# Properties

 Label 7360.2.a.u.1.1 Level $7360$ Weight $2$ Character 7360.1 Self dual yes Analytic conductor $58.770$ 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] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.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: $$58.7698958877$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7360.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} -1.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} -4.00000 q^{19} +2.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +1.00000 q^{31} +2.00000 q^{35} +8.00000 q^{37} -1.00000 q^{39} -5.00000 q^{41} -6.00000 q^{43} -2.00000 q^{45} -9.00000 q^{47} -3.00000 q^{49} -4.00000 q^{51} -2.00000 q^{53} -4.00000 q^{57} -4.00000 q^{63} -1.00000 q^{65} +4.00000 q^{67} -1.00000 q^{69} -3.00000 q^{71} +7.00000 q^{73} +1.00000 q^{75} -4.00000 q^{79} +1.00000 q^{81} +8.00000 q^{83} -4.00000 q^{85} +3.00000 q^{87} -14.0000 q^{89} -2.00000 q^{91} +1.00000 q^{93} -4.00000 q^{95} -14.0000 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ −4.00000 −0.503953
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 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
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 1.00000 0.103695
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −5.00000 −0.450835
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 1.00000 0.0848189 0.0424094 0.999100i $$-0.486497\pi$$
0.0424094 + 0.999100i $$0.486497\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.00000 0.249136
$$146$$ 0 0
$$147$$ −3.00000 −0.247436
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 13.0000 1.05792 0.528962 0.848645i $$-0.322581\pi$$
0.528962 + 0.848645i $$0.322581\pi$$
$$152$$ 0 0
$$153$$ 8.00000 0.646762
$$154$$ 0 0
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −24.0000 −1.85718 −0.928588 0.371113i $$-0.878976\pi$$
−0.928588 + 0.371113i $$0.878976\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 0 0
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 8.00000 0.588172
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −10.0000 −0.727393
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ −9.00000 −0.647834 −0.323917 0.946085i $$-0.605000\pi$$
−0.323917 + 0.946085i $$0.605000\pi$$
$$194$$ 0 0
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ 13.0000 0.926212 0.463106 0.886303i $$-0.346735\pi$$
0.463106 + 0.886303i $$0.346735\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 6.00000 0.421117
$$204$$ 0 0
$$205$$ −5.00000 −0.349215
$$206$$ 0 0
$$207$$ 2.00000 0.139010
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ −3.00000 −0.205557
$$214$$ 0 0
$$215$$ −6.00000 −0.409197
$$216$$ 0 0
$$217$$ 2.00000 0.135769
$$218$$ 0 0
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ 10.0000 0.663723 0.331862 0.943328i $$-0.392323\pi$$
0.331862 + 0.943328i $$0.392323\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 0 0
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 0 0
$$259$$ 16.0000 0.994192
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ −14.0000 −0.856786
$$268$$ 0 0
$$269$$ −25.0000 −1.52428 −0.762138 0.647414i $$-0.775850\pi$$
−0.762138 + 0.647414i $$0.775850\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3.00000 −0.180253 −0.0901263 0.995930i $$-0.528727\pi$$
−0.0901263 + 0.995930i $$0.528727\pi$$
$$278$$ 0 0
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 0 0
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ −10.0000 −0.590281
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 0 0
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ −2.00000 −0.114897
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 9.00000 0.510343 0.255172 0.966896i $$-0.417868\pi$$
0.255172 + 0.966896i $$0.417868\pi$$
$$312$$ 0 0
$$313$$ −4.00000 −0.226093 −0.113047 0.993590i $$-0.536061\pi$$
−0.113047 + 0.993590i $$0.536061\pi$$
$$314$$ 0 0
$$315$$ −4.00000 −0.225374
$$316$$ 0 0
$$317$$ −26.0000 −1.46031 −0.730153 0.683284i $$-0.760551\pi$$
−0.730153 + 0.683284i $$0.760551\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 16.0000 0.890264
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ −18.0000 −0.992372
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 0 0
$$333$$ −16.0000 −0.876795
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −10.0000 −0.544735 −0.272367 0.962193i $$-0.587807\pi$$
−0.272367 + 0.962193i $$0.587807\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 0 0
$$353$$ 3.00000 0.159674 0.0798369 0.996808i $$-0.474560\pi$$
0.0798369 + 0.996808i $$0.474560\pi$$
$$354$$ 0 0
$$355$$ −3.00000 −0.159223
$$356$$ 0 0
$$357$$ −8.00000 −0.423405
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ 7.00000 0.366397
$$366$$ 0 0
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ −7.00000 −0.358621
$$382$$ 0 0
$$383$$ −2.00000 −0.102195 −0.0510976 0.998694i $$-0.516272\pi$$
−0.0510976 + 0.998694i $$0.516272\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 12.0000 0.609994
$$388$$ 0 0
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ 3.00000 0.151330
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ 0 0
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ −1.00000 −0.0498135
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ 1.00000 0.0489702
$$418$$ 0 0
$$419$$ −6.00000 −0.293119 −0.146560 0.989202i $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ 18.0000 0.875190
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 3.00000 0.143839
$$436$$ 0 0
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 6.00000 0.285714
$$442$$ 0 0
$$443$$ 5.00000 0.237557 0.118779 0.992921i $$-0.462102\pi$$
0.118779 + 0.992921i $$0.462102\pi$$
$$444$$ 0 0
$$445$$ −14.0000 −0.663664
$$446$$ 0 0
$$447$$ −18.0000 −0.851371
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 13.0000 0.610793
$$454$$ 0 0
$$455$$ −2.00000 −0.0937614
$$456$$ 0 0
$$457$$ −40.0000 −1.87112 −0.935561 0.353166i $$-0.885105\pi$$
−0.935561 + 0.353166i $$0.885105\pi$$
$$458$$ 0 0
$$459$$ 20.0000 0.933520
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ 0 0
$$465$$ 1.00000 0.0463739
$$466$$ 0 0
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 4.00000 0.183147
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ −2.00000 −0.0910032
$$484$$ 0 0
$$485$$ −14.0000 −0.635707
$$486$$ 0 0
$$487$$ 5.00000 0.226572 0.113286 0.993562i $$-0.463862\pi$$
0.113286 + 0.993562i $$0.463862\pi$$
$$488$$ 0 0
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ −27.0000 −1.21849 −0.609246 0.792981i $$-0.708528\pi$$
−0.609246 + 0.792981i $$0.708528\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ 19.0000 0.850557 0.425278 0.905063i $$-0.360176\pi$$
0.425278 + 0.905063i $$0.360176\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ −2.00000 −0.0889988
$$506$$ 0 0
$$507$$ −12.0000 −0.532939
$$508$$ 0 0
$$509$$ 19.0000 0.842160 0.421080 0.907023i $$-0.361651\pi$$
0.421080 + 0.907023i $$0.361651\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 0 0
$$513$$ 20.0000 0.883022
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ −38.0000 −1.66162 −0.830812 0.556553i $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$524$$ 0 0
$$525$$ 2.00000 0.0872872
$$526$$ 0 0
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 5.00000 0.216574
$$534$$ 0 0
$$535$$ 4.00000 0.172935
$$536$$ 0 0
$$537$$ 9.00000 0.388379
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.0000 −0.730887 −0.365444 0.930834i $$-0.619083\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 0 0
$$543$$ 12.0000 0.514969
$$544$$ 0 0
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ −1.00000 −0.0427569 −0.0213785 0.999771i $$-0.506805\pi$$
−0.0213785 + 0.999771i $$0.506805\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ 8.00000 0.339581
$$556$$ 0 0
$$557$$ −34.0000 −1.44063 −0.720313 0.693649i $$-0.756002\pi$$
−0.720313 + 0.693649i $$0.756002\pi$$
$$558$$ 0 0
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 2.00000 0.0839921
$$568$$ 0 0
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 0 0
$$571$$ 24.0000 1.00437 0.502184 0.864761i $$-0.332530\pi$$
0.502184 + 0.864761i $$0.332530\pi$$
$$572$$ 0 0
$$573$$ 10.0000 0.417756
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 23.0000 0.957503 0.478751 0.877951i $$-0.341090\pi$$
0.478751 + 0.877951i $$0.341090\pi$$
$$578$$ 0 0
$$579$$ −9.00000 −0.374027
$$580$$ 0 0
$$581$$ 16.0000 0.663792
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ 29.0000 1.19696 0.598479 0.801138i $$-0.295772\pi$$
0.598479 + 0.801138i $$0.295772\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 13.0000 0.534749
$$592$$ 0 0
$$593$$ 26.0000 1.06769 0.533846 0.845582i $$-0.320746\pi$$
0.533846 + 0.845582i $$0.320746\pi$$
$$594$$ 0 0
$$595$$ −8.00000 −0.327968
$$596$$ 0 0
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 45.0000 1.83559 0.917794 0.397057i $$-0.129968\pi$$
0.917794 + 0.397057i $$0.129968\pi$$
$$602$$ 0 0
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ −11.0000 −0.447214
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ 9.00000 0.364101
$$612$$ 0 0
$$613$$ 4.00000 0.161558 0.0807792 0.996732i $$-0.474259\pi$$
0.0807792 + 0.996732i $$0.474259\pi$$
$$614$$ 0 0
$$615$$ −5.00000 −0.201619
$$616$$ 0 0
$$617$$ −28.0000 −1.12724 −0.563619 0.826035i $$-0.690591\pi$$
−0.563619 + 0.826035i $$0.690591\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 5.00000 0.200643
$$622$$ 0 0
$$623$$ −28.0000 −1.12180
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −32.0000 −1.27592
$$630$$ 0 0
$$631$$ −26.0000 −1.03504 −0.517522 0.855670i $$-0.673145\pi$$
−0.517522 + 0.855670i $$0.673145\pi$$
$$632$$ 0 0
$$633$$ −8.00000 −0.317971
$$634$$ 0 0
$$635$$ −7.00000 −0.277787
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ −2.00000 −0.0788723 −0.0394362 0.999222i $$-0.512556\pi$$
−0.0394362 + 0.999222i $$0.512556\pi$$
$$644$$ 0 0
$$645$$ −6.00000 −0.236250
$$646$$ 0 0
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 2.00000 0.0783862
$$652$$ 0 0
$$653$$ 35.0000 1.36966 0.684828 0.728705i $$-0.259877\pi$$
0.684828 + 0.728705i $$0.259877\pi$$
$$654$$ 0 0
$$655$$ 3.00000 0.117220
$$656$$ 0 0
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 0 0
$$663$$ 4.00000 0.155347
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ −3.00000 −0.116160
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −43.0000 −1.65753 −0.828764 0.559598i $$-0.810955\pi$$
−0.828764 + 0.559598i $$0.810955\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ 0 0
$$679$$ −28.0000 −1.07454
$$680$$ 0 0
$$681$$ 10.0000 0.383201
$$682$$ 0 0
$$683$$ −9.00000 −0.344375 −0.172188 0.985064i $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ 0 0
$$689$$ 2.00000 0.0761939
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.00000 0.0379322
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 0 0
$$699$$ 21.0000 0.794293
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ −32.0000 −1.20690
$$704$$ 0 0
$$705$$ −9.00000 −0.338960
$$706$$ 0 0
$$707$$ −4.00000 −0.150435
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ −1.00000 −0.0374503
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 21.0000 0.784259
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ 22.0000 0.818189
$$724$$ 0 0
$$725$$ 3.00000 0.111417
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 28.0000 1.03420 0.517102 0.855924i $$-0.327011\pi$$
0.517102 + 0.855924i $$0.327011\pi$$
$$734$$ 0 0
$$735$$ −3.00000 −0.110657
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 0 0
$$747$$ −16.0000 −0.585409
$$748$$ 0 0
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ −2.00000 −0.0729810 −0.0364905 0.999334i $$-0.511618\pi$$
−0.0364905 + 0.999334i $$0.511618\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 13.0000 0.473118
$$756$$ 0 0
$$757$$ 40.0000 1.45382 0.726912 0.686730i $$-0.240955\pi$$
0.726912 + 0.686730i $$0.240955\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 5.00000 0.181250 0.0906249 0.995885i $$-0.471114\pi$$
0.0906249 + 0.995885i $$0.471114\pi$$
$$762$$ 0 0
$$763$$ −4.00000 −0.144810
$$764$$ 0 0
$$765$$ 8.00000 0.289241
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ −3.00000 −0.108042
$$772$$ 0 0
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 0 0
$$775$$ 1.00000 0.0359211
$$776$$ 0 0
$$777$$ 16.0000 0.573997
$$778$$ 0 0
$$779$$ 20.0000 0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −15.0000 −0.536056
$$784$$ 0 0
$$785$$ 2.00000 0.0713831
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 0 0
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −2.00000 −0.0709327
$$796$$ 0 0
$$797$$ 24.0000 0.850124 0.425062 0.905164i $$-0.360252\pi$$
0.425062 + 0.905164i $$0.360252\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ 0 0
$$801$$ 28.0000 0.989331
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −2.00000 −0.0704907
$$806$$ 0 0
$$807$$ −25.0000 −0.880042
$$808$$ 0 0
$$809$$ −46.0000 −1.61727 −0.808637 0.588308i $$-0.799794\pi$$
−0.808637 + 0.588308i $$0.799794\pi$$
$$810$$ 0 0
$$811$$ 39.0000 1.36948 0.684738 0.728790i $$-0.259917\pi$$
0.684738 + 0.728790i $$0.259917\pi$$
$$812$$ 0 0
$$813$$ 28.0000 0.982003
$$814$$ 0 0
$$815$$ 11.0000 0.385313
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ 0 0
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 0 0
$$823$$ 3.00000 0.104573 0.0522867 0.998632i $$-0.483349\pi$$
0.0522867 + 0.998632i $$0.483349\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ −3.00000 −0.104069
$$832$$ 0 0
$$833$$ 12.0000 0.415775
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ 0 0
$$837$$ −5.00000 −0.172825
$$838$$ 0 0
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 6.00000 0.206651
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ −22.0000 −0.755929
$$848$$ 0 0
$$849$$ −10.0000 −0.343199
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ −7.00000 −0.239115 −0.119558 0.992827i $$-0.538148\pi$$
−0.119558 + 0.992827i $$0.538148\pi$$
$$858$$ 0 0
$$859$$ −45.0000 −1.53538 −0.767690 0.640821i $$-0.778594\pi$$
−0.767690 + 0.640821i $$0.778594\pi$$
$$860$$ 0 0
$$861$$ −10.0000 −0.340799
$$862$$ 0 0
$$863$$ −11.0000 −0.374444 −0.187222 0.982318i $$-0.559948\pi$$
−0.187222 + 0.982318i $$0.559948\pi$$
$$864$$ 0 0
$$865$$ −14.0000 −0.476014
$$866$$ 0 0
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ 28.0000 0.947656
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 0 0
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ 56.0000 1.88669 0.943344 0.331816i $$-0.107661\pi$$
0.943344 + 0.331816i $$0.107661\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 3.00000 0.100730 0.0503651 0.998731i $$-0.483962\pi$$
0.0503651 + 0.998731i $$0.483962\pi$$
$$888$$ 0 0
$$889$$ −14.0000 −0.469545
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 36.0000 1.20469
$$894$$ 0 0
$$895$$ 9.00000 0.300837
$$896$$ 0 0
$$897$$ 1.00000 0.0333890
$$898$$ 0 0
$$899$$ 3.00000 0.100056
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ 0 0
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ 12.0000 0.398893
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 0 0
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 6.00000 0.198137
$$918$$ 0 0
$$919$$ −46.0000 −1.51740 −0.758700 0.651440i $$-0.774165\pi$$
−0.758700 + 0.651440i $$0.774165\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 3.00000 0.0987462
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ −45.0000 −1.47640 −0.738201 0.674581i $$-0.764324\pi$$
−0.738201 + 0.674581i $$0.764324\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 0 0
$$933$$ 9.00000 0.294647
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 36.0000 1.17607 0.588034 0.808836i $$-0.299902\pi$$
0.588034 + 0.808836i $$0.299902\pi$$
$$938$$ 0 0
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ −48.0000 −1.56476 −0.782378 0.622804i $$-0.785993\pi$$
−0.782378 + 0.622804i $$0.785993\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 0 0
$$945$$ −10.0000 −0.325300
$$946$$ 0 0
$$947$$ 45.0000 1.46230 0.731152 0.682215i $$-0.238983\pi$$
0.731152 + 0.682215i $$0.238983\pi$$
$$948$$ 0 0
$$949$$ −7.00000 −0.227230
$$950$$ 0 0
$$951$$ −26.0000 −0.843108
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 10.0000 0.323592
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ 0 0
$$965$$ −9.00000 −0.289720
$$966$$ 0 0
$$967$$ −23.0000 −0.739630 −0.369815 0.929105i $$-0.620579\pi$$
−0.369815 + 0.929105i $$0.620579\pi$$
$$968$$ 0 0
$$969$$ 16.0000 0.513994
$$970$$ 0 0
$$971$$ 50.0000 1.60458 0.802288 0.596937i $$-0.203616\pi$$
0.802288 + 0.596937i $$0.203616\pi$$
$$972$$ 0 0
$$973$$ 2.00000 0.0641171
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 30.0000 0.956851 0.478426 0.878128i $$-0.341208\pi$$
0.478426 + 0.878128i $$0.341208\pi$$
$$984$$ 0 0
$$985$$ 13.0000 0.414214
$$986$$ 0 0
$$987$$ −18.0000 −0.572946
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 0 0
$$993$$ −19.0000 −0.602947
$$994$$ 0 0
$$995$$ 20.0000 0.634043
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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 7360.2.a.u.1.1 1
4.3 odd 2 7360.2.a.j.1.1 1
8.3 odd 2 920.2.a.d.1.1 1
8.5 even 2 1840.2.a.b.1.1 1
24.11 even 2 8280.2.a.o.1.1 1
40.3 even 4 4600.2.e.g.4049.2 2
40.19 odd 2 4600.2.a.f.1.1 1
40.27 even 4 4600.2.e.g.4049.1 2
40.29 even 2 9200.2.a.z.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.d.1.1 1 8.3 odd 2
1840.2.a.b.1.1 1 8.5 even 2
4600.2.a.f.1.1 1 40.19 odd 2
4600.2.e.g.4049.1 2 40.27 even 4
4600.2.e.g.4049.2 2 40.3 even 4
7360.2.a.j.1.1 1 4.3 odd 2
7360.2.a.u.1.1 1 1.1 even 1 trivial
8280.2.a.o.1.1 1 24.11 even 2
9200.2.a.z.1.1 1 40.29 even 2