# Properties

 Label 2304.4.a.d.1.1 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ 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] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$135.940400653$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-12.0000 q^{5} +32.0000 q^{7} +O(q^{10})$$ $$q-12.0000 q^{5} +32.0000 q^{7} +8.00000 q^{11} -20.0000 q^{13} +98.0000 q^{17} -88.0000 q^{19} +32.0000 q^{23} +19.0000 q^{25} -172.000 q^{29} -256.000 q^{31} -384.000 q^{35} +92.0000 q^{37} -102.000 q^{41} -296.000 q^{43} +320.000 q^{47} +681.000 q^{49} -76.0000 q^{53} -96.0000 q^{55} -408.000 q^{59} +636.000 q^{61} +240.000 q^{65} +552.000 q^{67} -416.000 q^{71} +138.000 q^{73} +256.000 q^{77} -64.0000 q^{79} -392.000 q^{83} -1176.00 q^{85} +582.000 q^{89} -640.000 q^{91} +1056.00 q^{95} +238.000 q^{97} +O(q^{100})$$

## 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
$$5$$ −12.0000 −1.07331 −0.536656 0.843801i $$-0.680313\pi$$
−0.536656 + 0.843801i $$0.680313\pi$$
$$6$$ 0 0
$$7$$ 32.0000 1.72784 0.863919 0.503631i $$-0.168003\pi$$
0.863919 + 0.503631i $$0.168003\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 8.00000 0.219281 0.109640 0.993971i $$-0.465030\pi$$
0.109640 + 0.993971i $$0.465030\pi$$
$$12$$ 0 0
$$13$$ −20.0000 −0.426692 −0.213346 0.976977i $$-0.568436\pi$$
−0.213346 + 0.976977i $$0.568436\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 98.0000 1.39815 0.699073 0.715050i $$-0.253596\pi$$
0.699073 + 0.715050i $$0.253596\pi$$
$$18$$ 0 0
$$19$$ −88.0000 −1.06256 −0.531279 0.847197i $$-0.678288\pi$$
−0.531279 + 0.847197i $$0.678288\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 32.0000 0.290107 0.145054 0.989424i $$-0.453665\pi$$
0.145054 + 0.989424i $$0.453665\pi$$
$$24$$ 0 0
$$25$$ 19.0000 0.152000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −172.000 −1.10137 −0.550683 0.834715i $$-0.685633\pi$$
−0.550683 + 0.834715i $$0.685633\pi$$
$$30$$ 0 0
$$31$$ −256.000 −1.48319 −0.741596 0.670847i $$-0.765931\pi$$
−0.741596 + 0.670847i $$0.765931\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −384.000 −1.85451
$$36$$ 0 0
$$37$$ 92.0000 0.408776 0.204388 0.978890i $$-0.434480\pi$$
0.204388 + 0.978890i $$0.434480\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −102.000 −0.388530 −0.194265 0.980949i $$-0.562232\pi$$
−0.194265 + 0.980949i $$0.562232\pi$$
$$42$$ 0 0
$$43$$ −296.000 −1.04976 −0.524879 0.851177i $$-0.675889\pi$$
−0.524879 + 0.851177i $$0.675889\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 320.000 0.993123 0.496562 0.868001i $$-0.334596\pi$$
0.496562 + 0.868001i $$0.334596\pi$$
$$48$$ 0 0
$$49$$ 681.000 1.98542
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −76.0000 −0.196970 −0.0984849 0.995139i $$-0.531400\pi$$
−0.0984849 + 0.995139i $$0.531400\pi$$
$$54$$ 0 0
$$55$$ −96.0000 −0.235357
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −408.000 −0.900289 −0.450145 0.892956i $$-0.648628\pi$$
−0.450145 + 0.892956i $$0.648628\pi$$
$$60$$ 0 0
$$61$$ 636.000 1.33494 0.667471 0.744636i $$-0.267377\pi$$
0.667471 + 0.744636i $$0.267377\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 240.000 0.457974
$$66$$ 0 0
$$67$$ 552.000 1.00653 0.503265 0.864132i $$-0.332132\pi$$
0.503265 + 0.864132i $$0.332132\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −416.000 −0.695354 −0.347677 0.937614i $$-0.613029\pi$$
−0.347677 + 0.937614i $$0.613029\pi$$
$$72$$ 0 0
$$73$$ 138.000 0.221256 0.110628 0.993862i $$-0.464714\pi$$
0.110628 + 0.993862i $$0.464714\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 256.000 0.378882
$$78$$ 0 0
$$79$$ −64.0000 −0.0911464 −0.0455732 0.998961i $$-0.514511\pi$$
−0.0455732 + 0.998961i $$0.514511\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −392.000 −0.518405 −0.259202 0.965823i $$-0.583460\pi$$
−0.259202 + 0.965823i $$0.583460\pi$$
$$84$$ 0 0
$$85$$ −1176.00 −1.50065
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 582.000 0.693167 0.346584 0.938019i $$-0.387342\pi$$
0.346584 + 0.938019i $$0.387342\pi$$
$$90$$ 0 0
$$91$$ −640.000 −0.737255
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1056.00 1.14046
$$96$$ 0 0
$$97$$ 238.000 0.249126 0.124563 0.992212i $$-0.460247\pi$$
0.124563 + 0.992212i $$0.460247\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1468.00 −1.44625 −0.723126 0.690716i $$-0.757295\pi$$
−0.723126 + 0.690716i $$0.757295\pi$$
$$102$$ 0 0
$$103$$ −992.000 −0.948977 −0.474489 0.880262i $$-0.657367\pi$$
−0.474489 + 0.880262i $$0.657367\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 584.000 0.527639 0.263820 0.964572i $$-0.415018\pi$$
0.263820 + 0.964572i $$0.415018\pi$$
$$108$$ 0 0
$$109$$ −740.000 −0.650267 −0.325134 0.945668i $$-0.605409\pi$$
−0.325134 + 0.945668i $$0.605409\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 302.000 0.251414 0.125707 0.992067i $$-0.459880\pi$$
0.125707 + 0.992067i $$0.459880\pi$$
$$114$$ 0 0
$$115$$ −384.000 −0.311376
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3136.00 2.41577
$$120$$ 0 0
$$121$$ −1267.00 −0.951916
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1272.00 0.910169
$$126$$ 0 0
$$127$$ −1664.00 −1.16265 −0.581323 0.813673i $$-0.697465\pi$$
−0.581323 + 0.813673i $$0.697465\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2328.00 1.55266 0.776329 0.630327i $$-0.217079\pi$$
0.776329 + 0.630327i $$0.217079\pi$$
$$132$$ 0 0
$$133$$ −2816.00 −1.83593
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1734.00 −1.08135 −0.540677 0.841230i $$-0.681832\pi$$
−0.540677 + 0.841230i $$0.681832\pi$$
$$138$$ 0 0
$$139$$ 3032.00 1.85015 0.925075 0.379784i $$-0.124002\pi$$
0.925075 + 0.379784i $$0.124002\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −160.000 −0.0935655
$$144$$ 0 0
$$145$$ 2064.00 1.18211
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1788.00 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ 480.000 0.258688 0.129344 0.991600i $$-0.458713\pi$$
0.129344 + 0.991600i $$0.458713\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3072.00 1.59193
$$156$$ 0 0
$$157$$ 2300.00 1.16917 0.584586 0.811332i $$-0.301257\pi$$
0.584586 + 0.811332i $$0.301257\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1024.00 0.501258
$$162$$ 0 0
$$163$$ −1592.00 −0.765000 −0.382500 0.923955i $$-0.624937\pi$$
−0.382500 + 0.923955i $$0.624937\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2208.00 −1.02311 −0.511557 0.859249i $$-0.670931\pi$$
−0.511557 + 0.859249i $$0.670931\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −3948.00 −1.73503 −0.867517 0.497408i $$-0.834285\pi$$
−0.867517 + 0.497408i $$0.834285\pi$$
$$174$$ 0 0
$$175$$ 608.000 0.262631
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2104.00 0.878549 0.439275 0.898353i $$-0.355235\pi$$
0.439275 + 0.898353i $$0.355235\pi$$
$$180$$ 0 0
$$181$$ −1412.00 −0.579852 −0.289926 0.957049i $$-0.593631\pi$$
−0.289926 + 0.957049i $$0.593631\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1104.00 −0.438744
$$186$$ 0 0
$$187$$ 784.000 0.306587
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3712.00 −1.40624 −0.703118 0.711074i $$-0.748209\pi$$
−0.703118 + 0.711074i $$0.748209\pi$$
$$192$$ 0 0
$$193$$ 1614.00 0.601960 0.300980 0.953630i $$-0.402686\pi$$
0.300980 + 0.953630i $$0.402686\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −684.000 −0.247376 −0.123688 0.992321i $$-0.539472\pi$$
−0.123688 + 0.992321i $$0.539472\pi$$
$$198$$ 0 0
$$199$$ −4064.00 −1.44769 −0.723843 0.689965i $$-0.757626\pi$$
−0.723843 + 0.689965i $$0.757626\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −5504.00 −1.90298
$$204$$ 0 0
$$205$$ 1224.00 0.417014
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −704.000 −0.232999
$$210$$ 0 0
$$211$$ 2120.00 0.691691 0.345846 0.938291i $$-0.387592\pi$$
0.345846 + 0.938291i $$0.387592\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 3552.00 1.12672
$$216$$ 0 0
$$217$$ −8192.00 −2.56272
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1960.00 −0.596579
$$222$$ 0 0
$$223$$ 2816.00 0.845620 0.422810 0.906218i $$-0.361044\pi$$
0.422810 + 0.906218i $$0.361044\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −3848.00 −1.12511 −0.562557 0.826759i $$-0.690182\pi$$
−0.562557 + 0.826759i $$0.690182\pi$$
$$228$$ 0 0
$$229$$ 652.000 0.188146 0.0940729 0.995565i $$-0.470011\pi$$
0.0940729 + 0.995565i $$0.470011\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3050.00 −0.857563 −0.428781 0.903408i $$-0.641057\pi$$
−0.428781 + 0.903408i $$0.641057\pi$$
$$234$$ 0 0
$$235$$ −3840.00 −1.06593
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6336.00 1.71482 0.857410 0.514635i $$-0.172072\pi$$
0.857410 + 0.514635i $$0.172072\pi$$
$$240$$ 0 0
$$241$$ −4610.00 −1.23218 −0.616092 0.787674i $$-0.711285\pi$$
−0.616092 + 0.787674i $$0.711285\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −8172.00 −2.13098
$$246$$ 0 0
$$247$$ 1760.00 0.453385
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −792.000 −0.199166 −0.0995829 0.995029i $$-0.531751\pi$$
−0.0995829 + 0.995029i $$0.531751\pi$$
$$252$$ 0 0
$$253$$ 256.000 0.0636149
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5374.00 1.30436 0.652181 0.758063i $$-0.273854\pi$$
0.652181 + 0.758063i $$0.273854\pi$$
$$258$$ 0 0
$$259$$ 2944.00 0.706298
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3488.00 −0.817792 −0.408896 0.912581i $$-0.634086\pi$$
−0.408896 + 0.912581i $$0.634086\pi$$
$$264$$ 0 0
$$265$$ 912.000 0.211410
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −4764.00 −1.07980 −0.539900 0.841729i $$-0.681538\pi$$
−0.539900 + 0.841729i $$0.681538\pi$$
$$270$$ 0 0
$$271$$ 1344.00 0.301263 0.150631 0.988590i $$-0.451869\pi$$
0.150631 + 0.988590i $$0.451869\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 152.000 0.0333307
$$276$$ 0 0
$$277$$ −8596.00 −1.86456 −0.932281 0.361735i $$-0.882184\pi$$
−0.932281 + 0.361735i $$0.882184\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2874.00 −0.610137 −0.305068 0.952330i $$-0.598679\pi$$
−0.305068 + 0.952330i $$0.598679\pi$$
$$282$$ 0 0
$$283$$ −2888.00 −0.606621 −0.303311 0.952892i $$-0.598092\pi$$
−0.303311 + 0.952892i $$0.598092\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3264.00 −0.671316
$$288$$ 0 0
$$289$$ 4691.00 0.954814
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6540.00 −1.30400 −0.651998 0.758221i $$-0.726069\pi$$
−0.651998 + 0.758221i $$0.726069\pi$$
$$294$$ 0 0
$$295$$ 4896.00 0.966292
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −640.000 −0.123786
$$300$$ 0 0
$$301$$ −9472.00 −1.81381
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7632.00 −1.43281
$$306$$ 0 0
$$307$$ −10584.0 −1.96762 −0.983812 0.179202i $$-0.942649\pi$$
−0.983812 + 0.179202i $$0.942649\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6368.00 −1.16108 −0.580540 0.814231i $$-0.697159\pi$$
−0.580540 + 0.814231i $$0.697159\pi$$
$$312$$ 0 0
$$313$$ 8758.00 1.58157 0.790785 0.612094i $$-0.209673\pi$$
0.790785 + 0.612094i $$0.209673\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −716.000 −0.126860 −0.0634299 0.997986i $$-0.520204\pi$$
−0.0634299 + 0.997986i $$0.520204\pi$$
$$318$$ 0 0
$$319$$ −1376.00 −0.241508
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −8624.00 −1.48561
$$324$$ 0 0
$$325$$ −380.000 −0.0648573
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 10240.0 1.71596
$$330$$ 0 0
$$331$$ 4408.00 0.731981 0.365990 0.930619i $$-0.380730\pi$$
0.365990 + 0.930619i $$0.380730\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −6624.00 −1.08032
$$336$$ 0 0
$$337$$ 1202.00 0.194294 0.0971471 0.995270i $$-0.469028\pi$$
0.0971471 + 0.995270i $$0.469028\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2048.00 −0.325236
$$342$$ 0 0
$$343$$ 10816.0 1.70265
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5160.00 0.798280 0.399140 0.916890i $$-0.369309\pi$$
0.399140 + 0.916890i $$0.369309\pi$$
$$348$$ 0 0
$$349$$ 4876.00 0.747869 0.373935 0.927455i $$-0.378008\pi$$
0.373935 + 0.927455i $$0.378008\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −4834.00 −0.728861 −0.364430 0.931231i $$-0.618736\pi$$
−0.364430 + 0.931231i $$0.618736\pi$$
$$354$$ 0 0
$$355$$ 4992.00 0.746332
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4128.00 −0.606873 −0.303437 0.952852i $$-0.598134\pi$$
−0.303437 + 0.952852i $$0.598134\pi$$
$$360$$ 0 0
$$361$$ 885.000 0.129028
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1656.00 −0.237477
$$366$$ 0 0
$$367$$ −4416.00 −0.628102 −0.314051 0.949406i $$-0.601686\pi$$
−0.314051 + 0.949406i $$0.601686\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2432.00 −0.340332
$$372$$ 0 0
$$373$$ −4180.00 −0.580247 −0.290124 0.956989i $$-0.593696\pi$$
−0.290124 + 0.956989i $$0.593696\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3440.00 0.469944
$$378$$ 0 0
$$379$$ −13736.0 −1.86166 −0.930832 0.365446i $$-0.880916\pi$$
−0.930832 + 0.365446i $$0.880916\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −512.000 −0.0683080 −0.0341540 0.999417i $$-0.510874\pi$$
−0.0341540 + 0.999417i $$0.510874\pi$$
$$384$$ 0 0
$$385$$ −3072.00 −0.406659
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 1732.00 0.225748 0.112874 0.993609i $$-0.463994\pi$$
0.112874 + 0.993609i $$0.463994\pi$$
$$390$$ 0 0
$$391$$ 3136.00 0.405612
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 768.000 0.0978285
$$396$$ 0 0
$$397$$ −10436.0 −1.31931 −0.659657 0.751567i $$-0.729299\pi$$
−0.659657 + 0.751567i $$0.729299\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12130.0 1.51058 0.755291 0.655390i $$-0.227496\pi$$
0.755291 + 0.655390i $$0.227496\pi$$
$$402$$ 0 0
$$403$$ 5120.00 0.632867
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 736.000 0.0896368
$$408$$ 0 0
$$409$$ 5014.00 0.606177 0.303088 0.952962i $$-0.401982\pi$$
0.303088 + 0.952962i $$0.401982\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −13056.0 −1.55555
$$414$$ 0 0
$$415$$ 4704.00 0.556410
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 8024.00 0.935556 0.467778 0.883846i $$-0.345055\pi$$
0.467778 + 0.883846i $$0.345055\pi$$
$$420$$ 0 0
$$421$$ 2348.00 0.271816 0.135908 0.990721i $$-0.456605\pi$$
0.135908 + 0.990721i $$0.456605\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1862.00 0.212518
$$426$$ 0 0
$$427$$ 20352.0 2.30656
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1728.00 0.193120 0.0965601 0.995327i $$-0.469216\pi$$
0.0965601 + 0.995327i $$0.469216\pi$$
$$432$$ 0 0
$$433$$ 62.0000 0.00688113 0.00344057 0.999994i $$-0.498905\pi$$
0.00344057 + 0.999994i $$0.498905\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2816.00 −0.308255
$$438$$ 0 0
$$439$$ −14112.0 −1.53423 −0.767117 0.641507i $$-0.778310\pi$$
−0.767117 + 0.641507i $$0.778310\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4488.00 0.481335 0.240667 0.970608i $$-0.422634\pi$$
0.240667 + 0.970608i $$0.422634\pi$$
$$444$$ 0 0
$$445$$ −6984.00 −0.743985
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2482.00 0.260875 0.130437 0.991457i $$-0.458362\pi$$
0.130437 + 0.991457i $$0.458362\pi$$
$$450$$ 0 0
$$451$$ −816.000 −0.0851972
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 7680.00 0.791305
$$456$$ 0 0
$$457$$ 5894.00 0.603303 0.301652 0.953418i $$-0.402462\pi$$
0.301652 + 0.953418i $$0.402462\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −7068.00 −0.714077 −0.357039 0.934090i $$-0.616214\pi$$
−0.357039 + 0.934090i $$0.616214\pi$$
$$462$$ 0 0
$$463$$ 7616.00 0.764461 0.382231 0.924067i $$-0.375156\pi$$
0.382231 + 0.924067i $$0.375156\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 13080.0 1.29608 0.648041 0.761606i $$-0.275589\pi$$
0.648041 + 0.761606i $$0.275589\pi$$
$$468$$ 0 0
$$469$$ 17664.0 1.73912
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −2368.00 −0.230192
$$474$$ 0 0
$$475$$ −1672.00 −0.161509
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 13568.0 1.29423 0.647117 0.762391i $$-0.275975\pi$$
0.647117 + 0.762391i $$0.275975\pi$$
$$480$$ 0 0
$$481$$ −1840.00 −0.174422
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2856.00 −0.267390
$$486$$ 0 0
$$487$$ 1696.00 0.157809 0.0789046 0.996882i $$-0.474858\pi$$
0.0789046 + 0.996882i $$0.474858\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3096.00 −0.284563 −0.142282 0.989826i $$-0.545444\pi$$
−0.142282 + 0.989826i $$0.545444\pi$$
$$492$$ 0 0
$$493$$ −16856.0 −1.53987
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −13312.0 −1.20146
$$498$$ 0 0
$$499$$ 19208.0 1.72318 0.861591 0.507603i $$-0.169468\pi$$
0.861591 + 0.507603i $$0.169468\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −16224.0 −1.43816 −0.719078 0.694929i $$-0.755436\pi$$
−0.719078 + 0.694929i $$0.755436\pi$$
$$504$$ 0 0
$$505$$ 17616.0 1.55228
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −11292.0 −0.983318 −0.491659 0.870788i $$-0.663609\pi$$
−0.491659 + 0.870788i $$0.663609\pi$$
$$510$$ 0 0
$$511$$ 4416.00 0.382294
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 11904.0 1.01855
$$516$$ 0 0
$$517$$ 2560.00 0.217773
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5178.00 0.435417 0.217709 0.976014i $$-0.430142\pi$$
0.217709 + 0.976014i $$0.430142\pi$$
$$522$$ 0 0
$$523$$ −6856.00 −0.573216 −0.286608 0.958048i $$-0.592528\pi$$
−0.286608 + 0.958048i $$0.592528\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −25088.0 −2.07372
$$528$$ 0 0
$$529$$ −11143.0 −0.915838
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2040.00 0.165783
$$534$$ 0 0
$$535$$ −7008.00 −0.566322
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 5448.00 0.435365
$$540$$ 0 0
$$541$$ −13732.0 −1.09128 −0.545642 0.838018i $$-0.683714\pi$$
−0.545642 + 0.838018i $$0.683714\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 8880.00 0.697940
$$546$$ 0 0
$$547$$ −10968.0 −0.857327 −0.428663 0.903464i $$-0.641015\pi$$
−0.428663 + 0.903464i $$0.641015\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 15136.0 1.17026
$$552$$ 0 0
$$553$$ −2048.00 −0.157486
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −25612.0 −1.94832 −0.974161 0.225855i $$-0.927482\pi$$
−0.974161 + 0.225855i $$0.927482\pi$$
$$558$$ 0 0
$$559$$ 5920.00 0.447924
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −9768.00 −0.731212 −0.365606 0.930770i $$-0.619138\pi$$
−0.365606 + 0.930770i $$0.619138\pi$$
$$564$$ 0 0
$$565$$ −3624.00 −0.269846
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −22838.0 −1.68263 −0.841317 0.540542i $$-0.818219\pi$$
−0.841317 + 0.540542i $$0.818219\pi$$
$$570$$ 0 0
$$571$$ 9208.00 0.674856 0.337428 0.941351i $$-0.390443\pi$$
0.337428 + 0.941351i $$0.390443\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 608.000 0.0440963
$$576$$ 0 0
$$577$$ −10878.0 −0.784848 −0.392424 0.919785i $$-0.628363\pi$$
−0.392424 + 0.919785i $$0.628363\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12544.0 −0.895719
$$582$$ 0 0
$$583$$ −608.000 −0.0431917
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −18040.0 −1.26847 −0.634234 0.773141i $$-0.718684\pi$$
−0.634234 + 0.773141i $$0.718684\pi$$
$$588$$ 0 0
$$589$$ 22528.0 1.57598
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −26994.0 −1.86933 −0.934663 0.355534i $$-0.884299\pi$$
−0.934663 + 0.355534i $$0.884299\pi$$
$$594$$ 0 0
$$595$$ −37632.0 −2.59288
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 18336.0 1.25073 0.625366 0.780331i $$-0.284950\pi$$
0.625366 + 0.780331i $$0.284950\pi$$
$$600$$ 0 0
$$601$$ −9286.00 −0.630256 −0.315128 0.949049i $$-0.602047\pi$$
−0.315128 + 0.949049i $$0.602047\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 15204.0 1.02170
$$606$$ 0 0
$$607$$ −17536.0 −1.17259 −0.586297 0.810096i $$-0.699415\pi$$
−0.586297 + 0.810096i $$0.699415\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6400.00 −0.423758
$$612$$ 0 0
$$613$$ 5868.00 0.386633 0.193317 0.981136i $$-0.438076\pi$$
0.193317 + 0.981136i $$0.438076\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 19286.0 1.25839 0.629194 0.777248i $$-0.283385\pi$$
0.629194 + 0.777248i $$0.283385\pi$$
$$618$$ 0 0
$$619$$ 5240.00 0.340248 0.170124 0.985423i $$-0.445583\pi$$
0.170124 + 0.985423i $$0.445583\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 18624.0 1.19768
$$624$$ 0 0
$$625$$ −17639.0 −1.12890
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 9016.00 0.571529
$$630$$ 0 0
$$631$$ −15520.0 −0.979147 −0.489573 0.871962i $$-0.662847\pi$$
−0.489573 + 0.871962i $$0.662847\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 19968.0 1.24788
$$636$$ 0 0
$$637$$ −13620.0 −0.847165
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −654.000 −0.0402987 −0.0201493 0.999797i $$-0.506414\pi$$
−0.0201493 + 0.999797i $$0.506414\pi$$
$$642$$ 0 0
$$643$$ 8232.00 0.504881 0.252440 0.967612i $$-0.418767\pi$$
0.252440 + 0.967612i $$0.418767\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24672.0 1.49916 0.749580 0.661914i $$-0.230256\pi$$
0.749580 + 0.661914i $$0.230256\pi$$
$$648$$ 0 0
$$649$$ −3264.00 −0.197416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 22052.0 1.32153 0.660767 0.750591i $$-0.270231\pi$$
0.660767 + 0.750591i $$0.270231\pi$$
$$654$$ 0 0
$$655$$ −27936.0 −1.66649
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 12024.0 0.710757 0.355378 0.934723i $$-0.384352\pi$$
0.355378 + 0.934723i $$0.384352\pi$$
$$660$$ 0 0
$$661$$ 19100.0 1.12391 0.561955 0.827168i $$-0.310050\pi$$
0.561955 + 0.827168i $$0.310050\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 33792.0 1.97052
$$666$$ 0 0
$$667$$ −5504.00 −0.319514
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 5088.00 0.292727
$$672$$ 0 0
$$673$$ 9902.00 0.567153 0.283577 0.958950i $$-0.408479\pi$$
0.283577 + 0.958950i $$0.408479\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 19684.0 1.11746 0.558728 0.829351i $$-0.311290\pi$$
0.558728 + 0.829351i $$0.311290\pi$$
$$678$$ 0 0
$$679$$ 7616.00 0.430450
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −19864.0 −1.11285 −0.556424 0.830899i $$-0.687827\pi$$
−0.556424 + 0.830899i $$0.687827\pi$$
$$684$$ 0 0
$$685$$ 20808.0 1.16063
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1520.00 0.0840456
$$690$$ 0 0
$$691$$ −3256.00 −0.179253 −0.0896267 0.995975i $$-0.528567\pi$$
−0.0896267 + 0.995975i $$0.528567\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −36384.0 −1.98579
$$696$$ 0 0
$$697$$ −9996.00 −0.543222
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2876.00 −0.154957 −0.0774786 0.996994i $$-0.524687\pi$$
−0.0774786 + 0.996994i $$0.524687\pi$$
$$702$$ 0 0
$$703$$ −8096.00 −0.434348
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −46976.0 −2.49889
$$708$$ 0 0
$$709$$ −7300.00 −0.386682 −0.193341 0.981132i $$-0.561932\pi$$
−0.193341 + 0.981132i $$0.561932\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −8192.00 −0.430284
$$714$$ 0 0
$$715$$ 1920.00 0.100425
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2880.00 0.149382 0.0746912 0.997207i $$-0.476203\pi$$
0.0746912 + 0.997207i $$0.476203\pi$$
$$720$$ 0 0
$$721$$ −31744.0 −1.63968
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −3268.00 −0.167408
$$726$$ 0 0
$$727$$ 8800.00 0.448933 0.224466 0.974482i $$-0.427936\pi$$
0.224466 + 0.974482i $$0.427936\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −29008.0 −1.46771
$$732$$ 0 0
$$733$$ −21076.0 −1.06202 −0.531009 0.847366i $$-0.678187\pi$$
−0.531009 + 0.847366i $$0.678187\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4416.00 0.220713
$$738$$ 0 0
$$739$$ 19336.0 0.962498 0.481249 0.876584i $$-0.340183\pi$$
0.481249 + 0.876584i $$0.340183\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 13664.0 0.674675 0.337338 0.941384i $$-0.390474\pi$$
0.337338 + 0.941384i $$0.390474\pi$$
$$744$$ 0 0
$$745$$ 21456.0 1.05515
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 18688.0 0.911675
$$750$$ 0 0
$$751$$ 19520.0 0.948462 0.474231 0.880400i $$-0.342726\pi$$
0.474231 + 0.880400i $$0.342726\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −5760.00 −0.277653
$$756$$ 0 0
$$757$$ −20004.0 −0.960446 −0.480223 0.877146i $$-0.659444\pi$$
−0.480223 + 0.877146i $$0.659444\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −31478.0 −1.49944 −0.749722 0.661753i $$-0.769813\pi$$
−0.749722 + 0.661753i $$0.769813\pi$$
$$762$$ 0 0
$$763$$ −23680.0 −1.12356
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8160.00 0.384147
$$768$$ 0 0
$$769$$ 7054.00 0.330785 0.165393 0.986228i $$-0.447111\pi$$
0.165393 + 0.986228i $$0.447111\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 9604.00 0.446872 0.223436 0.974719i $$-0.428273\pi$$
0.223436 + 0.974719i $$0.428273\pi$$
$$774$$ 0 0
$$775$$ −4864.00 −0.225445
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 8976.00 0.412835
$$780$$ 0 0
$$781$$ −3328.00 −0.152478
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −27600.0 −1.25489
$$786$$ 0 0
$$787$$ 3144.00 0.142403 0.0712017 0.997462i $$-0.477317\pi$$
0.0712017 + 0.997462i $$0.477317\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9664.00 0.434402
$$792$$ 0 0
$$793$$ −12720.0 −0.569610
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 22084.0 0.981500 0.490750 0.871300i $$-0.336723\pi$$
0.490750 + 0.871300i $$0.336723\pi$$
$$798$$ 0 0
$$799$$ 31360.0 1.38853
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 1104.00 0.0485172
$$804$$ 0 0
$$805$$ −12288.0 −0.538006
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −14950.0 −0.649708 −0.324854 0.945764i $$-0.605315\pi$$
−0.324854 + 0.945764i $$0.605315\pi$$
$$810$$ 0 0
$$811$$ −23432.0 −1.01456 −0.507280 0.861781i $$-0.669349\pi$$
−0.507280 + 0.861781i $$0.669349\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 19104.0 0.821085
$$816$$ 0 0
$$817$$ 26048.0 1.11543
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1044.00 0.0443798 0.0221899 0.999754i $$-0.492936\pi$$
0.0221899 + 0.999754i $$0.492936\pi$$
$$822$$ 0 0
$$823$$ −18208.0 −0.771192 −0.385596 0.922668i $$-0.626004\pi$$
−0.385596 + 0.922668i $$0.626004\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12488.0 0.525091 0.262546 0.964920i $$-0.415438\pi$$
0.262546 + 0.964920i $$0.415438\pi$$
$$828$$ 0 0
$$829$$ 30172.0 1.26407 0.632037 0.774938i $$-0.282219\pi$$
0.632037 + 0.774938i $$0.282219\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 66738.0 2.77591
$$834$$ 0 0
$$835$$ 26496.0 1.09812
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −32544.0 −1.33915 −0.669573 0.742746i $$-0.733523\pi$$
−0.669573 + 0.742746i $$0.733523\pi$$
$$840$$ 0 0
$$841$$ 5195.00 0.213006
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 21564.0 0.877898
$$846$$ 0 0
$$847$$ −40544.0 −1.64476
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2944.00 0.118589
$$852$$ 0 0
$$853$$ 26156.0 1.04990 0.524950 0.851133i $$-0.324084\pi$$
0.524950 + 0.851133i $$0.324084\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −18646.0 −0.743215 −0.371607 0.928390i $$-0.621193\pi$$
−0.371607 + 0.928390i $$0.621193\pi$$
$$858$$ 0 0
$$859$$ −5800.00 −0.230377 −0.115188 0.993344i $$-0.536747\pi$$
−0.115188 + 0.993344i $$0.536747\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 25088.0 0.989578 0.494789 0.869013i $$-0.335245\pi$$
0.494789 + 0.869013i $$0.335245\pi$$
$$864$$ 0 0
$$865$$ 47376.0 1.86223
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −512.000 −0.0199867
$$870$$ 0 0
$$871$$ −11040.0 −0.429479
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 40704.0 1.57262
$$876$$ 0 0
$$877$$ 3004.00 0.115665 0.0578323 0.998326i $$-0.481581\pi$$
0.0578323 + 0.998326i $$0.481581\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −43282.0 −1.65517 −0.827587 0.561338i $$-0.810287\pi$$
−0.827587 + 0.561338i $$0.810287\pi$$
$$882$$ 0 0
$$883$$ 27880.0 1.06256 0.531278 0.847198i $$-0.321712\pi$$
0.531278 + 0.847198i $$0.321712\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −7392.00 −0.279819 −0.139909 0.990164i $$-0.544681\pi$$
−0.139909 + 0.990164i $$0.544681\pi$$
$$888$$ 0 0
$$889$$ −53248.0 −2.00886
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −28160.0 −1.05525
$$894$$ 0 0
$$895$$ −25248.0 −0.942958
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 44032.0 1.63354
$$900$$ 0 0
$$901$$ −7448.00 −0.275393
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 16944.0 0.622362
$$906$$ 0 0
$$907$$ 29080.0 1.06459 0.532296 0.846558i $$-0.321329\pi$$
0.532296 + 0.846558i $$0.321329\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −26688.0 −0.970596 −0.485298 0.874349i $$-0.661289\pi$$
−0.485298 + 0.874349i $$0.661289\pi$$
$$912$$ 0 0
$$913$$ −3136.00 −0.113676
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 74496.0 2.68274
$$918$$ 0 0
$$919$$ 19680.0 0.706402 0.353201 0.935547i $$-0.385093\pi$$
0.353201 + 0.935547i $$0.385093\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 8320.00 0.296702
$$924$$ 0 0
$$925$$ 1748.00 0.0621339
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 48466.0 1.71164 0.855822 0.517270i $$-0.173052\pi$$
0.855822 + 0.517270i $$0.173052\pi$$
$$930$$ 0 0
$$931$$ −59928.0 −2.10962
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −9408.00 −0.329064
$$936$$ 0 0
$$937$$ 13610.0 0.474514 0.237257 0.971447i $$-0.423752\pi$$
0.237257 + 0.971447i $$0.423752\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 30692.0 1.06326 0.531632 0.846976i $$-0.321579\pi$$
0.531632 + 0.846976i $$0.321579\pi$$
$$942$$ 0 0
$$943$$ −3264.00 −0.112715
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4824.00 0.165532 0.0827661 0.996569i $$-0.473625\pi$$
0.0827661 + 0.996569i $$0.473625\pi$$
$$948$$ 0 0
$$949$$ −2760.00 −0.0944082
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 22986.0 0.781311 0.390656 0.920537i $$-0.372248\pi$$
0.390656 + 0.920537i $$0.372248\pi$$
$$954$$ 0 0
$$955$$ 44544.0 1.50933
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −55488.0 −1.86841
$$960$$ 0 0
$$961$$ 35745.0 1.19986
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −19368.0 −0.646091
$$966$$ 0 0
$$967$$ 17184.0 0.571458 0.285729 0.958310i $$-0.407764\pi$$
0.285729 + 0.958310i $$0.407764\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2920.00 0.0965059 0.0482530 0.998835i $$-0.484635\pi$$
0.0482530 + 0.998835i $$0.484635\pi$$
$$972$$ 0 0
$$973$$ 97024.0 3.19676
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 27042.0 0.885517 0.442759 0.896641i $$-0.354000\pi$$
0.442759 + 0.896641i $$0.354000\pi$$
$$978$$ 0 0
$$979$$ 4656.00 0.151998
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 44192.0 1.43388 0.716941 0.697134i $$-0.245542\pi$$
0.716941 + 0.697134i $$0.245542\pi$$
$$984$$ 0 0
$$985$$ 8208.00 0.265511
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9472.00 −0.304542
$$990$$ 0 0
$$991$$ 29824.0 0.955995 0.477997 0.878361i $$-0.341363\pi$$
0.477997 + 0.878361i $$0.341363\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 48768.0 1.55382
$$996$$ 0 0
$$997$$ 11612.0 0.368862 0.184431 0.982845i $$-0.440956\pi$$
0.184431 + 0.982845i $$0.440956\pi$$
$$998$$ 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 2304.4.a.d.1.1 1
3.2 odd 2 256.4.a.g.1.1 1
4.3 odd 2 2304.4.a.c.1.1 1
8.3 odd 2 2304.4.a.m.1.1 1
8.5 even 2 2304.4.a.n.1.1 1
12.11 even 2 256.4.a.c.1.1 1
16.3 odd 4 1152.4.d.h.577.2 2
16.5 even 4 1152.4.d.a.577.1 2
16.11 odd 4 1152.4.d.h.577.1 2
16.13 even 4 1152.4.d.a.577.2 2
24.5 odd 2 256.4.a.b.1.1 1
24.11 even 2 256.4.a.f.1.1 1
48.5 odd 4 128.4.b.a.65.1 2
48.11 even 4 128.4.b.d.65.2 yes 2
48.29 odd 4 128.4.b.a.65.2 yes 2
48.35 even 4 128.4.b.d.65.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.a.65.1 2 48.5 odd 4
128.4.b.a.65.2 yes 2 48.29 odd 4
128.4.b.d.65.1 yes 2 48.35 even 4
128.4.b.d.65.2 yes 2 48.11 even 4
256.4.a.b.1.1 1 24.5 odd 2
256.4.a.c.1.1 1 12.11 even 2
256.4.a.f.1.1 1 24.11 even 2
256.4.a.g.1.1 1 3.2 odd 2
1152.4.d.a.577.1 2 16.5 even 4
1152.4.d.a.577.2 2 16.13 even 4
1152.4.d.h.577.1 2 16.11 odd 4
1152.4.d.h.577.2 2 16.3 odd 4
2304.4.a.c.1.1 1 4.3 odd 2
2304.4.a.d.1.1 1 1.1 even 1 trivial
2304.4.a.m.1.1 1 8.3 odd 2
2304.4.a.n.1.1 1 8.5 even 2