# Properties

 Label 1440.4.a.p.1.1 Level $1440$ Weight $4$ Character 1440.1 Self dual yes Analytic conductor $84.963$ 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] = [1440,4,Mod(1,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, 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("1440.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1440.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: $$84.9627504083$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1440.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} +8.00000 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} +8.00000 q^{7} +4.00000 q^{11} -6.00000 q^{13} +2.00000 q^{17} +16.0000 q^{19} -60.0000 q^{23} +25.0000 q^{25} +142.000 q^{29} +176.000 q^{31} +40.0000 q^{35} -214.000 q^{37} +278.000 q^{41} +68.0000 q^{43} +116.000 q^{47} -279.000 q^{49} +350.000 q^{53} +20.0000 q^{55} +684.000 q^{59} -394.000 q^{61} -30.0000 q^{65} -108.000 q^{67} -96.0000 q^{71} -398.000 q^{73} +32.0000 q^{77} -136.000 q^{79} +436.000 q^{83} +10.0000 q^{85} +750.000 q^{89} -48.0000 q^{91} +80.0000 q^{95} +82.0000 q^{97} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 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
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000 0.109640 0.0548202 0.998496i $$-0.482541\pi$$
0.0548202 + 0.998496i $$0.482541\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −0.128008 −0.0640039 0.997950i $$-0.520387\pi$$
−0.0640039 + 0.997950i $$0.520387\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000 0.0285336 0.0142668 0.999898i $$-0.495459\pi$$
0.0142668 + 0.999898i $$0.495459\pi$$
$$18$$ 0 0
$$19$$ 16.0000 0.193192 0.0965961 0.995324i $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −60.0000 −0.543951 −0.271975 0.962304i $$-0.587677\pi$$
−0.271975 + 0.962304i $$0.587677\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 142.000 0.909267 0.454633 0.890679i $$-0.349770\pi$$
0.454633 + 0.890679i $$0.349770\pi$$
$$30$$ 0 0
$$31$$ 176.000 1.01969 0.509847 0.860265i $$-0.329702\pi$$
0.509847 + 0.860265i $$0.329702\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 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
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 278.000 1.05893 0.529467 0.848330i $$-0.322392\pi$$
0.529467 + 0.848330i $$0.322392\pi$$
$$42$$ 0 0
$$43$$ 68.0000 0.241161 0.120580 0.992704i $$-0.461524\pi$$
0.120580 + 0.992704i $$0.461524\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 116.000 0.360007 0.180004 0.983666i $$-0.442389\pi$$
0.180004 + 0.983666i $$0.442389\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 350.000 0.907098 0.453549 0.891231i $$-0.350158\pi$$
0.453549 + 0.891231i $$0.350158\pi$$
$$54$$ 0 0
$$55$$ 20.0000 0.0490327
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 684.000 1.50931 0.754654 0.656123i $$-0.227805\pi$$
0.754654 + 0.656123i $$0.227805\pi$$
$$60$$ 0 0
$$61$$ −394.000 −0.826992 −0.413496 0.910506i $$-0.635692\pi$$
−0.413496 + 0.910506i $$0.635692\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −96.0000 −0.160466 −0.0802331 0.996776i $$-0.525566\pi$$
−0.0802331 + 0.996776i $$0.525566\pi$$
$$72$$ 0 0
$$73$$ −398.000 −0.638115 −0.319057 0.947735i $$-0.603366\pi$$
−0.319057 + 0.947735i $$0.603366\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 436.000 0.576593 0.288296 0.957541i $$-0.406911\pi$$
0.288296 + 0.957541i $$0.406911\pi$$
$$84$$ 0 0
$$85$$ 10.0000 0.0127606
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 750.000 0.893257 0.446628 0.894720i $$-0.352625\pi$$
0.446628 + 0.894720i $$0.352625\pi$$
$$90$$ 0 0
$$91$$ −48.0000 −0.0552941
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −154.000 −0.151719 −0.0758593 0.997119i $$-0.524170\pi$$
−0.0758593 + 0.997119i $$0.524170\pi$$
$$102$$ 0 0
$$103$$ 1216.00 1.16326 0.581631 0.813453i $$-0.302415\pi$$
0.581631 + 0.813453i $$0.302415\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 732.000 0.661356 0.330678 0.943744i $$-0.392723\pi$$
0.330678 + 0.943744i $$0.392723\pi$$
$$108$$ 0 0
$$109$$ −554.000 −0.486822 −0.243411 0.969923i $$-0.578266\pi$$
−0.243411 + 0.969923i $$0.578266\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 474.000 0.394603 0.197302 0.980343i $$-0.436782\pi$$
0.197302 + 0.980343i $$0.436782\pi$$
$$114$$ 0 0
$$115$$ −300.000 −0.243262
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 16.0000 0.0123254
$$120$$ 0 0
$$121$$ −1315.00 −0.987979
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1484.00 0.989753 0.494877 0.868963i $$-0.335213\pi$$
0.494877 + 0.868963i $$0.335213\pi$$
$$132$$ 0 0
$$133$$ 128.000 0.0834512
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1006.00 −0.627360 −0.313680 0.949529i $$-0.601562\pi$$
−0.313680 + 0.949529i $$0.601562\pi$$
$$138$$ 0 0
$$139$$ −1400.00 −0.854291 −0.427146 0.904183i $$-0.640481\pi$$
−0.427146 + 0.904183i $$0.640481\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −24.0000 −0.0140348
$$144$$ 0 0
$$145$$ 710.000 0.406636
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2.00000 −0.00109964 −0.000549820 1.00000i $$-0.500175\pi$$
−0.000549820 1.00000i $$0.500175\pi$$
$$150$$ 0 0
$$151$$ 2248.00 1.21152 0.605760 0.795647i $$-0.292869\pi$$
0.605760 + 0.795647i $$0.292869\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 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
$$159$$ 0 0
$$160$$ 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
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −884.000 −0.409617 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$168$$ 0 0
$$169$$ −2161.00 −0.983614
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1398.00 0.614381 0.307191 0.951648i $$-0.400611\pi$$
0.307191 + 0.951648i $$0.400611\pi$$
$$174$$ 0 0
$$175$$ 200.000 0.0863919
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3612.00 1.50823 0.754116 0.656741i $$-0.228066\pi$$
0.754116 + 0.656741i $$0.228066\pi$$
$$180$$ 0 0
$$181$$ 3150.00 1.29358 0.646789 0.762669i $$-0.276111\pi$$
0.646789 + 0.762669i $$0.276111\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1070.00 −0.425232
$$186$$ 0 0
$$187$$ 8.00000 0.00312844
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3040.00 1.15166 0.575829 0.817570i $$-0.304679\pi$$
0.575829 + 0.817570i $$0.304679\pi$$
$$192$$ 0 0
$$193$$ 1994.00 0.743685 0.371843 0.928296i $$-0.378726\pi$$
0.371843 + 0.928296i $$0.378726\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4066.00 −1.47051 −0.735255 0.677790i $$-0.762938\pi$$
−0.735255 + 0.677790i $$0.762938\pi$$
$$198$$ 0 0
$$199$$ 3904.00 1.39069 0.695345 0.718676i $$-0.255252\pi$$
0.695345 + 0.718676i $$0.255252\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1136.00 0.392766
$$204$$ 0 0
$$205$$ 1390.00 0.473570
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 340.000 0.107850
$$216$$ 0 0
$$217$$ 1408.00 0.440467
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5124.00 1.49820 0.749101 0.662456i $$-0.230486\pi$$
0.749101 + 0.662456i $$0.230486\pi$$
$$228$$ 0 0
$$229$$ 1190.00 0.343395 0.171697 0.985150i $$-0.445075\pi$$
0.171697 + 0.985150i $$0.445075\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3862.00 −1.08587 −0.542936 0.839774i $$-0.682687\pi$$
−0.542936 + 0.839774i $$0.682687\pi$$
$$234$$ 0 0
$$235$$ 580.000 0.161000
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1640.00 0.443861 0.221931 0.975062i $$-0.428764\pi$$
0.221931 + 0.975062i $$0.428764\pi$$
$$240$$ 0 0
$$241$$ −2334.00 −0.623843 −0.311921 0.950108i $$-0.600973\pi$$
−0.311921 + 0.950108i $$0.600973\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1395.00 −0.363768
$$246$$ 0 0
$$247$$ −96.0000 −0.0247301
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −772.000 −0.194136 −0.0970681 0.995278i $$-0.530946\pi$$
−0.0970681 + 0.995278i $$0.530946\pi$$
$$252$$ 0 0
$$253$$ −240.000 −0.0596390
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6214.00 −1.50824 −0.754122 0.656734i $$-0.771937\pi$$
−0.754122 + 0.656734i $$0.771937\pi$$
$$258$$ 0 0
$$259$$ −1712.00 −0.410728
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4948.00 1.16010 0.580051 0.814580i $$-0.303033\pi$$
0.580051 + 0.814580i $$0.303033\pi$$
$$264$$ 0 0
$$265$$ 1750.00 0.405667
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 7038.00 1.59522 0.797610 0.603173i $$-0.206097\pi$$
0.797610 + 0.603173i $$0.206097\pi$$
$$270$$ 0 0
$$271$$ 728.000 0.163184 0.0815920 0.996666i $$-0.474000\pi$$
0.0815920 + 0.996666i $$0.474000\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1714.00 −0.363874 −0.181937 0.983310i $$-0.558237\pi$$
−0.181937 + 0.983310i $$0.558237\pi$$
$$282$$ 0 0
$$283$$ 4316.00 0.906571 0.453285 0.891365i $$-0.350252\pi$$
0.453285 + 0.891365i $$0.350252\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2224.00 0.457417
$$288$$ 0 0
$$289$$ −4909.00 −0.999186
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 6078.00 1.21188 0.605940 0.795511i $$-0.292797\pi$$
0.605940 + 0.795511i $$0.292797\pi$$
$$294$$ 0 0
$$295$$ 3420.00 0.674983
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 360.000 0.0696299
$$300$$ 0 0
$$301$$ 544.000 0.104172
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 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
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5216.00 0.951036 0.475518 0.879706i $$-0.342261\pi$$
0.475518 + 0.879706i $$0.342261\pi$$
$$312$$ 0 0
$$313$$ −5790.00 −1.04559 −0.522796 0.852458i $$-0.675111\pi$$
−0.522796 + 0.852458i $$0.675111\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3946.00 −0.699146 −0.349573 0.936909i $$-0.613673\pi$$
−0.349573 + 0.936909i $$0.613673\pi$$
$$318$$ 0 0
$$319$$ 568.000 0.0996925
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 32.0000 0.00551247
$$324$$ 0 0
$$325$$ −150.000 −0.0256015
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 704.000 0.111800
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −5580.00 −0.863257 −0.431628 0.902052i $$-0.642061\pi$$
−0.431628 + 0.902052i $$0.642061\pi$$
$$348$$ 0 0
$$349$$ −4154.00 −0.637130 −0.318565 0.947901i $$-0.603201\pi$$
−0.318565 + 0.947901i $$0.603201\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2802.00 0.422480 0.211240 0.977434i $$-0.432250\pi$$
0.211240 + 0.977434i $$0.432250\pi$$
$$354$$ 0 0
$$355$$ −480.000 −0.0717627
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11296.0 −1.66067 −0.830334 0.557265i $$-0.811851\pi$$
−0.830334 + 0.557265i $$0.811851\pi$$
$$360$$ 0 0
$$361$$ −6603.00 −0.962677
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 7916.00 1.05611 0.528053 0.849211i $$-0.322922\pi$$
0.528053 + 0.849211i $$0.322922\pi$$
$$384$$ 0 0
$$385$$ 160.000 0.0211801
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 9950.00 1.29688 0.648438 0.761267i $$-0.275422\pi$$
0.648438 + 0.761267i $$0.275422\pi$$
$$390$$ 0 0
$$391$$ −120.000 −0.0155209
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 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
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6390.00 0.795764 0.397882 0.917437i $$-0.369745\pi$$
0.397882 + 0.917437i $$0.369745\pi$$
$$402$$ 0 0
$$403$$ −1056.00 −0.130529
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 5472.00 0.651960
$$414$$ 0 0
$$415$$ 2180.00 0.257860
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 2100.00 0.244849 0.122424 0.992478i $$-0.460933\pi$$
0.122424 + 0.992478i $$0.460933\pi$$
$$420$$ 0 0
$$421$$ 3478.00 0.402630 0.201315 0.979527i $$-0.435478\pi$$
0.201315 + 0.979527i $$0.435478\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 50.0000 0.00570672
$$426$$ 0 0
$$427$$ −3152.00 −0.357227
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8504.00 0.950402 0.475201 0.879877i $$-0.342375\pi$$
0.475201 + 0.879877i $$0.342375\pi$$
$$432$$ 0 0
$$433$$ −16102.0 −1.78710 −0.893548 0.448967i $$-0.851792\pi$$
−0.893548 + 0.448967i $$0.851792\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −11076.0 −1.18789 −0.593947 0.804505i $$-0.702431\pi$$
−0.593947 + 0.804505i $$0.702431\pi$$
$$444$$ 0 0
$$445$$ 3750.00 0.399477
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9910.00 1.04161 0.520804 0.853676i $$-0.325632\pi$$
0.520804 + 0.853676i $$0.325632\pi$$
$$450$$ 0 0
$$451$$ 1112.00 0.116102
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 8406.00 0.849255 0.424627 0.905368i $$-0.360405\pi$$
0.424627 + 0.905368i $$0.360405\pi$$
$$462$$ 0 0
$$463$$ −18040.0 −1.81078 −0.905389 0.424584i $$-0.860420\pi$$
−0.905389 + 0.424584i $$0.860420\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −8484.00 −0.840670 −0.420335 0.907369i $$-0.638087\pi$$
−0.420335 + 0.907369i $$0.638087\pi$$
$$468$$ 0 0
$$469$$ −864.000 −0.0850657
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 272.000 0.0264410
$$474$$ 0 0
$$475$$ 400.000 0.0386384
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −2136.00 −0.203750 −0.101875 0.994797i $$-0.532484\pi$$
−0.101875 + 0.994797i $$0.532484\pi$$
$$480$$ 0 0
$$481$$ 1284.00 0.121716
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −11812.0 −1.08568 −0.542839 0.839837i $$-0.682651\pi$$
−0.542839 + 0.839837i $$0.682651\pi$$
$$492$$ 0 0
$$493$$ 284.000 0.0259447
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1300.00 −0.115237 −0.0576184 0.998339i $$-0.518351\pi$$
−0.0576184 + 0.998339i $$0.518351\pi$$
$$504$$ 0 0
$$505$$ −770.000 −0.0678506
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 4398.00 0.382982 0.191491 0.981494i $$-0.438668\pi$$
0.191491 + 0.981494i $$0.438668\pi$$
$$510$$ 0 0
$$511$$ −3184.00 −0.275640
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 6080.00 0.520227
$$516$$ 0 0
$$517$$ 464.000 0.0394714
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1650.00 −0.138748 −0.0693741 0.997591i $$-0.522100\pi$$
−0.0693741 + 0.997591i $$0.522100\pi$$
$$522$$ 0 0
$$523$$ 17276.0 1.44441 0.722205 0.691679i $$-0.243129\pi$$
0.722205 + 0.691679i $$0.243129\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 352.000 0.0290956
$$528$$ 0 0
$$529$$ −8567.00 −0.704118
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1668.00 −0.135552
$$534$$ 0 0
$$535$$ 3660.00 0.295767
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2272.00 0.175663
$$552$$ 0 0
$$553$$ −1088.00 −0.0836645
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −13546.0 −1.03045 −0.515227 0.857054i $$-0.672292\pi$$
−0.515227 + 0.857054i $$0.672292\pi$$
$$558$$ 0 0
$$559$$ −408.000 −0.0308704
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −16884.0 −1.26390 −0.631950 0.775009i $$-0.717745\pi$$
−0.631950 + 0.775009i $$0.717745\pi$$
$$564$$ 0 0
$$565$$ 2370.00 0.176472
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −13746.0 −1.01276 −0.506382 0.862309i $$-0.669017\pi$$
−0.506382 + 0.862309i $$0.669017\pi$$
$$570$$ 0 0
$$571$$ 15176.0 1.11225 0.556126 0.831098i $$-0.312287\pi$$
0.556126 + 0.831098i $$0.312287\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3488.00 0.249065
$$582$$ 0 0
$$583$$ 1400.00 0.0994547
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −17332.0 −1.21869 −0.609343 0.792907i $$-0.708567\pi$$
−0.609343 + 0.792907i $$0.708567\pi$$
$$588$$ 0 0
$$589$$ 2816.00 0.196997
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1290.00 0.0893321 0.0446661 0.999002i $$-0.485778\pi$$
0.0446661 + 0.999002i $$0.485778\pi$$
$$594$$ 0 0
$$595$$ 80.0000 0.00551207
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −20544.0 −1.40134 −0.700672 0.713484i $$-0.747116\pi$$
−0.700672 + 0.713484i $$0.747116\pi$$
$$600$$ 0 0
$$601$$ −2630.00 −0.178502 −0.0892512 0.996009i $$-0.528447\pi$$
−0.0892512 + 0.996009i $$0.528447\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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
$$609$$ 0 0
$$610$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −8934.00 −0.582932 −0.291466 0.956581i $$-0.594143\pi$$
−0.291466 + 0.956581i $$0.594143\pi$$
$$618$$ 0 0
$$619$$ −20408.0 −1.32515 −0.662574 0.748996i $$-0.730536\pi$$
−0.662574 + 0.748996i $$0.730536\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 6000.00 0.385851
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 4200.00 0.262475
$$636$$ 0 0
$$637$$ 1674.00 0.104123
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −15810.0 −0.974193 −0.487096 0.873348i $$-0.661944\pi$$
−0.487096 + 0.873348i $$0.661944\pi$$
$$642$$ 0 0
$$643$$ −7716.00 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −27324.0 −1.66030 −0.830152 0.557536i $$-0.811747\pi$$
−0.830152 + 0.557536i $$0.811747\pi$$
$$648$$ 0 0
$$649$$ 2736.00 0.165481
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −9450.00 −0.566320 −0.283160 0.959073i $$-0.591383\pi$$
−0.283160 + 0.959073i $$0.591383\pi$$
$$654$$ 0 0
$$655$$ 7420.00 0.442631
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 10804.0 0.638640 0.319320 0.947647i $$-0.396545\pi$$
0.319320 + 0.947647i $$0.396545\pi$$
$$660$$ 0 0
$$661$$ 1534.00 0.0902658 0.0451329 0.998981i $$-0.485629\pi$$
0.0451329 + 0.998981i $$0.485629\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 640.000 0.0373205
$$666$$ 0 0
$$667$$ −8520.00 −0.494596
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6398.00 0.363213 0.181606 0.983371i $$-0.441870\pi$$
0.181606 + 0.983371i $$0.441870\pi$$
$$678$$ 0 0
$$679$$ 656.000 0.0370765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −20396.0 −1.14265 −0.571326 0.820723i $$-0.693571\pi$$
−0.571326 + 0.820723i $$0.693571\pi$$
$$684$$ 0 0
$$685$$ −5030.00 −0.280564
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −7000.00 −0.382051
$$696$$ 0 0
$$697$$ 556.000 0.0302152
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3618.00 −0.194936 −0.0974679 0.995239i $$-0.531074\pi$$
−0.0974679 + 0.995239i $$0.531074\pi$$
$$702$$ 0 0
$$703$$ −3424.00 −0.183696
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −10560.0 −0.554664
$$714$$ 0 0
$$715$$ −120.000 −0.00627657
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −13088.0 −0.678860 −0.339430 0.940631i $$-0.610234\pi$$
−0.339430 + 0.940631i $$0.610234\pi$$
$$720$$ 0 0
$$721$$ 9728.00 0.502482
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 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
$$735$$ 0 0
$$736$$ 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
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −14868.0 −0.734124 −0.367062 0.930197i $$-0.619636\pi$$
−0.367062 + 0.930197i $$0.619636\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.000491774 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −15890.0 −0.756915 −0.378457 0.925619i $$-0.623545\pi$$
−0.378457 + 0.925619i $$0.623545\pi$$
$$762$$ 0 0
$$763$$ −4432.00 −0.210287
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 3694.00 0.171881 0.0859405 0.996300i $$-0.472611\pi$$
0.0859405 + 0.996300i $$0.472611\pi$$
$$774$$ 0 0
$$775$$ 4400.00 0.203939
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4448.00 0.204578
$$780$$ 0 0
$$781$$ −384.000 −0.0175936
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 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
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3792.00 0.170453
$$792$$ 0 0
$$793$$ 2364.00 0.105861
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −3258.00 −0.144798 −0.0723992 0.997376i $$-0.523066\pi$$
−0.0723992 + 0.997376i $$0.523066\pi$$
$$798$$ 0 0
$$799$$ 232.000 0.0102723
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1592.00 −0.0699632
$$804$$ 0 0
$$805$$ −2400.00 −0.105079
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −29754.0 −1.29307 −0.646536 0.762884i $$-0.723783\pi$$
−0.646536 + 0.762884i $$0.723783\pi$$
$$810$$ 0 0
$$811$$ 42848.0 1.85524 0.927618 0.373530i $$-0.121853\pi$$
0.927618 + 0.373530i $$0.121853\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 1260.00 0.0541544
$$816$$ 0 0
$$817$$ 1088.00 0.0465903
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6454.00 0.274356 0.137178 0.990546i $$-0.456197\pi$$
0.137178 + 0.990546i $$0.456197\pi$$
$$822$$ 0 0
$$823$$ −19128.0 −0.810158 −0.405079 0.914282i $$-0.632756\pi$$
−0.405079 + 0.914282i $$0.632756\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7060.00 0.296856 0.148428 0.988923i $$-0.452579\pi$$
0.148428 + 0.988923i $$0.452579\pi$$
$$828$$ 0 0
$$829$$ −4666.00 −0.195485 −0.0977424 0.995212i $$-0.531162\pi$$
−0.0977424 + 0.995212i $$0.531162\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −558.000 −0.0232095
$$834$$ 0 0
$$835$$ −4420.00 −0.183186
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −33792.0 −1.39050 −0.695250 0.718768i $$-0.744706\pi$$
−0.695250 + 0.718768i $$0.744706\pi$$
$$840$$ 0 0
$$841$$ −4225.00 −0.173234
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −10805.0 −0.439886
$$846$$ 0 0
$$847$$ −10520.0 −0.426767
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 40698.0 1.62219 0.811095 0.584914i $$-0.198872\pi$$
0.811095 + 0.584914i $$0.198872\pi$$
$$858$$ 0 0
$$859$$ −40136.0 −1.59421 −0.797103 0.603844i $$-0.793635\pi$$
−0.797103 + 0.603844i $$0.793635\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1396.00 0.0550642 0.0275321 0.999621i $$-0.491235\pi$$
0.0275321 + 0.999621i $$0.491235\pi$$
$$864$$ 0 0
$$865$$ 6990.00 0.274760
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −544.000 −0.0212358
$$870$$ 0 0
$$871$$ 648.000 0.0252085
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 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
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 15302.0 0.585173 0.292587 0.956239i $$-0.405484\pi$$
0.292587 + 0.956239i $$0.405484\pi$$
$$882$$ 0 0
$$883$$ 26300.0 1.00234 0.501170 0.865349i $$-0.332903\pi$$
0.501170 + 0.865349i $$0.332903\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −12052.0 −0.456219 −0.228110 0.973635i $$-0.573254\pi$$
−0.228110 + 0.973635i $$0.573254\pi$$
$$888$$ 0 0
$$889$$ 6720.00 0.253523
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1856.00 0.0695506
$$894$$ 0 0
$$895$$ 18060.0 0.674502
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 24992.0 0.927174
$$900$$ 0 0
$$901$$ 700.000 0.0258828
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10280.0 −0.373866 −0.186933 0.982373i $$-0.559855\pi$$
−0.186933 + 0.982373i $$0.559855\pi$$
$$912$$ 0 0
$$913$$ 1744.00 0.0632179
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 576.000 0.0205409
$$924$$ 0 0
$$925$$ −5350.00 −0.190170
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −28266.0 −0.998253 −0.499127 0.866529i $$-0.666346\pi$$
−0.499127 + 0.866529i $$0.666346\pi$$
$$930$$ 0 0
$$931$$ −4464.00 −0.157145
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −5338.00 −0.184924 −0.0924622 0.995716i $$-0.529474\pi$$
−0.0924622 + 0.995716i $$0.529474\pi$$
$$942$$ 0 0
$$943$$ −16680.0 −0.576008
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −47644.0 −1.63487 −0.817435 0.576021i $$-0.804605\pi$$
−0.817435 + 0.576021i $$0.804605\pi$$
$$948$$ 0 0
$$949$$ 2388.00 0.0816836
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 13906.0 0.472675 0.236338 0.971671i $$-0.424053\pi$$
0.236338 + 0.971671i $$0.424053\pi$$
$$954$$ 0 0
$$955$$ 15200.0 0.515037
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −8048.00 −0.270994
$$960$$ 0 0
$$961$$ 1185.00 0.0397771
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22572.0 0.746004 0.373002 0.927831i $$-0.378328\pi$$
0.373002 + 0.927831i $$0.378328\pi$$
$$972$$ 0 0
$$973$$ −11200.0 −0.369019
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 51474.0 1.68557 0.842784 0.538253i $$-0.180915\pi$$
0.842784 + 0.538253i $$0.180915\pi$$
$$978$$ 0 0
$$979$$ 3000.00 0.0979371
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 6412.00 0.208048 0.104024 0.994575i $$-0.466828\pi$$
0.104024 + 0.994575i $$0.466828\pi$$
$$984$$ 0 0
$$985$$ −20330.0 −0.657632
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 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
$$999$$ 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 1440.4.a.p.1.1 1
3.2 odd 2 480.4.a.b.1.1 1
4.3 odd 2 1440.4.a.m.1.1 1
12.11 even 2 480.4.a.g.1.1 yes 1
15.14 odd 2 2400.4.a.q.1.1 1
24.5 odd 2 960.4.a.bh.1.1 1
24.11 even 2 960.4.a.m.1.1 1
60.59 even 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 3.2 odd 2
480.4.a.g.1.1 yes 1 12.11 even 2
960.4.a.m.1.1 1 24.11 even 2
960.4.a.bh.1.1 1 24.5 odd 2
1440.4.a.m.1.1 1 4.3 odd 2
1440.4.a.p.1.1 1 1.1 even 1 trivial
2400.4.a.f.1.1 1 60.59 even 2
2400.4.a.q.1.1 1 15.14 odd 2