$422$$ 0 0
$$423$$ −1044.00 −0.120002
$$424$$ 0 0
$$425$$ −50.0000 −0.00570672
$$426$$ 0 0
$$427$$ −3152.00 −0.357227
$$428$$ 0 0
$$429$$ −72.0000 −0.00810301
$$430$$ 0 0
$$431$$ −8504.00 −0.950402 −0.475201 0.879877i $$-0.657625\pi$$
−0.475201 + 0.879877i $$0.657625\pi$$
$$432$$ 0 0
$$433$$ −16102.0 −1.78710 −0.893548 0.448967i $$-0.851792\pi$$
−0.893548 + 0.448967i $$0.851792\pi$$
$$434$$ 0 0
$$435$$ −2130.00 −0.234772
$$436$$ 0 0
$$437$$ 960.000 0.105087
$$438$$ 0 0
$$439$$ 1072.00 0.116546 0.0582731 0.998301i $$-0.481441\pi$$
0.0582731 + 0.998301i $$0.481441\pi$$
$$440$$ 0 0
$$441$$ −2511.00 −0.271137
$$442$$ 0 0
$$443$$ 11076.0 1.18789 0.593947 0.804505i $$-0.297569\pi$$
0.593947 + 0.804505i $$0.297569\pi$$
$$444$$ 0 0
$$445$$ 3750.00 0.399477
$$446$$ 0 0
$$447$$ −6.00000 −0.000634878 0
$$448$$ 0 0
$$449$$ −9910.00 −1.04161 −0.520804 0.853676i $$-0.674368\pi$$
−0.520804 + 0.853676i $$0.674368\pi$$
$$450$$ 0 0
$$451$$ 1112.00 0.116102
$$452$$ 0 0
$$453$$ −6744.00 −0.699472
$$454$$ 0 0
$$455$$ 240.000 0.0247283
$$456$$ 0 0
$$457$$ −3798.00 −0.388759 −0.194380 0.980926i $$-0.562269\pi$$
−0.194380 + 0.980926i $$0.562269\pi$$
$$458$$ 0 0
$$459$$ 54.0000 0.00549129
$$460$$ 0 0
$$461$$ −8406.00 −0.849255 −0.424627 0.905368i $$-0.639595\pi$$
−0.424627 + 0.905368i $$0.639595\pi$$
$$462$$ 0 0
$$463$$ −18040.0 −1.81078 −0.905389 0.424584i $$-0.860420\pi$$
−0.905389 + 0.424584i $$0.860420\pi$$
$$464$$ 0 0
$$465$$ 2640.00 0.263284
$$466$$ 0 0
$$467$$ 8484.00 0.840670 0.420335 0.907369i $$-0.361913\pi$$
0.420335 + 0.907369i $$0.361913\pi$$
$$468$$ 0 0
$$469$$ −864.000 −0.0850657
$$470$$ 0 0
$$471$$ −8670.00 −0.848179
$$472$$ 0 0
$$473$$ −272.000 −0.0264410
$$474$$ 0 0
$$475$$ 400.000 0.0386384
$$476$$ 0 0
$$477$$ −3150.00 −0.302366
$$478$$ 0 0
$$479$$ 2136.00 0.203750 0.101875 0.994797i $$-0.467516\pi$$
0.101875 + 0.994797i $$0.467516\pi$$
$$480$$ 0 0
$$481$$ 1284.00 0.121716
$$482$$ 0 0
$$483$$ −1440.00 −0.135657
$$484$$ 0 0
$$485$$ −410.000 −0.0383859
$$486$$ 0 0
$$487$$ −3152.00 −0.293287 −0.146643 0.989189i $$-0.546847\pi$$
−0.146643 + 0.989189i $$0.546847\pi$$
$$488$$ 0 0
$$489$$ −756.000 −0.0699131
$$490$$ 0 0
$$491$$ 11812.0 1.08568 0.542839 0.839837i $$-0.317349\pi$$
0.542839 + 0.839837i $$0.317349\pi$$
$$492$$ 0 0
$$493$$ 284.000 0.0259447
$$494$$ 0 0
$$495$$ 180.000 0.0163442
$$496$$ 0 0
$$497$$ 768.000 0.0693149
$$498$$ 0 0
$$499$$ −10352.0 −0.928696 −0.464348 0.885653i $$-0.653711\pi$$
−0.464348 + 0.885653i $$0.653711\pi$$
$$500$$ 0 0
$$501$$ −2652.00 −0.236492
$$502$$ 0 0
$$503$$ 1300.00 0.115237 0.0576184 0.998339i $$-0.481649\pi$$
0.0576184 + 0.998339i $$0.481649\pi$$
$$504$$ 0 0
$$505$$ −770.000 −0.0678506
$$506$$ 0 0
$$507$$ 6483.00 0.567890
$$508$$ 0 0
$$509$$ −4398.00 −0.382982 −0.191491 0.981494i $$-0.561332\pi$$
−0.191491 + 0.981494i $$0.561332\pi$$
$$510$$ 0 0
$$511$$ −3184.00 −0.275640
$$512$$ 0 0
$$513$$ −432.000 −0.0371799
$$514$$ 0 0
$$515$$ −6080.00 −0.520227
$$516$$ 0 0
$$517$$ 464.000 0.0394714
$$518$$ 0 0
$$519$$ 4194.00 0.354713
$$520$$ 0 0
$$521$$ 1650.00 0.138748 0.0693741 0.997591i $$-0.477900\pi$$
0.0693741 + 0.997591i $$0.477900\pi$$
$$522$$ 0 0
$$523$$ 17276.0 1.44441 0.722205 0.691679i $$-0.243129\pi$$
0.722205 + 0.691679i $$0.243129\pi$$
$$524$$ 0 0
$$525$$ −600.000 −0.0498784
$$526$$ 0 0
$$527$$ −352.000 −0.0290956
$$528$$ 0 0
$$529$$ −8567.00 −0.704118
$$530$$ 0 0
$$531$$ −6156.00 −0.503103
$$532$$ 0 0
$$533$$ 1668.00 0.135552
$$534$$ 0 0
$$535$$ 3660.00 0.295767
$$536$$ 0 0
$$537$$ 10836.0 0.870778
$$538$$ 0 0
$$539$$ 1116.00 0.0891828
$$540$$ 0 0
$$541$$ 3878.00 0.308185 0.154093 0.988056i $$-0.450755\pi$$
0.154093 + 0.988056i $$0.450755\pi$$
$$542$$ 0 0
$$543$$ −9450.00 −0.746848
$$544$$ 0 0
$$545$$ 2770.00 0.217713
$$546$$ 0 0
$$547$$ −9604.00 −0.750708 −0.375354 0.926881i $$-0.622479\pi$$
−0.375354 + 0.926881i $$0.622479\pi$$
$$548$$ 0 0
$$549$$ −3546.00 −0.275664
$$550$$ 0 0
$$551$$ −2272.00 −0.175663
$$552$$ 0 0
$$553$$ −1088.00 −0.0836645
$$554$$ 0 0
$$555$$ −3210.00 −0.245508
$$556$$ 0 0
$$557$$ 13546.0 1.03045 0.515227 0.857054i $$-0.327708\pi$$
0.515227 + 0.857054i $$0.327708\pi$$
$$558$$ 0 0
$$559$$ −408.000 −0.0308704
$$560$$ 0 0
$$561$$ −24.0000 −0.00180620
$$562$$ 0 0
$$563$$ 16884.0 1.26390 0.631950 0.775009i $$-0.282255\pi$$
0.631950 + 0.775009i $$0.282255\pi$$
$$564$$ 0 0
$$565$$ 2370.00 0.176472
$$566$$ 0 0
$$567$$ 648.000 0.0479955
$$568$$ 0 0
$$569$$ 13746.0 1.01276 0.506382 0.862309i $$-0.330983\pi$$
0.506382 + 0.862309i $$0.330983\pi$$
$$570$$ 0 0
$$571$$ 15176.0 1.11225 0.556126 0.831098i $$-0.312287\pi$$
0.556126 + 0.831098i $$0.312287\pi$$
$$572$$ 0 0
$$573$$ 9120.00 0.664910
$$574$$ 0 0
$$575$$ 1500.00 0.108790
$$576$$ 0 0
$$577$$ 15106.0 1.08990 0.544949 0.838469i $$-0.316549\pi$$
0.544949 + 0.838469i $$0.316549\pi$$
$$578$$ 0 0
$$579$$ −5982.00 −0.429367
$$580$$ 0 0
$$581$$ −3488.00 −0.249065
$$582$$ 0 0
$$583$$ 1400.00 0.0994547
$$584$$ 0 0
$$585$$ 270.000 0.0190823
$$586$$ 0 0
$$587$$ 17332.0 1.21869 0.609343 0.792907i $$-0.291433\pi$$
0.609343 + 0.792907i $$0.291433\pi$$
$$588$$ 0 0
$$589$$ 2816.00 0.196997
$$590$$ 0 0
$$591$$ −12198.0 −0.848999
$$592$$ 0 0
$$593$$ −1290.00 −0.0893321 −0.0446661 0.999002i $$-0.514222\pi$$
−0.0446661 + 0.999002i $$0.514222\pi$$
$$594$$ 0 0
$$595$$ 80.0000 0.00551207
$$596$$ 0 0
$$597$$ −11712.0 −0.802915
$$598$$ 0 0
$$599$$ 20544.0 1.40134 0.700672 0.713484i $$-0.252884\pi$$
0.700672 + 0.713484i $$0.252884\pi$$
$$600$$ 0 0
$$601$$ −2630.00 −0.178502 −0.0892512 0.996009i $$-0.528447\pi$$
−0.0892512 + 0.996009i $$0.528447\pi$$
$$602$$ 0 0
$$603$$ −972.000 −0.0656433
$$604$$ 0 0
$$605$$ 6575.00 0.441838
$$606$$ 0 0
$$607$$ −20320.0 −1.35875 −0.679377 0.733790i $$-0.737750\pi$$
−0.679377 + 0.733790i $$0.737750\pi$$
$$608$$ 0 0
$$609$$ 3408.00 0.226764
$$610$$ 0 0
$$611$$ 696.000 0.0460837
$$612$$ 0 0
$$613$$ 15386.0 1.01376 0.506880 0.862017i $$-0.330799\pi$$
0.506880 + 0.862017i $$0.330799\pi$$
$$614$$ 0 0
$$615$$ −4170.00 −0.273416
$$616$$ 0 0
$$617$$ 8934.00 0.582932 0.291466 0.956581i $$-0.405857\pi$$
0.291466 + 0.956581i $$0.405857\pi$$
$$618$$ 0 0
$$619$$ −20408.0 −1.32515 −0.662574 0.748996i $$-0.730536\pi$$
−0.662574 + 0.748996i $$0.730536\pi$$
$$620$$ 0 0
$$621$$ −1620.00 −0.104683
$$622$$ 0 0
$$623$$ −6000.00 −0.385851
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 192.000 0.0122293
$$628$$ 0 0
$$629$$ 428.000 0.0271311
$$630$$ 0 0
$$631$$ 24000.0 1.51414 0.757072 0.653331i $$-0.226629\pi$$
0.757072 + 0.653331i $$0.226629\pi$$
$$632$$ 0 0
$$633$$ −1680.00 −0.105488
$$634$$ 0 0
$$635$$ −4200.00 −0.262475
$$636$$ 0 0
$$637$$ 1674.00 0.104123
$$638$$ 0 0
$$639$$ 864.000 0.0534888
$$640$$ 0 0
$$641$$ 15810.0 0.974193 0.487096 0.873348i $$-0.338056\pi$$
0.487096 + 0.873348i $$0.338056\pi$$
$$642$$ 0 0
$$643$$ −7716.00 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 0 0
$$645$$ 1020.00 0.0622674
$$646$$ 0 0
$$647$$ 27324.0 1.66030 0.830152 0.557536i $$-0.188253\pi$$
0.830152 + 0.557536i $$0.188253\pi$$
$$648$$ 0 0
$$649$$ 2736.00 0.165481
$$650$$ 0 0
$$651$$ −4224.00 −0.254304
$$652$$ 0 0
$$653$$ 9450.00 0.566320 0.283160 0.959073i $$-0.408617\pi$$
0.283160 + 0.959073i $$0.408617\pi$$
$$654$$ 0 0
$$655$$ 7420.00 0.442631
$$656$$ 0 0
$$657$$ −3582.00 −0.212705
$$658$$ 0 0
$$659$$ −10804.0 −0.638640 −0.319320 0.947647i $$-0.603455\pi$$
−0.319320 + 0.947647i $$0.603455\pi$$
$$660$$ 0 0
$$661$$ 1534.00 0.0902658 0.0451329 0.998981i $$-0.485629\pi$$
0.0451329 + 0.998981i $$0.485629\pi$$
$$662$$ 0 0
$$663$$ −36.0000 −0.00210878
$$664$$ 0 0
$$665$$ −640.000 −0.0373205
$$666$$ 0 0
$$667$$ −8520.00 −0.494596
$$668$$ 0 0
$$669$$ 1824.00 0.105411
$$670$$ 0 0
$$671$$ 1576.00 0.0906718
$$672$$ 0 0
$$673$$ 1306.00 0.0748033 0.0374016 0.999300i $$-0.488092\pi$$
0.0374016 + 0.999300i $$0.488092\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −6398.00 −0.363213 −0.181606 0.983371i $$-0.558130\pi$$
−0.181606 + 0.983371i $$0.558130\pi$$
$$678$$ 0 0
$$679$$ 656.000 0.0370765
$$680$$ 0 0
$$681$$ 15372.0 0.864987
$$682$$ 0 0
$$683$$ 20396.0 1.14265 0.571326 0.820723i $$-0.306429\pi$$
0.571326 + 0.820723i $$0.306429\pi$$
$$684$$ 0 0
$$685$$ −5030.00 −0.280564
$$686$$ 0 0
$$687$$ −3570.00 −0.198259
$$688$$ 0 0
$$689$$ 2100.00 0.116116
$$690$$ 0 0
$$691$$ 11592.0 0.638177 0.319089 0.947725i $$-0.396623\pi$$
0.319089 + 0.947725i $$0.396623\pi$$
$$692$$ 0 0
$$693$$ −288.000 −0.0157867
$$694$$ 0 0
$$695$$ 7000.00 0.382051
$$696$$ 0 0
$$697$$ 556.000 0.0302152
$$698$$ 0 0
$$699$$ −11586.0 −0.626928
$$700$$ 0 0
$$701$$ 3618.00 0.194936 0.0974679 0.995239i $$-0.468926\pi$$
0.0974679 + 0.995239i $$0.468926\pi$$
$$702$$ 0 0
$$703$$ −3424.00 −0.183696
$$704$$ 0 0
$$705$$ −1740.00 −0.0929535
$$706$$ 0 0
$$707$$ 1232.00 0.0655362
$$708$$ 0 0
$$709$$ 30518.0 1.61654 0.808270 0.588811i $$-0.200404\pi$$
0.808270 + 0.588811i $$0.200404\pi$$
$$710$$ 0 0
$$711$$ −1224.00 −0.0645620
$$712$$ 0 0
$$713$$ 10560.0 0.554664
$$714$$ 0 0
$$715$$ −120.000 −0.00627657
$$716$$ 0 0
$$717$$ 4920.00 0.256263
$$718$$ 0 0
$$719$$ 13088.0 0.678860 0.339430 0.940631i $$-0.389766\pi$$
0.339430 + 0.940631i $$0.389766\pi$$
$$720$$ 0 0
$$721$$ 9728.00 0.502482
$$722$$ 0 0
$$723$$ 7002.00 0.360176
$$724$$ 0 0
$$725$$ −3550.00 −0.181853
$$726$$ 0 0
$$727$$ −6064.00 −0.309355 −0.154678 0.987965i $$-0.549434\pi$$
−0.154678 + 0.987965i $$0.549434\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −136.000 −0.00688118
$$732$$ 0 0
$$733$$ 3522.00 0.177473 0.0887367 0.996055i $$-0.471717\pi$$
0.0887367 + 0.996055i $$0.471717\pi$$
$$734$$ 0 0
$$735$$ −4185.00 −0.210022
$$736$$ 0 0
$$737$$ 432.000 0.0215915
$$738$$ 0 0
$$739$$ −39040.0 −1.94331 −0.971657 0.236394i $$-0.924035\pi$$
−0.971657 + 0.236394i $$0.924035\pi$$
$$740$$ 0 0
$$741$$ 288.000 0.0142779
$$742$$ 0 0
$$743$$ 14868.0 0.734124 0.367062 0.930197i $$-0.380364\pi$$
0.367062 + 0.930197i $$0.380364\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.000491774 0
$$746$$ 0 0
$$747$$ −3924.00 −0.192198
$$748$$ 0 0
$$749$$ −5856.00 −0.285679
$$750$$ 0 0
$$751$$ 4016.00 0.195134 0.0975672 0.995229i $$-0.468894\pi$$
0.0975672 + 0.995229i $$0.468894\pi$$
$$752$$ 0 0
$$753$$ −2316.00 −0.112085
$$754$$ 0 0
$$755$$ −11240.0 −0.541809
$$756$$ 0 0
$$757$$ −7958.00 −0.382085 −0.191043 0.981582i $$-0.561187\pi$$
−0.191043 + 0.981582i $$0.561187\pi$$
$$758$$ 0 0
$$759$$ 720.000 0.0344326
$$760$$ 0 0
$$761$$ 15890.0 0.756915 0.378457 0.925619i $$-0.376455\pi$$
0.378457 + 0.925619i $$0.376455\pi$$
$$762$$ 0 0
$$763$$ −4432.00 −0.210287
$$764$$ 0 0
$$765$$ 90.0000 0.00425354
$$766$$ 0 0
$$767$$ 4104.00 0.193203
$$768$$ 0 0
$$769$$ 6834.00 0.320469 0.160234 0.987079i $$-0.448775\pi$$
0.160234 + 0.987079i $$0.448775\pi$$
$$770$$ 0 0
$$771$$ −18642.0 −0.870785
$$772$$ 0 0
$$773$$ −3694.00 −0.171881 −0.0859405 0.996300i $$-0.527389\pi$$
−0.0859405 + 0.996300i $$0.527389\pi$$
$$774$$ 0 0
$$775$$ 4400.00 0.203939
$$776$$ 0 0
$$777$$ 5136.00 0.237134
$$778$$ 0 0
$$779$$ −4448.00 −0.204578
$$780$$ 0 0
$$781$$ −384.000 −0.0175936
$$782$$ 0 0
$$783$$ 3834.00 0.174988
$$784$$ 0 0
$$785$$ −14450.0 −0.656997
$$786$$ 0 0
$$787$$ −38284.0 −1.73402 −0.867012 0.498287i $$-0.833963\pi$$
−0.867012 + 0.498287i $$0.833963\pi$$
$$788$$ 0 0
$$789$$ 14844.0 0.669785
$$790$$ 0 0
$$791$$ −3792.00 −0.170453
$$792$$ 0 0
$$793$$ 2364.00 0.105861
$$794$$ 0 0
$$795$$ −5250.00 −0.234212
$$796$$ 0 0
$$797$$ 3258.00 0.144798 0.0723992 0.997376i $$-0.476934\pi$$
0.0723992 + 0.997376i $$0.476934\pi$$
$$798$$ 0 0
$$799$$ 232.000 0.0102723
$$800$$ 0 0
$$801$$ −6750.00 −0.297752
$$802$$ 0 0
$$803$$ 1592.00 0.0699632
$$804$$ 0 0
$$805$$ −2400.00 −0.105079
$$806$$ 0 0
$$807$$ 21114.0 0.921001
$$808$$ 0 0
$$809$$ 29754.0 1.29307 0.646536 0.762884i $$-0.276217\pi$$
0.646536 + 0.762884i $$0.276217\pi$$
$$810$$ 0 0
$$811$$ 42848.0 1.85524 0.927618 0.373530i $$-0.121853\pi$$
0.927618 + 0.373530i $$0.121853\pi$$
$$812$$ 0 0
$$813$$ −2184.00 −0.0942143
$$814$$ 0 0
$$815$$ −1260.00 −0.0541544
$$816$$ 0 0
$$817$$ 1088.00 0.0465903
$$818$$ 0 0
$$819$$ −432.000 −0.0184314
$$820$$ 0 0
$$821$$ −6454.00 −0.274356 −0.137178 0.990546i $$-0.543803\pi$$
−0.137178 + 0.990546i $$0.543803\pi$$
$$822$$ 0 0
$$823$$ −19128.0 −0.810158 −0.405079 0.914282i $$-0.632756\pi$$
−0.405079 + 0.914282i $$0.632756\pi$$
$$824$$ 0 0
$$825$$ 300.000 0.0126602
$$826$$ 0 0
$$827$$ −7060.00 −0.296856 −0.148428 0.988923i $$-0.547421\pi$$
−0.148428 + 0.988923i $$0.547421\pi$$
$$828$$ 0 0
$$829$$ −4666.00 −0.195485 −0.0977424 0.995212i $$-0.531162\pi$$
−0.0977424 + 0.995212i $$0.531162\pi$$
$$830$$ 0 0
$$831$$ −21822.0 −0.910947
$$832$$ 0 0
$$833$$ 558.000 0.0232095
$$834$$ 0 0
$$835$$ −4420.00 −0.183186
$$836$$ 0 0
$$837$$ −4752.00 −0.196240
$$838$$ 0 0
$$839$$ 33792.0 1.39050 0.695250 0.718768i $$-0.255294\pi$$
0.695250 + 0.718768i $$0.255294\pi$$
$$840$$ 0 0
$$841$$ −4225.00 −0.173234
$$842$$ 0 0
$$843$$ −5142.00 −0.210083
$$844$$ 0 0
$$845$$ 10805.0 0.439886
$$846$$ 0 0
$$847$$ −10520.0 −0.426767
$$848$$ 0 0
$$849$$ −12948.0 −0.523409
$$850$$ 0 0
$$851$$ −12840.0 −0.517214
$$852$$ 0 0
$$853$$ 23146.0 0.929078 0.464539 0.885553i $$-0.346220\pi$$
0.464539 + 0.885553i $$0.346220\pi$$
$$854$$ 0 0
$$855$$ −720.000 −0.0287994
$$856$$ 0 0
$$857$$ −40698.0 −1.62219 −0.811095 0.584914i $$-0.801128\pi$$
−0.811095 + 0.584914i $$0.801128\pi$$
$$858$$ 0 0
$$859$$ −40136.0 −1.59421 −0.797103 0.603844i $$-0.793635\pi$$
−0.797103 + 0.603844i $$0.793635\pi$$
$$860$$ 0 0
$$861$$ 6672.00 0.264090
$$862$$ 0 0
$$863$$ −1396.00 −0.0550642 −0.0275321 0.999621i $$-0.508765\pi$$
−0.0275321 + 0.999621i $$0.508765\pi$$
$$864$$ 0 0
$$865$$ 6990.00 0.274760
$$866$$ 0 0
$$867$$ 14727.0 0.576880
$$868$$ 0 0
$$869$$ 544.000 0.0212358
$$870$$ 0 0
$$871$$ 648.000 0.0252085
$$872$$ 0 0
$$873$$ 738.000 0.0286111
$$874$$ 0 0
$$875$$ −1000.00 −0.0386356
$$876$$ 0 0
$$877$$ −34646.0 −1.33399 −0.666997 0.745061i $$-0.732421\pi$$
−0.666997 + 0.745061i $$0.732421\pi$$
$$878$$ 0 0
$$879$$ 18234.0 0.699679
$$880$$ 0 0
$$881$$ −15302.0 −0.585173 −0.292587 0.956239i $$-0.594516\pi$$
−0.292587 + 0.956239i $$0.594516\pi$$
$$882$$ 0 0
$$883$$ 26300.0 1.00234 0.501170 0.865349i $$-0.332903\pi$$
0.501170 + 0.865349i $$0.332903\pi$$
$$884$$ 0 0
$$885$$ −10260.0 −0.389702
$$886$$ 0 0
$$887$$ 12052.0 0.456219 0.228110 0.973635i $$-0.426746\pi$$
0.228110 + 0.973635i $$0.426746\pi$$
$$888$$ 0 0
$$889$$ 6720.00 0.253523
$$890$$ 0 0
$$891$$ −324.000 −0.0121823
$$892$$ 0 0
$$893$$ −1856.00 −0.0695506
$$894$$ 0 0
$$895$$ 18060.0 0.674502
$$896$$ 0 0
$$897$$ 1080.00 0.0402008
$$898$$ 0 0
$$899$$ −24992.0 −0.927174
$$900$$ 0 0
$$901$$ 700.000 0.0258828
$$902$$ 0 0
$$903$$ −1632.00 −0.0601435
$$904$$ 0 0
$$905$$ −15750.0 −0.578506
$$906$$ 0 0
$$907$$ −18260.0 −0.668482 −0.334241 0.942488i $$-0.608480\pi$$
−0.334241 + 0.942488i $$0.608480\pi$$
$$908$$ 0 0
$$909$$ 1386.00 0.0505728
$$910$$ 0 0
$$911$$ 10280.0 0.373866 0.186933 0.982373i $$-0.440145\pi$$
0.186933 + 0.982373i $$0.440145\pi$$
$$912$$ 0 0
$$913$$ 1744.00 0.0632179
$$914$$ 0 0
$$915$$ −5910.00 −0.213528
$$916$$ 0 0
$$917$$ −11872.0 −0.427533
$$918$$ 0 0
$$919$$ 34504.0 1.23850 0.619250 0.785194i $$-0.287437\pi$$
0.619250 + 0.785194i $$0.287437\pi$$
$$920$$ 0 0
$$921$$ −4692.00 −0.167868
$$922$$ 0 0
$$923$$ −576.000 −0.0205409
$$924$$ 0 0
$$925$$ −5350.00 −0.190170
$$926$$ 0 0
$$927$$ 10944.0 0.387754
$$928$$ 0 0
$$929$$ 28266.0 0.998253 0.499127 0.866529i $$-0.333654\pi$$
0.499127 + 0.866529i $$0.333654\pi$$
$$930$$ 0 0
$$931$$ −4464.00 −0.157145
$$932$$ 0 0
$$933$$ 15648.0 0.549081
$$934$$ 0 0
$$935$$ −40.0000 −0.00139908
$$936$$ 0 0
$$937$$ −51494.0 −1.79534 −0.897671 0.440666i $$-0.854742\pi$$
−0.897671 + 0.440666i $$0.854742\pi$$
$$938$$ 0 0
$$939$$ 17370.0 0.603673
$$940$$ 0 0
$$941$$ 5338.00 0.184924 0.0924622 0.995716i $$-0.470526\pi$$
0.0924622 + 0.995716i $$0.470526\pi$$
$$942$$ 0 0
$$943$$ −16680.0 −0.576008
$$944$$ 0 0
$$945$$ 1080.00 0.0371771
$$946$$ 0 0
$$947$$ 47644.0 1.63487 0.817435 0.576021i $$-0.195395\pi$$
0.817435 + 0.576021i $$0.195395\pi$$
$$948$$ 0 0
$$949$$ 2388.00 0.0816836
$$950$$ 0 0
$$951$$ −11838.0 −0.403652
$$952$$ 0 0
$$953$$ −13906.0 −0.472675 −0.236338 0.971671i $$-0.575947\pi$$
−0.236338 + 0.971671i $$0.575947\pi$$
$$954$$ 0 0
$$955$$ 15200.0 0.515037
$$956$$ 0 0
$$957$$ −1704.00 −0.0575575
$$958$$ 0 0
$$959$$ 8048.00 0.270994
$$960$$ 0 0
$$961$$ 1185.00 0.0397771
$$962$$ 0 0
$$963$$ −6588.00 −0.220452
$$964$$ 0 0
$$965$$ −9970.00 −0.332586
$$966$$ 0 0
$$967$$ −8536.00 −0.283867 −0.141933 0.989876i $$-0.545332\pi$$
−0.141933 + 0.989876i $$0.545332\pi$$
$$968$$ 0 0
$$969$$ 96.0000 0.00318263
$$970$$ 0 0
$$971$$ −22572.0 −0.746004 −0.373002 0.927831i $$-0.621672\pi$$
−0.373002 + 0.927831i $$0.621672\pi$$
$$972$$ 0 0
$$973$$ −11200.0 −0.369019
$$974$$ 0 0
$$975$$ 450.000 0.0147811
$$976$$ 0 0
$$977$$ −51474.0 −1.68557 −0.842784 0.538253i $$-0.819085\pi$$
−0.842784 + 0.538253i $$0.819085\pi$$
$$978$$ 0 0
$$979$$ 3000.00 0.0979371
$$980$$ 0 0
$$981$$ −4986.00 −0.162274
$$982$$ 0 0
$$983$$ −6412.00 −0.208048 −0.104024 0.994575i $$-0.533172\pi$$
−0.104024 + 0.994575i $$0.533172\pi$$
$$984$$ 0 0
$$985$$ −20330.0 −0.657632
$$986$$ 0 0
$$987$$ 2784.00 0.0897829
$$988$$ 0 0
$$989$$ 4080.00 0.131179
$$990$$ 0 0
$$991$$ 54008.0 1.73120 0.865601 0.500735i $$-0.166937\pi$$
0.865601 + 0.500735i $$0.166937\pi$$
$$992$$ 0 0
$$993$$ −29976.0 −0.957965
$$994$$ 0 0
$$995$$ −19520.0 −0.621935
$$996$$ 0 0
$$997$$ −30846.0 −0.979842 −0.489921 0.871767i $$-0.662974\pi$$
−0.489921 + 0.871767i $$0.662974\pi$$
$$998$$ 0 0
$$999$$ 5778.00 0.182991
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.b.1.1 1
3.2 odd 2 1440.4.a.p.1.1 1
4.3 odd 2 480.4.a.g.1.1 yes 1
5.4 even 2 2400.4.a.q.1.1 1
8.3 odd 2 960.4.a.m.1.1 1
8.5 even 2 960.4.a.bh.1.1 1
12.11 even 2 1440.4.a.m.1.1 1
20.19 odd 2 2400.4.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.b.1.1 1 1.1 even 1 trivial
480.4.a.g.1.1 yes 1 4.3 odd 2
960.4.a.m.1.1 1 8.3 odd 2
960.4.a.bh.1.1 1 8.5 even 2
1440.4.a.m.1.1 1 12.11 even 2
1440.4.a.p.1.1 1 3.2 odd 2
2400.4.a.f.1.1 1 20.19 odd 2
2400.4.a.q.1.1 1 5.4 even 2