# Properties

 Label 8281.2.a.f.1.1 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ 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] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.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: $$66.1241179138$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{8} -3.00000 q^{9} +3.00000 q^{11} -1.00000 q^{16} +7.00000 q^{17} +3.00000 q^{18} +7.00000 q^{19} -3.00000 q^{22} -6.00000 q^{23} -5.00000 q^{25} -5.00000 q^{29} -5.00000 q^{32} -7.00000 q^{34} +3.00000 q^{36} -8.00000 q^{37} -7.00000 q^{38} +2.00000 q^{43} -3.00000 q^{44} +6.00000 q^{46} -7.00000 q^{47} +5.00000 q^{50} -3.00000 q^{53} +5.00000 q^{58} +7.00000 q^{59} -7.00000 q^{61} +7.00000 q^{64} +3.00000 q^{67} -7.00000 q^{68} +5.00000 q^{71} -9.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} -7.00000 q^{76} -6.00000 q^{79} +9.00000 q^{81} -2.00000 q^{86} +9.00000 q^{88} +6.00000 q^{92} +7.00000 q^{94} +14.0000 q^{97} -9.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 3.00000 1.06066
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 3.00000 0.707107
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ −7.00000 −1.20049
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ −7.00000 −1.13555
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 5.00000 0.656532
$$59$$ 7.00000 0.911322 0.455661 0.890153i $$-0.349403\pi$$
0.455661 + 0.890153i $$0.349403\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −7.00000 −0.848875
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ −9.00000 −1.06066
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 9.00000 0.959403
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ −9.00000 −0.904534
$$100$$ 5.00000 0.500000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 5.00000 0.464238
$$117$$ 0 0
$$118$$ −7.00000 −0.644402
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 7.00000 0.633750
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ 21.0000 1.80074
$$137$$ −4.00000 −0.341743 −0.170872 0.985293i $$-0.554658\pi$$
−0.170872 + 0.985293i $$0.554658\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5.00000 −0.419591
$$143$$ 0 0
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ 8.00000 0.657596
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 3.00000 0.244137 0.122068 0.992522i $$-0.461047\pi$$
0.122068 + 0.992522i $$0.461047\pi$$
$$152$$ 21.0000 1.70332
$$153$$ −21.0000 −1.69775
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 6.00000 0.477334
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −9.00000 −0.707107
$$163$$ 13.0000 1.01824 0.509119 0.860696i $$-0.329971\pi$$
0.509119 + 0.860696i $$0.329971\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.00000 0.541676 0.270838 0.962625i $$-0.412699\pi$$
0.270838 + 0.962625i $$0.412699\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −21.0000 −1.60591
$$172$$ −2.00000 −0.152499
$$173$$ 7.00000 0.532200 0.266100 0.963945i $$-0.414265\pi$$
0.266100 + 0.963945i $$0.414265\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −18.0000 −1.32698
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 21.0000 1.53567
$$188$$ 7.00000 0.510527
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 9.00000 0.639602
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ −15.0000 −1.06066
$$201$$ 0 0
$$202$$ 14.0000 0.985037
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 18.0000 1.25109
$$208$$ 0 0
$$209$$ 21.0000 1.45260
$$210$$ 0 0
$$211$$ −26.0000 −1.78991 −0.894957 0.446153i $$-0.852794\pi$$
−0.894957 + 0.446153i $$0.852794\pi$$
$$212$$ 3.00000 0.206041
$$213$$ 0 0
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −21.0000 −1.40626 −0.703132 0.711059i $$-0.748216\pi$$
−0.703132 + 0.711059i $$0.748216\pi$$
$$224$$ 0 0
$$225$$ 15.0000 1.00000
$$226$$ −9.00000 −0.598671
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −15.0000 −0.984798
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −7.00000 −0.455661
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 19.0000 1.22901 0.614504 0.788914i $$-0.289356\pi$$
0.614504 + 0.788914i $$0.289356\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 0 0
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 0 0
$$253$$ −18.0000 −1.13165
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 15.0000 0.928477
$$262$$ −14.0000 −0.864923
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −3.00000 −0.183254
$$269$$ −21.0000 −1.28039 −0.640196 0.768211i $$-0.721147\pi$$
−0.640196 + 0.768211i $$0.721147\pi$$
$$270$$ 0 0
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ −7.00000 −0.424437
$$273$$ 0 0
$$274$$ 4.00000 0.241649
$$275$$ −15.0000 −0.904534
$$276$$ 0 0
$$277$$ −17.0000 −1.02143 −0.510716 0.859750i $$-0.670619\pi$$
−0.510716 + 0.859750i $$0.670619\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −5.00000 −0.296695
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 15.0000 0.883883
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 14.0000 0.819288
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −24.0000 −1.39497
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −3.00000 −0.172631
$$303$$ 0 0
$$304$$ −7.00000 −0.401478
$$305$$ 0 0
$$306$$ 21.0000 1.20049
$$307$$ −21.0000 −1.19853 −0.599267 0.800549i $$-0.704541\pi$$
−0.599267 + 0.800549i $$0.704541\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ −7.00000 −0.395033
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ −15.0000 −0.839839
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 49.0000 2.72643
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −13.0000 −0.720003
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 24.0000 1.31519
$$334$$ −7.00000 −0.383023
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 21.0000 1.13555
$$343$$ 0 0
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ −7.00000 −0.376322
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −15.0000 −0.799503
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 10.0000 0.528516
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 7.00000 0.367912
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 15.0000 0.776671 0.388335 0.921518i $$-0.373050\pi$$
0.388335 + 0.921518i $$0.373050\pi$$
$$374$$ −21.0000 −1.08588
$$375$$ 0 0
$$376$$ −21.0000 −1.08299
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 20.0000 1.02329
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −6.00000 −0.304997
$$388$$ −14.0000 −0.710742
$$389$$ −3.00000 −0.152106 −0.0760530 0.997104i $$-0.524232\pi$$
−0.0760530 + 0.997104i $$0.524232\pi$$
$$390$$ 0 0
$$391$$ −42.0000 −2.12403
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 9.00000 0.452267
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 14.0000 0.701757
$$399$$ 0 0
$$400$$ 5.00000 0.250000
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 28.0000 1.38451 0.692255 0.721653i $$-0.256617\pi$$
0.692255 + 0.721653i $$0.256617\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ 0 0
$$414$$ −18.0000 −0.884652
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −21.0000 −1.02714
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 26.0000 1.26566
$$423$$ 21.0000 1.02105
$$424$$ −9.00000 −0.437079
$$425$$ −35.0000 −1.69775
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 21.0000 1.00920 0.504598 0.863355i $$-0.331641\pi$$
0.504598 + 0.863355i $$0.331641\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 4.00000 0.191565
$$437$$ −42.0000 −2.00913
$$438$$ 0 0
$$439$$ 14.0000 0.668184 0.334092 0.942541i $$-0.391570\pi$$
0.334092 + 0.942541i $$0.391570\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 21.0000 0.994379
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ −15.0000 −0.707107
$$451$$ 0 0
$$452$$ −9.00000 −0.423324
$$453$$ 0 0
$$454$$ −28.0000 −1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 28.0000 1.30409 0.652045 0.758180i $$-0.273911\pi$$
0.652045 + 0.758180i $$0.273911\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ 27.0000 1.25075
$$467$$ 14.0000 0.647843 0.323921 0.946084i $$-0.394999\pi$$
0.323921 + 0.946084i $$0.394999\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 21.0000 0.966603
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ −35.0000 −1.60591
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ −19.0000 −0.869040
$$479$$ −7.00000 −0.319838 −0.159919 0.987130i $$-0.551123\pi$$
−0.159919 + 0.987130i $$0.551123\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 28.0000 1.27537
$$483$$ 0 0
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −25.0000 −1.13286 −0.566429 0.824110i $$-0.691675\pi$$
−0.566429 + 0.824110i $$0.691675\pi$$
$$488$$ −21.0000 −0.950625
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 30.0000 1.35388 0.676941 0.736038i $$-0.263305\pi$$
0.676941 + 0.736038i $$0.263305\pi$$
$$492$$ 0 0
$$493$$ −35.0000 −1.57632
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −8.00000 −0.358129 −0.179065 0.983837i $$-0.557307\pi$$
−0.179065 + 0.983837i $$0.557307\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 14.0000 0.624851
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 18.0000 0.800198
$$507$$ 0 0
$$508$$ −2.00000 −0.0887357
$$509$$ 28.0000 1.24108 0.620539 0.784176i $$-0.286914\pi$$
0.620539 + 0.784176i $$0.286914\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −21.0000 −0.923579
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ −15.0000 −0.656532
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −21.0000 −0.911322
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 9.00000 0.388741
$$537$$ 0 0
$$538$$ 21.0000 0.905374
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ −7.00000 −0.300676
$$543$$ 0 0
$$544$$ −35.0000 −1.50061
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ 4.00000 0.170872
$$549$$ 21.0000 0.896258
$$550$$ 15.0000 0.639602
$$551$$ −35.0000 −1.49105
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 17.0000 0.722261
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −12.0000 −0.506189
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ 0 0
$$568$$ 15.0000 0.629386
$$569$$ 1.00000 0.0419222 0.0209611 0.999780i $$-0.493327\pi$$
0.0209611 + 0.999780i $$0.493327\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 30.0000 1.25109
$$576$$ −21.0000 −0.875000
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ −42.0000 −1.73797
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ −21.0000 −0.866763 −0.433381 0.901211i $$-0.642680\pi$$
−0.433381 + 0.901211i $$0.642680\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ 0 0
$$603$$ −9.00000 −0.366508
$$604$$ −3.00000 −0.122068
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ −35.0000 −1.41944
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 21.0000 0.848875
$$613$$ −32.0000 −1.29247 −0.646234 0.763139i $$-0.723657\pi$$
−0.646234 + 0.763139i $$0.723657\pi$$
$$614$$ 21.0000 0.847491
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ −7.00000 −0.279330
$$629$$ −56.0000 −2.23287
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −18.0000 −0.716002
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 15.0000 0.593856
$$639$$ −15.0000 −0.593391
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 7.00000 0.276053 0.138027 0.990429i $$-0.455924\pi$$
0.138027 + 0.990429i $$0.455924\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −49.0000 −1.92788
$$647$$ −42.0000 −1.65119 −0.825595 0.564263i $$-0.809160\pi$$
−0.825595 + 0.564263i $$0.809160\pi$$
$$648$$ 27.0000 1.06066
$$649$$ 21.0000 0.824322
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −13.0000 −0.509119
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 42.0000 1.63858
$$658$$ 0 0
$$659$$ −40.0000 −1.55818 −0.779089 0.626913i $$-0.784318\pi$$
−0.779089 + 0.626913i $$0.784318\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −24.0000 −0.929981
$$667$$ 30.0000 1.16160
$$668$$ −7.00000 −0.270838
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −21.0000 −0.810696
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −23.0000 −0.885927
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −35.0000 −1.34516 −0.672580 0.740025i $$-0.734814\pi$$
−0.672580 + 0.740025i $$0.734814\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 21.0000 0.802955
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −2.00000 −0.0762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −35.0000 −1.33146 −0.665731 0.746191i $$-0.731880\pi$$
−0.665731 + 0.746191i $$0.731880\pi$$
$$692$$ −7.00000 −0.266100
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 14.0000 0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −56.0000 −2.11208
$$704$$ 21.0000 0.791467
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −50.0000 −1.87779 −0.938895 0.344204i $$-0.888149\pi$$
−0.938895 + 0.344204i $$0.888149\pi$$
$$710$$ 0 0
$$711$$ 18.0000 0.675053
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −30.0000 −1.11648
$$723$$ 0 0
$$724$$ 7.00000 0.260153
$$725$$ 25.0000 0.928477
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 14.0000 0.517809
$$732$$ 0 0
$$733$$ −42.0000 −1.55131 −0.775653 0.631160i $$-0.782579\pi$$
−0.775653 + 0.631160i $$0.782579\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ 30.0000 1.10581
$$737$$ 9.00000 0.331519
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −15.0000 −0.549189
$$747$$ 0 0
$$748$$ −21.0000 −0.767836
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 7.00000 0.255264
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 9.00000 0.327111 0.163555 0.986534i $$-0.447704\pi$$
0.163555 + 0.986534i $$0.447704\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 20.0000 0.723575
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.00000 0.143963
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ 42.0000 1.50771
$$777$$ 0 0
$$778$$ 3.00000 0.107555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 15.0000 0.536742
$$782$$ 42.0000 1.50192
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −27.0000 −0.959403
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ −49.0000 −1.73350
$$800$$ 25.0000 0.883883
$$801$$ 0 0
$$802$$ 22.0000 0.776847
$$803$$ −42.0000 −1.48215
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −42.0000 −1.47755
$$809$$ −31.0000 −1.08990 −0.544951 0.838468i $$-0.683452\pi$$
−0.544951 + 0.838468i $$0.683452\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 14.0000 0.489798
$$818$$ −28.0000 −0.978997
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −36.0000 −1.25641 −0.628204 0.778048i $$-0.716210\pi$$
−0.628204 + 0.778048i $$0.716210\pi$$
$$822$$ 0 0
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 42.0000 1.46314
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 5.00000 0.173867 0.0869335 0.996214i $$-0.472293\pi$$
0.0869335 + 0.996214i $$0.472293\pi$$
$$828$$ −18.0000 −0.625543
$$829$$ 35.0000 1.21560 0.607800 0.794090i $$-0.292052\pi$$
0.607800 + 0.794090i $$0.292052\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −21.0000 −0.726300
$$837$$ 0 0
$$838$$ −14.0000 −0.483622
$$839$$ −7.00000 −0.241667 −0.120833 0.992673i $$-0.538557\pi$$
−0.120833 + 0.992673i $$0.538557\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 30.0000 1.03387
$$843$$ 0 0
$$844$$ 26.0000 0.894957
$$845$$ 0 0
$$846$$ −21.0000 −0.721995
$$847$$ 0 0
$$848$$ 3.00000 0.103020
$$849$$ 0 0
$$850$$ 35.0000 1.20049
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 24.0000 0.820303
$$857$$ 21.0000 0.717346 0.358673 0.933463i $$-0.383229\pi$$
0.358673 + 0.933463i $$0.383229\pi$$
$$858$$ 0 0
$$859$$ −56.0000 −1.91070 −0.955348 0.295484i $$-0.904519\pi$$
−0.955348 + 0.295484i $$0.904519\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 20.0000 0.680808 0.340404 0.940279i $$-0.389436\pi$$
0.340404 + 0.940279i $$0.389436\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −21.0000 −0.713609
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −18.0000 −0.610608
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −12.0000 −0.406371
$$873$$ −42.0000 −1.42148
$$874$$ 42.0000 1.42067
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 48.0000 1.62084 0.810422 0.585846i $$-0.199238\pi$$
0.810422 + 0.585846i $$0.199238\pi$$
$$878$$ −14.0000 −0.472477
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ −40.0000 −1.34611 −0.673054 0.739594i $$-0.735018\pi$$
−0.673054 + 0.739594i $$0.735018\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 27.0000 0.904534
$$892$$ 21.0000 0.703132
$$893$$ −49.0000 −1.63972
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ 0 0
$$900$$ −15.0000 −0.500000
$$901$$ −21.0000 −0.699611
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 27.0000 0.898007
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −38.0000 −1.26177 −0.630885 0.775877i $$-0.717308\pi$$
−0.630885 + 0.775877i $$0.717308\pi$$
$$908$$ −28.0000 −0.929213
$$909$$ 42.0000 1.39305
$$910$$ 0 0
$$911$$ −54.0000 −1.78910 −0.894550 0.446968i $$-0.852504\pi$$
−0.894550 + 0.446968i $$0.852504\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 50.0000 1.64935 0.824674 0.565608i $$-0.191359\pi$$
0.824674 + 0.565608i $$0.191359\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −28.0000 −0.922131
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 40.0000 1.31519
$$926$$ 16.0000 0.525793
$$927$$ −42.0000 −1.37946
$$928$$ 25.0000 0.820665
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 27.0000 0.884414
$$933$$ 0 0
$$934$$ −14.0000 −0.458094
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −28.0000 −0.912774 −0.456387 0.889781i $$-0.650857\pi$$
−0.456387 + 0.889781i $$0.650857\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −7.00000 −0.227831
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 35.0000 1.13555
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.00000 0.291539 0.145769 0.989319i $$-0.453434\pi$$
0.145769 + 0.989319i $$0.453434\pi$$
$$954$$ −9.00000 −0.291386
$$955$$ 0 0
$$956$$ −19.0000 −0.614504
$$957$$ 0 0
$$958$$ 7.00000 0.226160
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −24.0000 −0.773389
$$964$$ 28.0000 0.901819
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 5.00000 0.160789 0.0803946 0.996763i $$-0.474382\pi$$
0.0803946 + 0.996763i $$0.474382\pi$$
$$968$$ −6.00000 −0.192847
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 25.0000 0.801052
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ 38.0000 1.21573 0.607864 0.794041i $$-0.292027\pi$$
0.607864 + 0.794041i $$0.292027\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ −30.0000 −0.957338
$$983$$ 7.00000 0.223265 0.111633 0.993750i $$-0.464392\pi$$
0.111633 + 0.993750i $$0.464392\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 35.0000 1.11463
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −7.00000 −0.221692 −0.110846 0.993838i $$-0.535356\pi$$
−0.110846 + 0.993838i $$0.535356\pi$$
$$998$$ 8.00000 0.253236
$$999$$ 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 8281.2.a.f.1.1 1
7.2 even 3 1183.2.e.b.508.1 2
7.4 even 3 1183.2.e.b.170.1 2
7.6 odd 2 8281.2.a.e.1.1 1
13.12 even 2 637.2.a.c.1.1 1
39.38 odd 2 5733.2.a.c.1.1 1
91.12 odd 6 637.2.e.a.508.1 2
91.25 even 6 91.2.e.a.79.1 yes 2
91.38 odd 6 637.2.e.a.79.1 2
91.51 even 6 91.2.e.a.53.1 2
91.90 odd 2 637.2.a.d.1.1 1
273.116 odd 6 819.2.j.b.352.1 2
273.233 odd 6 819.2.j.b.235.1 2
273.272 even 2 5733.2.a.d.1.1 1
364.51 odd 6 1456.2.r.g.417.1 2
364.207 odd 6 1456.2.r.g.625.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.a.53.1 2 91.51 even 6
91.2.e.a.79.1 yes 2 91.25 even 6
637.2.a.c.1.1 1 13.12 even 2
637.2.a.d.1.1 1 91.90 odd 2
637.2.e.a.79.1 2 91.38 odd 6
637.2.e.a.508.1 2 91.12 odd 6
819.2.j.b.235.1 2 273.233 odd 6
819.2.j.b.352.1 2 273.116 odd 6
1183.2.e.b.170.1 2 7.4 even 3
1183.2.e.b.508.1 2 7.2 even 3
1456.2.r.g.417.1 2 364.51 odd 6
1456.2.r.g.625.1 2 364.207 odd 6
5733.2.a.c.1.1 1 39.38 odd 2
5733.2.a.d.1.1 1 273.272 even 2
8281.2.a.e.1.1 1 7.6 odd 2
8281.2.a.f.1.1 1 1.1 even 1 trivial