# Properties

 Label 108.4.a.a.1.1 Level $108$ Weight $4$ Character 108.1 Self dual yes Analytic conductor $6.372$ 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] = [108,4,Mod(1,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("108.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.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: $$6.37220628062$$ 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 $$\chi$$ $$=$$ 108.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-9.00000 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q-9.00000 q^{5} -1.00000 q^{7} -63.0000 q^{11} -28.0000 q^{13} -72.0000 q^{17} +98.0000 q^{19} -126.000 q^{23} -44.0000 q^{25} +126.000 q^{29} -259.000 q^{31} +9.00000 q^{35} +386.000 q^{37} +450.000 q^{41} -34.0000 q^{43} +54.0000 q^{47} -342.000 q^{49} +693.000 q^{53} +567.000 q^{55} -180.000 q^{59} -280.000 q^{61} +252.000 q^{65} -586.000 q^{67} -504.000 q^{71} +161.000 q^{73} +63.0000 q^{77} +440.000 q^{79} -999.000 q^{83} +648.000 q^{85} -882.000 q^{89} +28.0000 q^{91} -882.000 q^{95} -721.000 q^{97} +O(q^{100})$$

## 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
$$5$$ −9.00000 −0.804984 −0.402492 0.915423i $$-0.631856\pi$$
−0.402492 + 0.915423i $$0.631856\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.0539949 −0.0269975 0.999636i $$-0.508595\pi$$
−0.0269975 + 0.999636i $$0.508595\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −63.0000 −1.72684 −0.863419 0.504488i $$-0.831681\pi$$
−0.863419 + 0.504488i $$0.831681\pi$$
$$12$$ 0 0
$$13$$ −28.0000 −0.597369 −0.298685 0.954352i $$-0.596548\pi$$
−0.298685 + 0.954352i $$0.596548\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −72.0000 −1.02721 −0.513605 0.858027i $$-0.671690\pi$$
−0.513605 + 0.858027i $$0.671690\pi$$
$$18$$ 0 0
$$19$$ 98.0000 1.18330 0.591651 0.806194i $$-0.298476\pi$$
0.591651 + 0.806194i $$0.298476\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −126.000 −1.14230 −0.571148 0.820847i $$-0.693502\pi$$
−0.571148 + 0.820847i $$0.693502\pi$$
$$24$$ 0 0
$$25$$ −44.0000 −0.352000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 126.000 0.806814 0.403407 0.915021i $$-0.367826\pi$$
0.403407 + 0.915021i $$0.367826\pi$$
$$30$$ 0 0
$$31$$ −259.000 −1.50057 −0.750287 0.661113i $$-0.770085\pi$$
−0.750287 + 0.661113i $$0.770085\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 9.00000 0.0434651
$$36$$ 0 0
$$37$$ 386.000 1.71508 0.857541 0.514416i $$-0.171991\pi$$
0.857541 + 0.514416i $$0.171991\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 450.000 1.71410 0.857051 0.515231i $$-0.172294\pi$$
0.857051 + 0.515231i $$0.172294\pi$$
$$42$$ 0 0
$$43$$ −34.0000 −0.120580 −0.0602901 0.998181i $$-0.519203\pi$$
−0.0602901 + 0.998181i $$0.519203\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 54.0000 0.167590 0.0837948 0.996483i $$-0.473296\pi$$
0.0837948 + 0.996483i $$0.473296\pi$$
$$48$$ 0 0
$$49$$ −342.000 −0.997085
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 693.000 1.79605 0.898027 0.439940i $$-0.145000\pi$$
0.898027 + 0.439940i $$0.145000\pi$$
$$54$$ 0 0
$$55$$ 567.000 1.39008
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −180.000 −0.397187 −0.198593 0.980082i $$-0.563637\pi$$
−0.198593 + 0.980082i $$0.563637\pi$$
$$60$$ 0 0
$$61$$ −280.000 −0.587710 −0.293855 0.955850i $$-0.594938\pi$$
−0.293855 + 0.955850i $$0.594938\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 252.000 0.480873
$$66$$ 0 0
$$67$$ −586.000 −1.06853 −0.534263 0.845318i $$-0.679411\pi$$
−0.534263 + 0.845318i $$0.679411\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −504.000 −0.842448 −0.421224 0.906957i $$-0.638399\pi$$
−0.421224 + 0.906957i $$0.638399\pi$$
$$72$$ 0 0
$$73$$ 161.000 0.258132 0.129066 0.991636i $$-0.458802\pi$$
0.129066 + 0.991636i $$0.458802\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 63.0000 0.0932405
$$78$$ 0 0
$$79$$ 440.000 0.626631 0.313316 0.949649i $$-0.398560\pi$$
0.313316 + 0.949649i $$0.398560\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −999.000 −1.32114 −0.660569 0.750765i $$-0.729685\pi$$
−0.660569 + 0.750765i $$0.729685\pi$$
$$84$$ 0 0
$$85$$ 648.000 0.826888
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −882.000 −1.05047 −0.525235 0.850957i $$-0.676023\pi$$
−0.525235 + 0.850957i $$0.676023\pi$$
$$90$$ 0 0
$$91$$ 28.0000 0.0322549
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −882.000 −0.952540
$$96$$ 0 0
$$97$$ −721.000 −0.754706 −0.377353 0.926070i $$-0.623166\pi$$
−0.377353 + 0.926070i $$0.623166\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 441.000 0.434467 0.217233 0.976120i $$-0.430297\pi$$
0.217233 + 0.976120i $$0.430297\pi$$
$$102$$ 0 0
$$103$$ −532.000 −0.508927 −0.254464 0.967082i $$-0.581899\pi$$
−0.254464 + 0.967082i $$0.581899\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −819.000 −0.739960 −0.369980 0.929040i $$-0.620635\pi$$
−0.369980 + 0.929040i $$0.620635\pi$$
$$108$$ 0 0
$$109$$ −1294.00 −1.13709 −0.568545 0.822652i $$-0.692493\pi$$
−0.568545 + 0.822652i $$0.692493\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1134.00 −0.944051 −0.472025 0.881585i $$-0.656477\pi$$
−0.472025 + 0.881585i $$0.656477\pi$$
$$114$$ 0 0
$$115$$ 1134.00 0.919531
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 72.0000 0.0554641
$$120$$ 0 0
$$121$$ 2638.00 1.98197
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1521.00 1.08834
$$126$$ 0 0
$$127$$ −1807.00 −1.26256 −0.631281 0.775554i $$-0.717470\pi$$
−0.631281 + 0.775554i $$0.717470\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2205.00 1.47062 0.735312 0.677729i $$-0.237036\pi$$
0.735312 + 0.677729i $$0.237036\pi$$
$$132$$ 0 0
$$133$$ −98.0000 −0.0638923
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1386.00 −0.864336 −0.432168 0.901793i $$-0.642251\pi$$
−0.432168 + 0.901793i $$0.642251\pi$$
$$138$$ 0 0
$$139$$ 476.000 0.290459 0.145229 0.989398i $$-0.453608\pi$$
0.145229 + 0.989398i $$0.453608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1764.00 1.03156
$$144$$ 0 0
$$145$$ −1134.00 −0.649473
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1575.00 0.865967 0.432983 0.901402i $$-0.357461\pi$$
0.432983 + 0.901402i $$0.357461\pi$$
$$150$$ 0 0
$$151$$ 449.000 0.241981 0.120990 0.992654i $$-0.461393\pi$$
0.120990 + 0.992654i $$0.461393\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 2331.00 1.20794
$$156$$ 0 0
$$157$$ 1820.00 0.925171 0.462585 0.886575i $$-0.346922\pi$$
0.462585 + 0.886575i $$0.346922\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 126.000 0.0616782
$$162$$ 0 0
$$163$$ −1828.00 −0.878405 −0.439202 0.898388i $$-0.644739\pi$$
−0.439202 + 0.898388i $$0.644739\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 810.000 0.375327 0.187664 0.982233i $$-0.439908\pi$$
0.187664 + 0.982233i $$0.439908\pi$$
$$168$$ 0 0
$$169$$ −1413.00 −0.643150
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1323.00 0.581421 0.290710 0.956811i $$-0.406108\pi$$
0.290710 + 0.956811i $$0.406108\pi$$
$$174$$ 0 0
$$175$$ 44.0000 0.0190062
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 315.000 0.131532 0.0657659 0.997835i $$-0.479051\pi$$
0.0657659 + 0.997835i $$0.479051\pi$$
$$180$$ 0 0
$$181$$ −2800.00 −1.14985 −0.574924 0.818207i $$-0.694968\pi$$
−0.574924 + 0.818207i $$0.694968\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −3474.00 −1.38061
$$186$$ 0 0
$$187$$ 4536.00 1.77382
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3276.00 1.24106 0.620532 0.784182i $$-0.286917\pi$$
0.620532 + 0.784182i $$0.286917\pi$$
$$192$$ 0 0
$$193$$ 3221.00 1.20131 0.600655 0.799509i $$-0.294907\pi$$
0.600655 + 0.799509i $$0.294907\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3339.00 −1.20758 −0.603792 0.797142i $$-0.706344\pi$$
−0.603792 + 0.797142i $$0.706344\pi$$
$$198$$ 0 0
$$199$$ 3689.00 1.31410 0.657051 0.753846i $$-0.271804\pi$$
0.657051 + 0.753846i $$0.271804\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −126.000 −0.0435639
$$204$$ 0 0
$$205$$ −4050.00 −1.37983
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6174.00 −2.04337
$$210$$ 0 0
$$211$$ −6022.00 −1.96479 −0.982397 0.186805i $$-0.940187\pi$$
−0.982397 + 0.186805i $$0.940187\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 306.000 0.0970652
$$216$$ 0 0
$$217$$ 259.000 0.0810233
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2016.00 0.613624
$$222$$ 0 0
$$223$$ −952.000 −0.285877 −0.142939 0.989732i $$-0.545655\pi$$
−0.142939 + 0.989732i $$0.545655\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5292.00 −1.54732 −0.773662 0.633599i $$-0.781577\pi$$
−0.773662 + 0.633599i $$0.781577\pi$$
$$228$$ 0 0
$$229$$ 2198.00 0.634270 0.317135 0.948380i $$-0.397279\pi$$
0.317135 + 0.948380i $$0.397279\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −5166.00 −1.45251 −0.726257 0.687423i $$-0.758742\pi$$
−0.726257 + 0.687423i $$0.758742\pi$$
$$234$$ 0 0
$$235$$ −486.000 −0.134907
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −3402.00 −0.920741 −0.460370 0.887727i $$-0.652283\pi$$
−0.460370 + 0.887727i $$0.652283\pi$$
$$240$$ 0 0
$$241$$ 1862.00 0.497684 0.248842 0.968544i $$-0.419950\pi$$
0.248842 + 0.968544i $$0.419950\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3078.00 0.802638
$$246$$ 0 0
$$247$$ −2744.00 −0.706869
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5472.00 1.37605 0.688027 0.725685i $$-0.258477\pi$$
0.688027 + 0.725685i $$0.258477\pi$$
$$252$$ 0 0
$$253$$ 7938.00 1.97256
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −5292.00 −1.28446 −0.642229 0.766513i $$-0.721990\pi$$
−0.642229 + 0.766513i $$0.721990\pi$$
$$258$$ 0 0
$$259$$ −386.000 −0.0926057
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −1638.00 −0.384043 −0.192022 0.981391i $$-0.561504\pi$$
−0.192022 + 0.981391i $$0.561504\pi$$
$$264$$ 0 0
$$265$$ −6237.00 −1.44580
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 1206.00 0.273350 0.136675 0.990616i $$-0.456358\pi$$
0.136675 + 0.990616i $$0.456358\pi$$
$$270$$ 0 0
$$271$$ 4319.00 0.968120 0.484060 0.875035i $$-0.339162\pi$$
0.484060 + 0.875035i $$0.339162\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2772.00 0.607847
$$276$$ 0 0
$$277$$ −2248.00 −0.487615 −0.243807 0.969824i $$-0.578396\pi$$
−0.243807 + 0.969824i $$0.578396\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2016.00 0.427987 0.213994 0.976835i $$-0.431353\pi$$
0.213994 + 0.976835i $$0.431353\pi$$
$$282$$ 0 0
$$283$$ −2338.00 −0.491094 −0.245547 0.969385i $$-0.578968\pi$$
−0.245547 + 0.969385i $$0.578968\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −450.000 −0.0925528
$$288$$ 0 0
$$289$$ 271.000 0.0551598
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 8514.00 1.69759 0.848794 0.528724i $$-0.177329\pi$$
0.848794 + 0.528724i $$0.177329\pi$$
$$294$$ 0 0
$$295$$ 1620.00 0.319729
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3528.00 0.682373
$$300$$ 0 0
$$301$$ 34.0000 0.00651072
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2520.00 0.473098
$$306$$ 0 0
$$307$$ 6104.00 1.13477 0.567384 0.823453i $$-0.307956\pi$$
0.567384 + 0.823453i $$0.307956\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4338.00 0.790950 0.395475 0.918477i $$-0.370580\pi$$
0.395475 + 0.918477i $$0.370580\pi$$
$$312$$ 0 0
$$313$$ −8155.00 −1.47268 −0.736338 0.676613i $$-0.763447\pi$$
−0.736338 + 0.676613i $$0.763447\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4977.00 0.881818 0.440909 0.897552i $$-0.354656\pi$$
0.440909 + 0.897552i $$0.354656\pi$$
$$318$$ 0 0
$$319$$ −7938.00 −1.39324
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −7056.00 −1.21550
$$324$$ 0 0
$$325$$ 1232.00 0.210274
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −54.0000 −0.00904899
$$330$$ 0 0
$$331$$ 5678.00 0.942873 0.471437 0.881900i $$-0.343736\pi$$
0.471437 + 0.881900i $$0.343736\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 5274.00 0.860147
$$336$$ 0 0
$$337$$ 2906.00 0.469733 0.234866 0.972028i $$-0.424535\pi$$
0.234866 + 0.972028i $$0.424535\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16317.0 2.59125
$$342$$ 0 0
$$343$$ 685.000 0.107832
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −6993.00 −1.08186 −0.540928 0.841069i $$-0.681927\pi$$
−0.540928 + 0.841069i $$0.681927\pi$$
$$348$$ 0 0
$$349$$ 7910.00 1.21322 0.606608 0.795001i $$-0.292530\pi$$
0.606608 + 0.795001i $$0.292530\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2466.00 −0.371819 −0.185909 0.982567i $$-0.559523\pi$$
−0.185909 + 0.982567i $$0.559523\pi$$
$$354$$ 0 0
$$355$$ 4536.00 0.678157
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −7182.00 −1.05585 −0.527927 0.849290i $$-0.677030\pi$$
−0.527927 + 0.849290i $$0.677030\pi$$
$$360$$ 0 0
$$361$$ 2745.00 0.400204
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1449.00 −0.207792
$$366$$ 0 0
$$367$$ −11431.0 −1.62587 −0.812934 0.582356i $$-0.802131\pi$$
−0.812934 + 0.582356i $$0.802131\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −693.000 −0.0969778
$$372$$ 0 0
$$373$$ −6616.00 −0.918401 −0.459200 0.888333i $$-0.651864\pi$$
−0.459200 + 0.888333i $$0.651864\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3528.00 −0.481966
$$378$$ 0 0
$$379$$ −9820.00 −1.33092 −0.665461 0.746433i $$-0.731765\pi$$
−0.665461 + 0.746433i $$0.731765\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1440.00 −0.192116 −0.0960582 0.995376i $$-0.530623\pi$$
−0.0960582 + 0.995376i $$0.530623\pi$$
$$384$$ 0 0
$$385$$ −567.000 −0.0750571
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −5985.00 −0.780081 −0.390041 0.920798i $$-0.627539\pi$$
−0.390041 + 0.920798i $$0.627539\pi$$
$$390$$ 0 0
$$391$$ 9072.00 1.17338
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −3960.00 −0.504428
$$396$$ 0 0
$$397$$ −11284.0 −1.42652 −0.713259 0.700900i $$-0.752782\pi$$
−0.713259 + 0.700900i $$0.752782\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7308.00 0.910085 0.455043 0.890470i $$-0.349624\pi$$
0.455043 + 0.890470i $$0.349624\pi$$
$$402$$ 0 0
$$403$$ 7252.00 0.896397
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24318.0 −2.96167
$$408$$ 0 0
$$409$$ 6335.00 0.765882 0.382941 0.923773i $$-0.374911\pi$$
0.382941 + 0.923773i $$0.374911\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 180.000 0.0214461
$$414$$ 0 0
$$415$$ 8991.00 1.06350
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −6372.00 −0.742942 −0.371471 0.928445i $$-0.621146\pi$$
−0.371471 + 0.928445i $$0.621146\pi$$
$$420$$ 0 0
$$421$$ 3320.00 0.384339 0.192170 0.981362i $$-0.438448\pi$$
0.192170 + 0.981362i $$0.438448\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3168.00 0.361578
$$426$$ 0 0
$$427$$ 280.000 0.0317334
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2898.00 0.323879 0.161939 0.986801i $$-0.448225\pi$$
0.161939 + 0.986801i $$0.448225\pi$$
$$432$$ 0 0
$$433$$ −4291.00 −0.476241 −0.238120 0.971236i $$-0.576531\pi$$
−0.238120 + 0.971236i $$0.576531\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −12348.0 −1.35168
$$438$$ 0 0
$$439$$ −8323.00 −0.904864 −0.452432 0.891799i $$-0.649443\pi$$
−0.452432 + 0.891799i $$0.649443\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3780.00 0.405402 0.202701 0.979241i $$-0.435028\pi$$
0.202701 + 0.979241i $$0.435028\pi$$
$$444$$ 0 0
$$445$$ 7938.00 0.845612
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −12474.0 −1.31110 −0.655551 0.755151i $$-0.727563\pi$$
−0.655551 + 0.755151i $$0.727563\pi$$
$$450$$ 0 0
$$451$$ −28350.0 −2.95998
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −252.000 −0.0259647
$$456$$ 0 0
$$457$$ 16679.0 1.70724 0.853622 0.520893i $$-0.174401\pi$$
0.853622 + 0.520893i $$0.174401\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 17271.0 1.74488 0.872441 0.488719i $$-0.162536\pi$$
0.872441 + 0.488719i $$0.162536\pi$$
$$462$$ 0 0
$$463$$ 17387.0 1.74523 0.872616 0.488407i $$-0.162422\pi$$
0.872616 + 0.488407i $$0.162422\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −3087.00 −0.305887 −0.152944 0.988235i $$-0.548875\pi$$
−0.152944 + 0.988235i $$0.548875\pi$$
$$468$$ 0 0
$$469$$ 586.000 0.0576950
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2142.00 0.208223
$$474$$ 0 0
$$475$$ −4312.00 −0.416522
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 5238.00 0.499646 0.249823 0.968292i $$-0.419628\pi$$
0.249823 + 0.968292i $$0.419628\pi$$
$$480$$ 0 0
$$481$$ −10808.0 −1.02454
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6489.00 0.607526
$$486$$ 0 0
$$487$$ −4384.00 −0.407922 −0.203961 0.978979i $$-0.565382\pi$$
−0.203961 + 0.978979i $$0.565382\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3843.00 0.353222 0.176611 0.984281i $$-0.443486\pi$$
0.176611 + 0.984281i $$0.443486\pi$$
$$492$$ 0 0
$$493$$ −9072.00 −0.828767
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 504.000 0.0454879
$$498$$ 0 0
$$499$$ 12566.0 1.12732 0.563659 0.826008i $$-0.309393\pi$$
0.563659 + 0.826008i $$0.309393\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 5238.00 0.464316 0.232158 0.972678i $$-0.425421\pi$$
0.232158 + 0.972678i $$0.425421\pi$$
$$504$$ 0 0
$$505$$ −3969.00 −0.349739
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6309.00 0.549394 0.274697 0.961531i $$-0.411422\pi$$
0.274697 + 0.961531i $$0.411422\pi$$
$$510$$ 0 0
$$511$$ −161.000 −0.0139378
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4788.00 0.409679
$$516$$ 0 0
$$517$$ −3402.00 −0.289400
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5868.00 0.493439 0.246720 0.969087i $$-0.420647\pi$$
0.246720 + 0.969087i $$0.420647\pi$$
$$522$$ 0 0
$$523$$ 6776.00 0.566527 0.283264 0.959042i $$-0.408583\pi$$
0.283264 + 0.959042i $$0.408583\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 18648.0 1.54140
$$528$$ 0 0
$$529$$ 3709.00 0.304841
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12600.0 −1.02395
$$534$$ 0 0
$$535$$ 7371.00 0.595656
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 21546.0 1.72180
$$540$$ 0 0
$$541$$ −9388.00 −0.746066 −0.373033 0.927818i $$-0.621682\pi$$
−0.373033 + 0.927818i $$0.621682\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 11646.0 0.915339
$$546$$ 0 0
$$547$$ 15056.0 1.17687 0.588435 0.808544i $$-0.299744\pi$$
0.588435 + 0.808544i $$0.299744\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 12348.0 0.954705
$$552$$ 0 0
$$553$$ −440.000 −0.0338349
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16569.0 −1.26041 −0.630207 0.776427i $$-0.717030\pi$$
−0.630207 + 0.776427i $$0.717030\pi$$
$$558$$ 0 0
$$559$$ 952.000 0.0720310
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −14553.0 −1.08941 −0.544703 0.838629i $$-0.683358\pi$$
−0.544703 + 0.838629i $$0.683358\pi$$
$$564$$ 0 0
$$565$$ 10206.0 0.759946
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4284.00 −0.315632 −0.157816 0.987469i $$-0.550445\pi$$
−0.157816 + 0.987469i $$0.550445\pi$$
$$570$$ 0 0
$$571$$ 692.000 0.0507168 0.0253584 0.999678i $$-0.491927\pi$$
0.0253584 + 0.999678i $$0.491927\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5544.00 0.402088
$$576$$ 0 0
$$577$$ 4886.00 0.352525 0.176262 0.984343i $$-0.443599\pi$$
0.176262 + 0.984343i $$0.443599\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 999.000 0.0713348
$$582$$ 0 0
$$583$$ −43659.0 −3.10149
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 11025.0 0.775214 0.387607 0.921825i $$-0.373302\pi$$
0.387607 + 0.921825i $$0.373302\pi$$
$$588$$ 0 0
$$589$$ −25382.0 −1.77563
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 14562.0 1.00841 0.504207 0.863583i $$-0.331785\pi$$
0.504207 + 0.863583i $$0.331785\pi$$
$$594$$ 0 0
$$595$$ −648.000 −0.0446477
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −11466.0 −0.782117 −0.391058 0.920366i $$-0.627891\pi$$
−0.391058 + 0.920366i $$0.627891\pi$$
$$600$$ 0 0
$$601$$ 10955.0 0.743534 0.371767 0.928326i $$-0.378752\pi$$
0.371767 + 0.928326i $$0.378752\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −23742.0 −1.59545
$$606$$ 0 0
$$607$$ 1232.00 0.0823811 0.0411906 0.999151i $$-0.486885\pi$$
0.0411906 + 0.999151i $$0.486885\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1512.00 −0.100113
$$612$$ 0 0
$$613$$ 25622.0 1.68819 0.844097 0.536191i $$-0.180137\pi$$
0.844097 + 0.536191i $$0.180137\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −21042.0 −1.37296 −0.686482 0.727147i $$-0.740846\pi$$
−0.686482 + 0.727147i $$0.740846\pi$$
$$618$$ 0 0
$$619$$ 6524.00 0.423621 0.211811 0.977311i $$-0.432064\pi$$
0.211811 + 0.977311i $$0.432064\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 882.000 0.0567200
$$624$$ 0 0
$$625$$ −8189.00 −0.524096
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −27792.0 −1.76175
$$630$$ 0 0
$$631$$ 20663.0 1.30361 0.651807 0.758384i $$-0.274011\pi$$
0.651807 + 0.758384i $$0.274011\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 16263.0 1.01634
$$636$$ 0 0
$$637$$ 9576.00 0.595628
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 882.000 0.0543477 0.0271739 0.999631i $$-0.491349\pi$$
0.0271739 + 0.999631i $$0.491349\pi$$
$$642$$ 0 0
$$643$$ −7252.00 −0.444776 −0.222388 0.974958i $$-0.571385\pi$$
−0.222388 + 0.974958i $$0.571385\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21168.0 −1.28624 −0.643122 0.765764i $$-0.722361\pi$$
−0.643122 + 0.765764i $$0.722361\pi$$
$$648$$ 0 0
$$649$$ 11340.0 0.685877
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −14301.0 −0.857031 −0.428516 0.903534i $$-0.640963\pi$$
−0.428516 + 0.903534i $$0.640963\pi$$
$$654$$ 0 0
$$655$$ −19845.0 −1.18383
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −15057.0 −0.890042 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$660$$ 0 0
$$661$$ −4690.00 −0.275976 −0.137988 0.990434i $$-0.544063\pi$$
−0.137988 + 0.990434i $$0.544063\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 882.000 0.0514323
$$666$$ 0 0
$$667$$ −15876.0 −0.921621
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 17640.0 1.01488
$$672$$ 0 0
$$673$$ −5203.00 −0.298010 −0.149005 0.988836i $$-0.547607\pi$$
−0.149005 + 0.988836i $$0.547607\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −6174.00 −0.350496 −0.175248 0.984524i $$-0.556073\pi$$
−0.175248 + 0.984524i $$0.556073\pi$$
$$678$$ 0 0
$$679$$ 721.000 0.0407503
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −756.000 −0.0423536 −0.0211768 0.999776i $$-0.506741\pi$$
−0.0211768 + 0.999776i $$0.506741\pi$$
$$684$$ 0 0
$$685$$ 12474.0 0.695777
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −19404.0 −1.07291
$$690$$ 0 0
$$691$$ 17948.0 0.988096 0.494048 0.869435i $$-0.335517\pi$$
0.494048 + 0.869435i $$0.335517\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4284.00 −0.233815
$$696$$ 0 0
$$697$$ −32400.0 −1.76074
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −34083.0 −1.83637 −0.918186 0.396149i $$-0.870346\pi$$
−0.918186 + 0.396149i $$0.870346\pi$$
$$702$$ 0 0
$$703$$ 37828.0 2.02946
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −441.000 −0.0234590
$$708$$ 0 0
$$709$$ −5284.00 −0.279894 −0.139947 0.990159i $$-0.544693\pi$$
−0.139947 + 0.990159i $$0.544693\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 32634.0 1.71410
$$714$$ 0 0
$$715$$ −15876.0 −0.830390
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 24516.0 1.27162 0.635808 0.771847i $$-0.280667\pi$$
0.635808 + 0.771847i $$0.280667\pi$$
$$720$$ 0 0
$$721$$ 532.000 0.0274795
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −5544.00 −0.283999
$$726$$ 0 0
$$727$$ −12481.0 −0.636719 −0.318359 0.947970i $$-0.603132\pi$$
−0.318359 + 0.947970i $$0.603132\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2448.00 0.123861
$$732$$ 0 0
$$733$$ −3094.00 −0.155907 −0.0779533 0.996957i $$-0.524838\pi$$
−0.0779533 + 0.996957i $$0.524838\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 36918.0 1.84517
$$738$$ 0 0
$$739$$ −376.000 −0.0187164 −0.00935818 0.999956i $$-0.502979\pi$$
−0.00935818 + 0.999956i $$0.502979\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −29484.0 −1.45580 −0.727902 0.685681i $$-0.759505\pi$$
−0.727902 + 0.685681i $$0.759505\pi$$
$$744$$ 0 0
$$745$$ −14175.0 −0.697090
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 819.000 0.0399541
$$750$$ 0 0
$$751$$ 11681.0 0.567571 0.283785 0.958888i $$-0.408410\pi$$
0.283785 + 0.958888i $$0.408410\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −4041.00 −0.194791
$$756$$ 0 0
$$757$$ 890.000 0.0427313 0.0213657 0.999772i $$-0.493199\pi$$
0.0213657 + 0.999772i $$0.493199\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 32832.0 1.56394 0.781970 0.623315i $$-0.214215\pi$$
0.781970 + 0.623315i $$0.214215\pi$$
$$762$$ 0 0
$$763$$ 1294.00 0.0613970
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 5040.00 0.237267
$$768$$ 0 0
$$769$$ 3185.00 0.149355 0.0746775 0.997208i $$-0.476207\pi$$
0.0746775 + 0.997208i $$0.476207\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 21222.0 0.987454 0.493727 0.869617i $$-0.335634\pi$$
0.493727 + 0.869617i $$0.335634\pi$$
$$774$$ 0 0
$$775$$ 11396.0 0.528202
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 44100.0 2.02830
$$780$$ 0 0
$$781$$ 31752.0 1.45477
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16380.0 −0.744748
$$786$$ 0 0
$$787$$ 12320.0 0.558019 0.279009 0.960288i $$-0.409994\pi$$
0.279009 + 0.960288i $$0.409994\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1134.00 0.0509740
$$792$$ 0 0
$$793$$ 7840.00 0.351080
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 11907.0 0.529194 0.264597 0.964359i $$-0.414761\pi$$
0.264597 + 0.964359i $$0.414761\pi$$
$$798$$ 0 0
$$799$$ −3888.00 −0.172150
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −10143.0 −0.445752
$$804$$ 0 0
$$805$$ −1134.00 −0.0496500
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 14994.0 0.651620 0.325810 0.945435i $$-0.394363\pi$$
0.325810 + 0.945435i $$0.394363\pi$$
$$810$$ 0 0
$$811$$ −38878.0 −1.68334 −0.841672 0.539990i $$-0.818428\pi$$
−0.841672 + 0.539990i $$0.818428\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 16452.0 0.707102
$$816$$ 0 0
$$817$$ −3332.00 −0.142683
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3906.00 −0.166042 −0.0830209 0.996548i $$-0.526457\pi$$
−0.0830209 + 0.996548i $$0.526457\pi$$
$$822$$ 0 0
$$823$$ −10207.0 −0.432313 −0.216157 0.976359i $$-0.569352\pi$$
−0.216157 + 0.976359i $$0.569352\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −8064.00 −0.339072 −0.169536 0.985524i $$-0.554227\pi$$
−0.169536 + 0.985524i $$0.554227\pi$$
$$828$$ 0 0
$$829$$ −10486.0 −0.439317 −0.219659 0.975577i $$-0.570494\pi$$
−0.219659 + 0.975577i $$0.570494\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 24624.0 1.02421
$$834$$ 0 0
$$835$$ −7290.00 −0.302133
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 33696.0 1.38655 0.693275 0.720673i $$-0.256167\pi$$
0.693275 + 0.720673i $$0.256167\pi$$
$$840$$ 0 0
$$841$$ −8513.00 −0.349051
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 12717.0 0.517726
$$846$$ 0 0
$$847$$ −2638.00 −0.107016
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −48636.0 −1.95913
$$852$$ 0 0
$$853$$ −15778.0 −0.633328 −0.316664 0.948538i $$-0.602563\pi$$
−0.316664 + 0.948538i $$0.602563\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 26136.0 1.04176 0.520880 0.853630i $$-0.325604\pi$$
0.520880 + 0.853630i $$0.325604\pi$$
$$858$$ 0 0
$$859$$ 23912.0 0.949787 0.474893 0.880043i $$-0.342487\pi$$
0.474893 + 0.880043i $$0.342487\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −17892.0 −0.705737 −0.352868 0.935673i $$-0.614794\pi$$
−0.352868 + 0.935673i $$0.614794\pi$$
$$864$$ 0 0
$$865$$ −11907.0 −0.468035
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −27720.0 −1.08209
$$870$$ 0 0
$$871$$ 16408.0 0.638305
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1521.00 −0.0587648
$$876$$ 0 0
$$877$$ 44162.0 1.70039 0.850197 0.526465i $$-0.176483\pi$$
0.850197 + 0.526465i $$0.176483\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 8820.00 0.337291 0.168645 0.985677i $$-0.446061\pi$$
0.168645 + 0.985677i $$0.446061\pi$$
$$882$$ 0 0
$$883$$ −4654.00 −0.177372 −0.0886861 0.996060i $$-0.528267\pi$$
−0.0886861 + 0.996060i $$0.528267\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −33084.0 −1.25237 −0.626185 0.779675i $$-0.715384\pi$$
−0.626185 + 0.779675i $$0.715384\pi$$
$$888$$ 0 0
$$889$$ 1807.00 0.0681719
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 5292.00 0.198309
$$894$$ 0 0
$$895$$ −2835.00 −0.105881
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −32634.0 −1.21068
$$900$$ 0 0
$$901$$ −49896.0 −1.84492
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 25200.0 0.925609
$$906$$ 0 0
$$907$$ 47258.0 1.73007 0.865036 0.501709i $$-0.167295\pi$$
0.865036 + 0.501709i $$0.167295\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −36666.0 −1.33348 −0.666739 0.745291i $$-0.732310\pi$$
−0.666739 + 0.745291i $$0.732310\pi$$
$$912$$ 0 0
$$913$$ 62937.0 2.28139
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −2205.00 −0.0794062
$$918$$ 0 0
$$919$$ −2995.00 −0.107504 −0.0537519 0.998554i $$-0.517118\pi$$
−0.0537519 + 0.998554i $$0.517118\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 14112.0 0.503253
$$924$$ 0 0
$$925$$ −16984.0 −0.603709
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 13284.0 0.469143 0.234572 0.972099i $$-0.424631\pi$$
0.234572 + 0.972099i $$0.424631\pi$$
$$930$$ 0 0
$$931$$ −33516.0 −1.17985
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −40824.0 −1.42790
$$936$$ 0 0
$$937$$ −2989.00 −0.104212 −0.0521059 0.998642i $$-0.516593\pi$$
−0.0521059 + 0.998642i $$0.516593\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −46485.0 −1.61038 −0.805190 0.593017i $$-0.797937\pi$$
−0.805190 + 0.593017i $$0.797937\pi$$
$$942$$ 0 0
$$943$$ −56700.0 −1.95801
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −11529.0 −0.395609 −0.197805 0.980241i $$-0.563381\pi$$
−0.197805 + 0.980241i $$0.563381\pi$$
$$948$$ 0 0
$$949$$ −4508.00 −0.154200
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −37044.0 −1.25915 −0.629577 0.776938i $$-0.716772\pi$$
−0.629577 + 0.776938i $$0.716772\pi$$
$$954$$ 0 0
$$955$$ −29484.0 −0.999036
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1386.00 0.0466697
$$960$$ 0 0
$$961$$ 37290.0 1.25172
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −28989.0 −0.967035
$$966$$ 0 0
$$967$$ 12455.0 0.414194 0.207097 0.978320i $$-0.433598\pi$$
0.207097 + 0.978320i $$0.433598\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20169.0 0.666585 0.333292 0.942823i $$-0.391840\pi$$
0.333292 + 0.942823i $$0.391840\pi$$
$$972$$ 0 0
$$973$$ −476.000 −0.0156833
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39942.0 1.30794 0.653970 0.756520i $$-0.273102\pi$$
0.653970 + 0.756520i $$0.273102\pi$$
$$978$$ 0 0
$$979$$ 55566.0 1.81399
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −54234.0 −1.75971 −0.879856 0.475241i $$-0.842361\pi$$
−0.879856 + 0.475241i $$0.842361\pi$$
$$984$$ 0 0
$$985$$ 30051.0 0.972086
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4284.00 0.137738
$$990$$ 0 0
$$991$$ −26137.0 −0.837809 −0.418905 0.908030i $$-0.637586\pi$$
−0.418905 + 0.908030i $$0.637586\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −33201.0 −1.05783
$$996$$ 0 0
$$997$$ −33460.0 −1.06288 −0.531439 0.847097i $$-0.678348\pi$$
−0.531439 + 0.847097i $$0.678348\pi$$
$$998$$ 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 108.4.a.a.1.1 1
3.2 odd 2 108.4.a.d.1.1 yes 1
4.3 odd 2 432.4.a.c.1.1 1
8.3 odd 2 1728.4.a.z.1.1 1
8.5 even 2 1728.4.a.y.1.1 1
9.2 odd 6 324.4.e.b.109.1 2
9.4 even 3 324.4.e.g.217.1 2
9.5 odd 6 324.4.e.b.217.1 2
9.7 even 3 324.4.e.g.109.1 2
12.11 even 2 432.4.a.l.1.1 1
24.5 odd 2 1728.4.a.g.1.1 1
24.11 even 2 1728.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.a.a.1.1 1 1.1 even 1 trivial
108.4.a.d.1.1 yes 1 3.2 odd 2
324.4.e.b.109.1 2 9.2 odd 6
324.4.e.b.217.1 2 9.5 odd 6
324.4.e.g.109.1 2 9.7 even 3
324.4.e.g.217.1 2 9.4 even 3
432.4.a.c.1.1 1 4.3 odd 2
432.4.a.l.1.1 1 12.11 even 2
1728.4.a.g.1.1 1 24.5 odd 2
1728.4.a.h.1.1 1 24.11 even 2
1728.4.a.y.1.1 1 8.5 even 2
1728.4.a.z.1.1 1 8.3 odd 2