# Properties

 Label 2760.2.a.i.1.1 Level $2760$ Weight $2$ Character 2760.1 Self dual yes Analytic conductor $22.039$ 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] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2760.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: $$22.0387109579$$ 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 $$\chi$$ $$=$$ 2760.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{13} -1.00000 q^{15} +3.00000 q^{17} -6.00000 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} -3.00000 q^{31} -6.00000 q^{33} -3.00000 q^{35} +3.00000 q^{37} -2.00000 q^{39} -3.00000 q^{41} -1.00000 q^{45} +4.00000 q^{47} +2.00000 q^{49} +3.00000 q^{51} -9.00000 q^{53} +6.00000 q^{55} -6.00000 q^{57} -3.00000 q^{59} -8.00000 q^{61} +3.00000 q^{63} +2.00000 q^{65} +3.00000 q^{67} -1.00000 q^{69} -9.00000 q^{71} +6.00000 q^{73} +1.00000 q^{75} -18.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +3.00000 q^{83} -3.00000 q^{85} -9.00000 q^{87} -8.00000 q^{89} -6.00000 q^{91} -3.00000 q^{93} +6.00000 q^{95} +18.0000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 6.00000 0.809040
$$56$$ 0 0
$$57$$ −6.00000 −0.794719
$$58$$ 0 0
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −9.00000 −1.06810 −0.534052 0.845452i $$-0.679331\pi$$
−0.534052 + 0.845452i $$0.679331\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −18.0000 −2.05129
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ −9.00000 −0.964901
$$88$$ 0 0
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 0 0
$$93$$ −3.00000 −0.311086
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 11.0000 1.09454 0.547270 0.836956i $$-0.315667\pi$$
0.547270 + 0.836956i $$0.315667\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −5.00000 −0.483368 −0.241684 0.970355i $$-0.577700\pi$$
−0.241684 + 0.970355i $$0.577700\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ −3.00000 −0.270501
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 0 0
$$133$$ −18.0000 −1.56080
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ 24.0000 1.96616 0.983078 0.183186i $$-0.0586410\pi$$
0.983078 + 0.183186i $$0.0586410\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 3.00000 0.242536
$$154$$ 0 0
$$155$$ 3.00000 0.240966
$$156$$ 0 0
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 0 0
$$165$$ 6.00000 0.467099
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ 0 0
$$173$$ −4.00000 −0.304114 −0.152057 0.988372i $$-0.548590\pi$$
−0.152057 + 0.988372i $$0.548590\pi$$
$$174$$ 0 0
$$175$$ 3.00000 0.226779
$$176$$ 0 0
$$177$$ −3.00000 −0.225494
$$178$$ 0 0
$$179$$ −8.00000 −0.597948 −0.298974 0.954261i $$-0.596644\pi$$
−0.298974 + 0.954261i $$0.596644\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ −8.00000 −0.591377
$$184$$ 0 0
$$185$$ −3.00000 −0.220564
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ 0 0
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 0 0
$$203$$ −27.0000 −1.89503
$$204$$ 0 0
$$205$$ 3.00000 0.209529
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 36.0000 2.49017
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ 0 0
$$213$$ −9.00000 −0.616670
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.00000 −0.610960
$$218$$ 0 0
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ −18.0000 −1.18431
$$232$$ 0 0
$$233$$ 20.0000 1.31024 0.655122 0.755523i $$-0.272617\pi$$
0.655122 + 0.755523i $$0.272617\pi$$
$$234$$ 0 0
$$235$$ −4.00000 −0.260931
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ 1.00000 0.0646846 0.0323423 0.999477i $$-0.489703\pi$$
0.0323423 + 0.999477i $$0.489703\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ 0 0
$$249$$ 3.00000 0.190117
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 0 0
$$255$$ −3.00000 −0.187867
$$256$$ 0 0
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 9.00000 0.559233
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 0 0
$$267$$ −8.00000 −0.489592
$$268$$ 0 0
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ 0 0
$$271$$ −5.00000 −0.303728 −0.151864 0.988401i $$-0.548528\pi$$
−0.151864 + 0.988401i $$0.548528\pi$$
$$272$$ 0 0
$$273$$ −6.00000 −0.363137
$$274$$ 0 0
$$275$$ −6.00000 −0.361814
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 0 0
$$283$$ 5.00000 0.297219 0.148610 0.988896i $$-0.452520\pi$$
0.148610 + 0.988896i $$0.452520\pi$$
$$284$$ 0 0
$$285$$ 6.00000 0.355409
$$286$$ 0 0
$$287$$ −9.00000 −0.531253
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ 0 0
$$293$$ 5.00000 0.292103 0.146052 0.989277i $$-0.453343\pi$$
0.146052 + 0.989277i $$0.453343\pi$$
$$294$$ 0 0
$$295$$ 3.00000 0.174667
$$296$$ 0 0
$$297$$ −6.00000 −0.348155
$$298$$ 0 0
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 11.0000 0.631933
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ 6.00000 0.342438 0.171219 0.985233i $$-0.445229\pi$$
0.171219 + 0.985233i $$0.445229\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −31.0000 −1.75222 −0.876112 0.482108i $$-0.839871\pi$$
−0.876112 + 0.482108i $$0.839871\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 0 0
$$317$$ −12.0000 −0.673987 −0.336994 0.941507i $$-0.609410\pi$$
−0.336994 + 0.941507i $$0.609410\pi$$
$$318$$ 0 0
$$319$$ 54.0000 3.02342
$$320$$ 0 0
$$321$$ −5.00000 −0.279073
$$322$$ 0 0
$$323$$ −18.0000 −1.00155
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 9.00000 0.494685 0.247342 0.968928i $$-0.420443\pi$$
0.247342 + 0.968928i $$0.420443\pi$$
$$332$$ 0 0
$$333$$ 3.00000 0.164399
$$334$$ 0 0
$$335$$ −3.00000 −0.163908
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ −1.00000 −0.0543125
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 1.00000 0.0538382
$$346$$ 0 0
$$347$$ −8.00000 −0.429463 −0.214731 0.976673i $$-0.568888\pi$$
−0.214731 + 0.976673i $$0.568888\pi$$
$$348$$ 0 0
$$349$$ 31.0000 1.65939 0.829696 0.558216i $$-0.188514\pi$$
0.829696 + 0.558216i $$0.188514\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ −4.00000 −0.212899 −0.106449 0.994318i $$-0.533948\pi$$
−0.106449 + 0.994318i $$0.533948\pi$$
$$354$$ 0 0
$$355$$ 9.00000 0.477670
$$356$$ 0 0
$$357$$ 9.00000 0.476331
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 25.0000 1.31216
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 0 0
$$369$$ −3.00000 −0.156174
$$370$$ 0 0
$$371$$ −27.0000 −1.40177
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 18.0000 0.927047
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ −27.0000 −1.37964 −0.689818 0.723983i $$-0.742309\pi$$
−0.689818 + 0.723983i $$0.742309\pi$$
$$384$$ 0 0
$$385$$ 18.0000 0.917365
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 34.0000 1.72387 0.861934 0.507020i $$-0.169253\pi$$
0.861934 + 0.507020i $$0.169253\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ 0 0
$$393$$ −20.0000 −1.00887
$$394$$ 0 0
$$395$$ 4.00000 0.201262
$$396$$ 0 0
$$397$$ 32.0000 1.60603 0.803017 0.595956i $$-0.203227\pi$$
0.803017 + 0.595956i $$0.203227\pi$$
$$398$$ 0 0
$$399$$ −18.0000 −0.901127
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −18.0000 −0.892227
$$408$$ 0 0
$$409$$ −35.0000 −1.73064 −0.865319 0.501221i $$-0.832884\pi$$
−0.865319 + 0.501221i $$0.832884\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ −9.00000 −0.442861
$$414$$ 0 0
$$415$$ −3.00000 −0.147264
$$416$$ 0 0
$$417$$ −9.00000 −0.440732
$$418$$ 0 0
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 4.00000 0.194487
$$424$$ 0 0
$$425$$ 3.00000 0.145521
$$426$$ 0 0
$$427$$ −24.0000 −1.16144
$$428$$ 0 0
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 19.0000 0.913082 0.456541 0.889702i $$-0.349088\pi$$
0.456541 + 0.889702i $$0.349088\pi$$
$$434$$ 0 0
$$435$$ 9.00000 0.431517
$$436$$ 0 0
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 8.00000 0.379236
$$446$$ 0 0
$$447$$ 24.0000 1.13516
$$448$$ 0 0
$$449$$ −33.0000 −1.55737 −0.778683 0.627417i $$-0.784112\pi$$
−0.778683 + 0.627417i $$0.784112\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ 0 0
$$453$$ 4.00000 0.187936
$$454$$ 0 0
$$455$$ 6.00000 0.281284
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 0 0
$$459$$ 3.00000 0.140028
$$460$$ 0 0
$$461$$ 38.0000 1.76984 0.884918 0.465746i $$-0.154214\pi$$
0.884918 + 0.465746i $$0.154214\pi$$
$$462$$ 0 0
$$463$$ 2.00000 0.0929479 0.0464739 0.998920i $$-0.485202\pi$$
0.0464739 + 0.998920i $$0.485202\pi$$
$$464$$ 0 0
$$465$$ 3.00000 0.139122
$$466$$ 0 0
$$467$$ −29.0000 −1.34196 −0.670980 0.741475i $$-0.734126\pi$$
−0.670980 + 0.741475i $$0.734126\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ 17.0000 0.783319
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ −9.00000 −0.412082
$$478$$ 0 0
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −18.0000 −0.817338
$$486$$ 0 0
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ 0 0
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ −33.0000 −1.48927 −0.744635 0.667472i $$-0.767376\pi$$
−0.744635 + 0.667472i $$0.767376\pi$$
$$492$$ 0 0
$$493$$ −27.0000 −1.21602
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 0 0
$$497$$ −27.0000 −1.21112
$$498$$ 0 0
$$499$$ 27.0000 1.20869 0.604343 0.796724i $$-0.293436\pi$$
0.604343 + 0.796724i $$0.293436\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 1.00000 0.0445878 0.0222939 0.999751i $$-0.492903\pi$$
0.0222939 + 0.999751i $$0.492903\pi$$
$$504$$ 0 0
$$505$$ −11.0000 −0.489494
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 18.0000 0.796273
$$512$$ 0 0
$$513$$ −6.00000 −0.264906
$$514$$ 0 0
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ 3.00000 0.130931
$$526$$ 0 0
$$527$$ −9.00000 −0.392046
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −3.00000 −0.130189
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ 5.00000 0.216169
$$536$$ 0 0
$$537$$ −8.00000 −0.345225
$$538$$ 0 0
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ −12.0000 −0.514969
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 24.0000 1.02617 0.513083 0.858339i $$-0.328503\pi$$
0.513083 + 0.858339i $$0.328503\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ 54.0000 2.30048
$$552$$ 0 0
$$553$$ −12.0000 −0.510292
$$554$$ 0 0
$$555$$ −3.00000 −0.127343
$$556$$ 0 0
$$557$$ −9.00000 −0.381342 −0.190671 0.981654i $$-0.561066\pi$$
−0.190671 + 0.981654i $$0.561066\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ 0 0
$$563$$ 13.0000 0.547885 0.273942 0.961746i $$-0.411672\pi$$
0.273942 + 0.961746i $$0.411672\pi$$
$$564$$ 0 0
$$565$$ 1.00000 0.0420703
$$566$$ 0 0
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ −20.0000 −0.835512
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −8.00000 −0.333044 −0.166522 0.986038i $$-0.553254\pi$$
−0.166522 + 0.986038i $$0.553254\pi$$
$$578$$ 0 0
$$579$$ −12.0000 −0.498703
$$580$$ 0 0
$$581$$ 9.00000 0.373383
$$582$$ 0 0
$$583$$ 54.0000 2.23645
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 0 0
$$589$$ 18.0000 0.741677
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 0 0
$$597$$ −14.0000 −0.572982
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 3.00000 0.122373 0.0611863 0.998126i $$-0.480512\pi$$
0.0611863 + 0.998126i $$0.480512\pi$$
$$602$$ 0 0
$$603$$ 3.00000 0.122169
$$604$$ 0 0
$$605$$ −25.0000 −1.01639
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ −27.0000 −1.09410
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 0 0
$$615$$ 3.00000 0.120972
$$616$$ 0 0
$$617$$ 45.0000 1.81163 0.905816 0.423672i $$-0.139259\pi$$
0.905816 + 0.423672i $$0.139259\pi$$
$$618$$ 0 0
$$619$$ −40.0000 −1.60774 −0.803868 0.594808i $$-0.797228\pi$$
−0.803868 + 0.594808i $$0.797228\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 36.0000 1.43770
$$628$$ 0 0
$$629$$ 9.00000 0.358854
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ 5.00000 0.198732
$$634$$ 0 0
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ −9.00000 −0.356034
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 0 0
$$643$$ 31.0000 1.22252 0.611260 0.791430i $$-0.290663\pi$$
0.611260 + 0.791430i $$0.290663\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ 18.0000 0.706562
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ 0 0
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 20.0000 0.781465
$$656$$ 0 0
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 0 0
$$663$$ −6.00000 −0.233021
$$664$$ 0 0
$$665$$ 18.0000 0.698010
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ 0 0
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ 48.0000 1.85302
$$672$$ 0 0
$$673$$ −8.00000 −0.308377 −0.154189 0.988041i $$-0.549276\pi$$
−0.154189 + 0.988041i $$0.549276\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 33.0000 1.26829 0.634147 0.773213i $$-0.281352\pi$$
0.634147 + 0.773213i $$0.281352\pi$$
$$678$$ 0 0
$$679$$ 54.0000 2.07233
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ 20.0000 0.763048
$$688$$ 0 0
$$689$$ 18.0000 0.685745
$$690$$ 0 0
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ 0 0
$$693$$ −18.0000 −0.683763
$$694$$ 0 0
$$695$$ 9.00000 0.341389
$$696$$ 0 0
$$697$$ −9.00000 −0.340899
$$698$$ 0 0
$$699$$ 20.0000 0.756469
$$700$$ 0 0
$$701$$ −44.0000 −1.66186 −0.830929 0.556379i $$-0.812190\pi$$
−0.830929 + 0.556379i $$0.812190\pi$$
$$702$$ 0 0
$$703$$ −18.0000 −0.678883
$$704$$ 0 0
$$705$$ −4.00000 −0.150649
$$706$$ 0 0
$$707$$ 33.0000 1.24109
$$708$$ 0 0
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 3.00000 0.112351
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 0 0
$$717$$ 1.00000 0.0373457
$$718$$ 0 0
$$719$$ −23.0000 −0.857755 −0.428878 0.903363i $$-0.641091\pi$$
−0.428878 + 0.903363i $$0.641091\pi$$
$$720$$ 0 0
$$721$$ 48.0000 1.78761
$$722$$ 0 0
$$723$$ −26.0000 −0.966950
$$724$$ 0 0
$$725$$ −9.00000 −0.334252
$$726$$ 0 0
$$727$$ −7.00000 −0.259616 −0.129808 0.991539i $$-0.541436\pi$$
−0.129808 + 0.991539i $$0.541436\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 43.0000 1.58824 0.794121 0.607760i $$-0.207932\pi$$
0.794121 + 0.607760i $$0.207932\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ −18.0000 −0.663039
$$738$$ 0 0
$$739$$ 5.00000 0.183928 0.0919640 0.995762i $$-0.470686\pi$$
0.0919640 + 0.995762i $$0.470686\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ 0 0
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ 0 0
$$747$$ 3.00000 0.109764
$$748$$ 0 0
$$749$$ −15.0000 −0.548088
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 0 0
$$753$$ −18.0000 −0.655956
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ 13.0000 0.472493 0.236247 0.971693i $$-0.424083\pi$$
0.236247 + 0.971693i $$0.424083\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ 49.0000 1.77625 0.888124 0.459603i $$-0.152008\pi$$
0.888124 + 0.459603i $$0.152008\pi$$
$$762$$ 0 0
$$763$$ −6.00000 −0.217215
$$764$$ 0 0
$$765$$ −3.00000 −0.108465
$$766$$ 0 0
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ −28.0000 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ 38.0000 1.36677 0.683383 0.730061i $$-0.260508\pi$$
0.683383 + 0.730061i $$0.260508\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 0 0
$$777$$ 9.00000 0.322873
$$778$$ 0 0
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ 54.0000 1.93227
$$782$$ 0 0
$$783$$ −9.00000 −0.321634
$$784$$ 0 0
$$785$$ −17.0000 −0.606756
$$786$$ 0 0
$$787$$ −49.0000 −1.74666 −0.873331 0.487128i $$-0.838045\pi$$
−0.873331 + 0.487128i $$0.838045\pi$$
$$788$$ 0 0
$$789$$ 21.0000 0.747620
$$790$$ 0 0
$$791$$ −3.00000 −0.106668
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ 0 0
$$795$$ 9.00000 0.319197
$$796$$ 0 0
$$797$$ 33.0000 1.16892 0.584460 0.811423i $$-0.301306\pi$$
0.584460 + 0.811423i $$0.301306\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 0 0
$$803$$ −36.0000 −1.27041
$$804$$ 0 0
$$805$$ 3.00000 0.105736
$$806$$ 0 0
$$807$$ 21.0000 0.739235
$$808$$ 0 0
$$809$$ 11.0000 0.386739 0.193370 0.981126i $$-0.438058\pi$$
0.193370 + 0.981126i $$0.438058\pi$$
$$810$$ 0 0
$$811$$ −19.0000 −0.667180 −0.333590 0.942718i $$-0.608260\pi$$
−0.333590 + 0.942718i $$0.608260\pi$$
$$812$$ 0 0
$$813$$ −5.00000 −0.175358
$$814$$ 0 0
$$815$$ 14.0000 0.490399
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ −6.00000 −0.208893
$$826$$ 0 0
$$827$$ −23.0000 −0.799788 −0.399894 0.916561i $$-0.630953\pi$$
−0.399894 + 0.916561i $$0.630953\pi$$
$$828$$ 0 0
$$829$$ 17.0000 0.590434 0.295217 0.955430i $$-0.404608\pi$$
0.295217 + 0.955430i $$0.404608\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ −3.00000 −0.103695
$$838$$ 0 0
$$839$$ 54.0000 1.86429 0.932144 0.362089i $$-0.117936\pi$$
0.932144 + 0.362089i $$0.117936\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 14.0000 0.482186
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 75.0000 2.57703
$$848$$ 0 0
$$849$$ 5.00000 0.171600
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 1.00000 0.0341196 0.0170598 0.999854i $$-0.494569\pi$$
0.0170598 + 0.999854i $$0.494569\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ 0 0
$$863$$ −46.0000 −1.56586 −0.782929 0.622111i $$-0.786275\pi$$
−0.782929 + 0.622111i $$0.786275\pi$$
$$864$$ 0 0
$$865$$ 4.00000 0.136004
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ −6.00000 −0.203302
$$872$$ 0 0
$$873$$ 18.0000 0.609208
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ 5.00000 0.168646
$$880$$ 0 0
$$881$$ −40.0000 −1.34763 −0.673817 0.738898i $$-0.735346\pi$$
−0.673817 + 0.738898i $$0.735346\pi$$
$$882$$ 0 0
$$883$$ −24.0000 −0.807664 −0.403832 0.914833i $$-0.632322\pi$$
−0.403832 + 0.914833i $$0.632322\pi$$
$$884$$ 0 0
$$885$$ 3.00000 0.100844
$$886$$ 0 0
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 0 0
$$893$$ −24.0000 −0.803129
$$894$$ 0 0
$$895$$ 8.00000 0.267411
$$896$$ 0 0
$$897$$ 2.00000 0.0667781
$$898$$ 0 0
$$899$$ 27.0000 0.900500
$$900$$ 0 0
$$901$$ −27.0000 −0.899500
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.0000 0.398893
$$906$$ 0 0
$$907$$ −5.00000 −0.166022 −0.0830111 0.996549i $$-0.526454\pi$$
−0.0830111 + 0.996549i $$0.526454\pi$$
$$908$$ 0 0
$$909$$ 11.0000 0.364847
$$910$$ 0 0
$$911$$ 52.0000 1.72284 0.861418 0.507896i $$-0.169577\pi$$
0.861418 + 0.507896i $$0.169577\pi$$
$$912$$ 0 0
$$913$$ −18.0000 −0.595713
$$914$$ 0 0
$$915$$ 8.00000 0.264472
$$916$$ 0 0
$$917$$ −60.0000 −1.98137
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 6.00000 0.197707
$$922$$ 0 0
$$923$$ 18.0000 0.592477
$$924$$ 0 0
$$925$$ 3.00000 0.0986394
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ 15.0000 0.492134 0.246067 0.969253i $$-0.420862\pi$$
0.246067 + 0.969253i $$0.420862\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 18.0000 0.588663
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −31.0000 −1.01165
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ 3.00000 0.0976934
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 0 0
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 0 0
$$955$$ 20.0000 0.647185
$$956$$ 0 0
$$957$$ 54.0000 1.74557
$$958$$ 0 0
$$959$$ −54.0000 −1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ −5.00000 −0.161123
$$964$$ 0 0
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ −18.0000 −0.578243
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −27.0000 −0.865580
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −23.0000 −0.735835 −0.367918 0.929858i $$-0.619929\pi$$
−0.367918 + 0.929858i $$0.619929\pi$$
$$978$$ 0 0
$$979$$ 48.0000 1.53409
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 0 0
$$983$$ 31.0000 0.988746 0.494373 0.869250i $$-0.335398\pi$$
0.494373 + 0.869250i $$0.335398\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 12.0000 0.381964
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −55.0000 −1.74713 −0.873566 0.486705i $$-0.838199\pi$$
−0.873566 + 0.486705i $$0.838199\pi$$
$$992$$ 0 0
$$993$$ 9.00000 0.285606
$$994$$ 0 0
$$995$$ 14.0000 0.443830
$$996$$ 0 0
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 0 0
$$999$$ 3.00000 0.0949158
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 2760.2.a.i.1.1 1
3.2 odd 2 8280.2.a.u.1.1 1
4.3 odd 2 5520.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.i.1.1 1 1.1 even 1 trivial
5520.2.a.c.1.1 1 4.3 odd 2
8280.2.a.u.1.1 1 3.2 odd 2