# Properties

 Label 7728.2.a.f.1.1 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $0$ 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] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.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: $$61.7083906820$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7728.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -6.00000 q^{13} -7.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} +10.0000 q^{31} +1.00000 q^{33} +6.00000 q^{37} +6.00000 q^{39} -5.00000 q^{41} -4.00000 q^{43} +1.00000 q^{47} +1.00000 q^{49} -3.00000 q^{53} +7.00000 q^{57} +3.00000 q^{59} +9.00000 q^{61} +1.00000 q^{63} +4.00000 q^{67} +1.00000 q^{69} +12.0000 q^{71} -4.00000 q^{73} +5.00000 q^{75} -1.00000 q^{77} +2.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +10.0000 q^{87} +18.0000 q^{89} -6.00000 q^{91} -10.0000 q^{93} -6.00000 q^{97} -1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.00000 0.145865 0.0729325 0.997337i $$-0.476764\pi$$
0.0729325 + 0.997337i $$0.476764\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 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
$$57$$ 7.00000 0.927173
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ 9.00000 1.15233 0.576166 0.817333i $$-0.304548\pi$$
0.576166 + 0.817333i $$0.304548\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 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
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ 5.00000 0.577350
$$76$$ 0 0
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 0 0
$$93$$ −10.0000 −1.03695
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −1.00000 −0.0995037 −0.0497519 0.998762i $$-0.515843\pi$$
−0.0497519 + 0.998762i $$0.515843\pi$$
$$102$$ 0 0
$$103$$ 19.0000 1.87213 0.936063 0.351833i $$-0.114441\pi$$
0.936063 + 0.351833i $$0.114441\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 20.0000 1.93347 0.966736 0.255774i $$-0.0823304\pi$$
0.966736 + 0.255774i $$0.0823304\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 5.00000 0.450835
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 1.00000 0.0873704 0.0436852 0.999045i $$-0.486090\pi$$
0.0436852 + 0.999045i $$0.486090\pi$$
$$132$$ 0 0
$$133$$ −7.00000 −0.606977
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 17.0000 1.45241 0.726204 0.687479i $$-0.241283\pi$$
0.726204 + 0.687479i $$0.241283\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −1.00000 −0.0842152
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ 5.00000 0.406894 0.203447 0.979086i $$-0.434786\pi$$
0.203447 + 0.979086i $$0.434786\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ 0 0
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ 1.00000 0.0783260 0.0391630 0.999233i $$-0.487531\pi$$
0.0391630 + 0.999233i $$0.487531\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.00000 0.386912 0.193456 0.981109i $$-0.438030\pi$$
0.193456 + 0.981109i $$0.438030\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ −5.00000 −0.377964
$$176$$ 0 0
$$177$$ −3.00000 −0.225494
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ −9.00000 −0.665299
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 0 0
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ −10.0000 −0.701862
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 7.00000 0.484200
$$210$$ 0 0
$$211$$ 25.0000 1.72107 0.860535 0.509390i $$-0.170129\pi$$
0.860535 + 0.509390i $$0.170129\pi$$
$$212$$ 0 0
$$213$$ −12.0000 −0.822226
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 10.0000 0.678844
$$218$$ 0 0
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ −6.00000 −0.398234 −0.199117 0.979976i $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ 0 0
$$229$$ −7.00000 −0.462573 −0.231287 0.972886i $$-0.574293\pi$$
−0.231287 + 0.972886i $$0.574293\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ 0 0
$$233$$ −16.0000 −1.04819 −0.524097 0.851658i $$-0.675597\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2.00000 −0.129914
$$238$$ 0 0
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 42.0000 2.67240
$$248$$ 0 0
$$249$$ 2.00000 0.126745
$$250$$ 0 0
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 0 0
$$253$$ 1.00000 0.0628695
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 15.0000 0.935674 0.467837 0.883815i $$-0.345033\pi$$
0.467837 + 0.883815i $$0.345033\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ 0 0
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −18.0000 −1.10158
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ 5.00000 0.301511
$$276$$ 0 0
$$277$$ 11.0000 0.660926 0.330463 0.943819i $$-0.392795\pi$$
0.330463 + 0.943819i $$0.392795\pi$$
$$278$$ 0 0
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −5.00000 −0.295141
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 0 0
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 0 0
$$303$$ 1.00000 0.0574485
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ −19.0000 −1.08087
$$310$$ 0 0
$$311$$ 15.0000 0.850572 0.425286 0.905059i $$-0.360174\pi$$
0.425286 + 0.905059i $$0.360174\pi$$
$$312$$ 0 0
$$313$$ −23.0000 −1.30004 −0.650018 0.759918i $$-0.725239\pi$$
−0.650018 + 0.759918i $$0.725239\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 10.0000 0.559893
$$320$$ 0 0
$$321$$ −20.0000 −1.11629
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 30.0000 1.66410
$$326$$ 0 0
$$327$$ 16.0000 0.884802
$$328$$ 0 0
$$329$$ 1.00000 0.0551318
$$330$$ 0 0
$$331$$ 25.0000 1.37412 0.687062 0.726599i $$-0.258900\pi$$
0.687062 + 0.726599i $$0.258900\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 0 0
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ 10.0000 0.524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ 0 0
$$369$$ −5.00000 −0.260290
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 60.0000 3.09016
$$378$$ 0 0
$$379$$ 38.0000 1.95193 0.975964 0.217930i $$-0.0699304\pi$$
0.975964 + 0.217930i $$0.0699304\pi$$
$$380$$ 0 0
$$381$$ 7.00000 0.358621
$$382$$ 0 0
$$383$$ −18.0000 −0.919757 −0.459879 0.887982i $$-0.652107\pi$$
−0.459879 + 0.887982i $$0.652107\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −1.00000 −0.0504433
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 0 0
$$399$$ 7.00000 0.350438
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 0 0
$$403$$ −60.0000 −2.98881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6.00000 −0.297409
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ −17.0000 −0.838548
$$412$$ 0 0
$$413$$ 3.00000 0.147620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ 0 0
$$423$$ 1.00000 0.0486217
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 9.00000 0.435541
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 11.0000 0.529851 0.264926 0.964269i $$-0.414653\pi$$
0.264926 + 0.964269i $$0.414653\pi$$
$$432$$ 0 0
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.00000 0.334855
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 9.00000 0.425685
$$448$$ 0 0
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ 5.00000 0.235441
$$452$$ 0 0
$$453$$ −5.00000 −0.234920
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ 5.00000 0.232370 0.116185 0.993228i $$-0.462933\pi$$
0.116185 + 0.993228i $$0.462933\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −5.00000 −0.230388
$$472$$ 0 0
$$473$$ 4.00000 0.183920
$$474$$ 0 0
$$475$$ 35.0000 1.60591
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 0 0
$$489$$ −1.00000 −0.0452216
$$490$$ 0 0
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −5.00000 −0.223384
$$502$$ 0 0
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −23.0000 −1.02147
$$508$$ 0 0
$$509$$ −1.00000 −0.0443242 −0.0221621 0.999754i $$-0.507055\pi$$
−0.0221621 + 0.999754i $$0.507055\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 0 0
$$513$$ 7.00000 0.309058
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.00000 −0.0439799
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 9.00000 0.393543 0.196771 0.980449i $$-0.436954\pi$$
0.196771 + 0.980449i $$0.436954\pi$$
$$524$$ 0 0
$$525$$ 5.00000 0.218218
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 3.00000 0.130189
$$532$$ 0 0
$$533$$ 30.0000 1.29944
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 20.0000 0.863064
$$538$$ 0 0
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ 7.00000 0.300954 0.150477 0.988614i $$-0.451919\pi$$
0.150477 + 0.988614i $$0.451919\pi$$
$$542$$ 0 0
$$543$$ −6.00000 −0.257485
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ 0 0
$$549$$ 9.00000 0.384111
$$550$$ 0 0
$$551$$ 70.0000 2.98210
$$552$$ 0 0
$$553$$ 2.00000 0.0850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10.0000 −0.423714 −0.211857 0.977301i $$-0.567951\pi$$
−0.211857 + 0.977301i $$0.567951\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −1.00000 −0.0419222 −0.0209611 0.999780i $$-0.506673\pi$$
−0.0209611 + 0.999780i $$0.506673\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 0 0
$$573$$ 15.0000 0.626634
$$574$$ 0 0
$$575$$ 5.00000 0.208514
$$576$$ 0 0
$$577$$ 6.00000 0.249783 0.124892 0.992170i $$-0.460142\pi$$
0.124892 + 0.992170i $$0.460142\pi$$
$$578$$ 0 0
$$579$$ 19.0000 0.789613
$$580$$ 0 0
$$581$$ −2.00000 −0.0829740
$$582$$ 0 0
$$583$$ 3.00000 0.124247
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 13.0000 0.536567 0.268284 0.963340i $$-0.413544\pi$$
0.268284 + 0.963340i $$0.413544\pi$$
$$588$$ 0 0
$$589$$ −70.0000 −2.88430
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −5.00000 −0.204636
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −20.0000 −0.811775 −0.405887 0.913923i $$-0.633038\pi$$
−0.405887 + 0.913923i $$0.633038\pi$$
$$608$$ 0 0
$$609$$ 10.0000 0.405220
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ 42.0000 1.69636 0.848182 0.529705i $$-0.177697\pi$$
0.848182 + 0.529705i $$0.177697\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 18.0000 0.721155
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ −7.00000 −0.279553
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −18.0000 −0.716569 −0.358284 0.933613i $$-0.616638\pi$$
−0.358284 + 0.933613i $$0.616638\pi$$
$$632$$ 0 0
$$633$$ −25.0000 −0.993661
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −1.00000 −0.0394976 −0.0197488 0.999805i $$-0.506287\pi$$
−0.0197488 + 0.999805i $$0.506287\pi$$
$$642$$ 0 0
$$643$$ 3.00000 0.118308 0.0591542 0.998249i $$-0.481160\pi$$
0.0591542 + 0.998249i $$0.481160\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ −3.00000 −0.117760
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ 0 0
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 32.0000 1.24654 0.623272 0.782006i $$-0.285803\pi$$
0.623272 + 0.782006i $$0.285803\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.0000 0.387202
$$668$$ 0 0
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ −9.00000 −0.347441
$$672$$ 0 0
$$673$$ 33.0000 1.27206 0.636028 0.771666i $$-0.280576\pi$$
0.636028 + 0.771666i $$0.280576\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ 28.0000 1.07613 0.538064 0.842904i $$-0.319156\pi$$
0.538064 + 0.842904i $$0.319156\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 6.00000 0.229920
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 7.00000 0.267067
$$688$$ 0 0
$$689$$ 18.0000 0.685745
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ −1.00000 −0.0379869
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 16.0000 0.605176
$$700$$ 0 0
$$701$$ −3.00000 −0.113308 −0.0566542 0.998394i $$-0.518043\pi$$
−0.0566542 + 0.998394i $$0.518043\pi$$
$$702$$ 0 0
$$703$$ −42.0000 −1.58406
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.00000 −0.0376089
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ 2.00000 0.0750059
$$712$$ 0 0
$$713$$ −10.0000 −0.374503
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −4.00000 −0.149383
$$718$$ 0 0
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ 19.0000 0.707597
$$722$$ 0 0
$$723$$ −7.00000 −0.260333
$$724$$ 0 0
$$725$$ 50.0000 1.85695
$$726$$ 0 0
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −42.0000 −1.54291
$$742$$ 0 0
$$743$$ −49.0000 −1.79764 −0.898818 0.438322i $$-0.855573\pi$$
−0.898818 + 0.438322i $$0.855573\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −2.00000 −0.0731762
$$748$$ 0 0
$$749$$ 20.0000 0.730784
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 10.0000 0.364420
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −20.0000 −0.726912 −0.363456 0.931611i $$-0.618403\pi$$
−0.363456 + 0.931611i $$0.618403\pi$$
$$758$$ 0 0
$$759$$ −1.00000 −0.0362977
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −18.0000 −0.649942
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −15.0000 −0.540212
$$772$$ 0 0
$$773$$ 52.0000 1.87031 0.935155 0.354239i $$-0.115260\pi$$
0.935155 + 0.354239i $$0.115260\pi$$
$$774$$ 0 0
$$775$$ −50.0000 −1.79605
$$776$$ 0 0
$$777$$ −6.00000 −0.215249
$$778$$ 0 0
$$779$$ 35.0000 1.25401
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ 0 0
$$783$$ 10.0000 0.357371
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −5.00000 −0.178231 −0.0891154 0.996021i $$-0.528404\pi$$
−0.0891154 + 0.996021i $$0.528404\pi$$
$$788$$ 0 0
$$789$$ 9.00000 0.320408
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −54.0000 −1.91760
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ 4.00000 0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ 34.0000 1.19538 0.597688 0.801729i $$-0.296086\pi$$
0.597688 + 0.801729i $$0.296086\pi$$
$$810$$ 0 0
$$811$$ 6.00000 0.210688 0.105344 0.994436i $$-0.466406\pi$$
0.105344 + 0.994436i $$0.466406\pi$$
$$812$$ 0 0
$$813$$ 24.0000 0.841717
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 28.0000 0.979596
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ 0 0
$$823$$ −21.0000 −0.732014 −0.366007 0.930612i $$-0.619275\pi$$
−0.366007 + 0.930612i $$0.619275\pi$$
$$824$$ 0 0
$$825$$ −5.00000 −0.174078
$$826$$ 0 0
$$827$$ −33.0000 −1.14752 −0.573761 0.819023i $$-0.694516\pi$$
−0.573761 + 0.819023i $$0.694516\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ −11.0000 −0.381586
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 0 0
$$839$$ −46.0000 −1.58810 −0.794048 0.607855i $$-0.792030\pi$$
−0.794048 + 0.607855i $$0.792030\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ 22.0000 0.757720
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −10.0000 −0.343604
$$848$$ 0 0
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ −24.0000 −0.821744 −0.410872 0.911693i $$-0.634776\pi$$
−0.410872 + 0.911693i $$0.634776\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −9.00000 −0.307434 −0.153717 0.988115i $$-0.549124\pi$$
−0.153717 + 0.988115i $$0.549124\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 5.00000 0.170400
$$862$$ 0 0
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ −2.00000 −0.0678454
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 0 0
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −29.0000 −0.979260 −0.489630 0.871930i $$-0.662868\pi$$
−0.489630 + 0.871930i $$0.662868\pi$$
$$878$$ 0 0
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ 8.00000 0.269221 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ −7.00000 −0.234772
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ −7.00000 −0.234246
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.00000 −0.200334
$$898$$ 0 0
$$899$$ −100.000 −3.33519
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 4.00000 0.133112
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ 0 0
$$909$$ −1.00000 −0.0331679
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 2.00000 0.0661903
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.00000 0.0330229
$$918$$ 0 0
$$919$$ 10.0000 0.329870 0.164935 0.986304i $$-0.447259\pi$$
0.164935 + 0.986304i $$0.447259\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ −72.0000 −2.36991
$$924$$ 0 0
$$925$$ −30.0000 −0.986394
$$926$$ 0 0
$$927$$ 19.0000 0.624042
$$928$$ 0 0
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ 0 0
$$933$$ −15.0000 −0.491078
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.00000 0.0980057 0.0490029 0.998799i $$-0.484396\pi$$
0.0490029 + 0.998799i $$0.484396\pi$$
$$938$$ 0 0
$$939$$ 23.0000 0.750577
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ 24.0000 0.779073
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ 41.0000 1.32812 0.664060 0.747679i $$-0.268832\pi$$
0.664060 + 0.747679i $$0.268832\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −10.0000 −0.323254
$$958$$ 0 0
$$959$$ 17.0000 0.548959
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ 20.0000 0.644491
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −54.0000 −1.73294 −0.866471 0.499227i $$-0.833617\pi$$
−0.866471 + 0.499227i $$0.833617\pi$$
$$972$$ 0 0
$$973$$ 12.0000 0.384702
$$974$$ 0 0
$$975$$ −30.0000 −0.960769
$$976$$ 0 0
$$977$$ −35.0000 −1.11975 −0.559875 0.828577i $$-0.689151\pi$$
−0.559875 + 0.828577i $$0.689151\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −1.00000 −0.0318304
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 45.0000 1.42947 0.714736 0.699394i $$-0.246547\pi$$
0.714736 + 0.699394i $$0.246547\pi$$
$$992$$ 0 0
$$993$$ −25.0000 −0.793351
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ 0 0
$$999$$ −6.00000 −0.189832
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 7728.2.a.f.1.1 1
4.3 odd 2 3864.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.c.1.1 1 4.3 odd 2
7728.2.a.f.1.1 1 1.1 even 1 trivial