# Properties

 Label 6080.2.a.bd.1.1 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3040) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -2.82843 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{5} -2.82843 q^{7} -3.00000 q^{9} -4.00000 q^{11} +0.828427 q^{13} -3.65685 q^{17} +1.00000 q^{19} -2.82843 q^{23} +1.00000 q^{25} -7.65685 q^{29} +5.65685 q^{31} +2.82843 q^{35} -10.4853 q^{37} +7.65685 q^{41} +8.48528 q^{43} +3.00000 q^{45} -10.8284 q^{47} +1.00000 q^{49} -12.8284 q^{53} +4.00000 q^{55} -1.65685 q^{59} -6.00000 q^{61} +8.48528 q^{63} -0.828427 q^{65} -11.3137 q^{67} +5.65685 q^{71} +4.34315 q^{73} +11.3137 q^{77} -5.65685 q^{79} +9.00000 q^{81} +10.8284 q^{83} +3.65685 q^{85} -0.343146 q^{89} -2.34315 q^{91} -1.00000 q^{95} +12.8284 q^{97} +12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 6 * q^9 $$2 q - 2 q^{5} - 6 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 4 q^{29} - 4 q^{37} + 4 q^{41} + 6 q^{45} - 16 q^{47} + 2 q^{49} - 20 q^{53} + 8 q^{55} + 8 q^{59} - 12 q^{61} + 4 q^{65} + 20 q^{73} + 18 q^{81} + 16 q^{83} - 4 q^{85} - 12 q^{89} - 16 q^{91} - 2 q^{95} + 20 q^{97} + 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 - 6 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 4 * q^29 - 4 * q^37 + 4 * q^41 + 6 * q^45 - 16 * q^47 + 2 * q^49 - 20 * q^53 + 8 * q^55 + 8 * q^59 - 12 * q^61 + 4 * q^65 + 20 * q^73 + 18 * q^81 + 16 * q^83 - 4 * q^85 - 12 * q^89 - 16 * q^91 - 2 * q^95 + 20 * q^97 + 24 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 0.828427 0.229764 0.114882 0.993379i $$-0.463351\pi$$
0.114882 + 0.993379i $$0.463351\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.65685 −0.886917 −0.443459 0.896295i $$-0.646249\pi$$
−0.443459 + 0.896295i $$0.646249\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.65685 −1.42184 −0.710921 0.703272i $$-0.751722\pi$$
−0.710921 + 0.703272i $$0.751722\pi$$
$$30$$ 0 0
$$31$$ 5.65685 1.01600 0.508001 0.861357i $$-0.330385\pi$$
0.508001 + 0.861357i $$0.330385\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.82843 0.478091
$$36$$ 0 0
$$37$$ −10.4853 −1.72377 −0.861885 0.507104i $$-0.830716\pi$$
−0.861885 + 0.507104i $$0.830716\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.65685 1.19580 0.597900 0.801571i $$-0.296002\pi$$
0.597900 + 0.801571i $$0.296002\pi$$
$$42$$ 0 0
$$43$$ 8.48528 1.29399 0.646997 0.762493i $$-0.276025\pi$$
0.646997 + 0.762493i $$0.276025\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ −10.8284 −1.57949 −0.789744 0.613436i $$-0.789787\pi$$
−0.789744 + 0.613436i $$0.789787\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −12.8284 −1.76212 −0.881060 0.473005i $$-0.843169\pi$$
−0.881060 + 0.473005i $$0.843169\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −1.65685 −0.215704 −0.107852 0.994167i $$-0.534397\pi$$
−0.107852 + 0.994167i $$0.534397\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ 8.48528 1.06904
$$64$$ 0 0
$$65$$ −0.828427 −0.102754
$$66$$ 0 0
$$67$$ −11.3137 −1.38219 −0.691095 0.722764i $$-0.742871\pi$$
−0.691095 + 0.722764i $$0.742871\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ 4.34315 0.508327 0.254163 0.967161i $$-0.418200\pi$$
0.254163 + 0.967161i $$0.418200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 11.3137 1.28932
$$78$$ 0 0
$$79$$ −5.65685 −0.636446 −0.318223 0.948016i $$-0.603086\pi$$
−0.318223 + 0.948016i $$0.603086\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 10.8284 1.18857 0.594287 0.804253i $$-0.297434\pi$$
0.594287 + 0.804253i $$0.297434\pi$$
$$84$$ 0 0
$$85$$ 3.65685 0.396642
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −0.343146 −0.0363734 −0.0181867 0.999835i $$-0.505789\pi$$
−0.0181867 + 0.999835i $$0.505789\pi$$
$$90$$ 0 0
$$91$$ −2.34315 −0.245628
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 12.8284 1.30253 0.651265 0.758851i $$-0.274239\pi$$
0.651265 + 0.758851i $$0.274239\pi$$
$$98$$ 0 0
$$99$$ 12.0000 1.20605
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 6.34315 0.625009 0.312504 0.949916i $$-0.398832\pi$$
0.312504 + 0.949916i $$0.398832\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.34315 0.226520 0.113260 0.993565i $$-0.463871\pi$$
0.113260 + 0.993565i $$0.463871\pi$$
$$108$$ 0 0
$$109$$ 17.3137 1.65835 0.829176 0.558987i $$-0.188810\pi$$
0.829176 + 0.558987i $$0.188810\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.48528 −0.610084 −0.305042 0.952339i $$-0.598670\pi$$
−0.305042 + 0.952339i $$0.598670\pi$$
$$114$$ 0 0
$$115$$ 2.82843 0.263752
$$116$$ 0 0
$$117$$ −2.48528 −0.229764
$$118$$ 0 0
$$119$$ 10.3431 0.948155
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 20.9706 1.86084 0.930418 0.366499i $$-0.119444\pi$$
0.930418 + 0.366499i $$0.119444\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1.65685 −0.144760 −0.0723800 0.997377i $$-0.523059\pi$$
−0.0723800 + 0.997377i $$0.523059\pi$$
$$132$$ 0 0
$$133$$ −2.82843 −0.245256
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.34315 0.371060 0.185530 0.982639i $$-0.440600\pi$$
0.185530 + 0.982639i $$0.440600\pi$$
$$138$$ 0 0
$$139$$ −20.9706 −1.77870 −0.889350 0.457227i $$-0.848843\pi$$
−0.889350 + 0.457227i $$0.848843\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.31371 −0.277106
$$144$$ 0 0
$$145$$ 7.65685 0.635867
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 19.3137 1.57173 0.785864 0.618400i $$-0.212219\pi$$
0.785864 + 0.618400i $$0.212219\pi$$
$$152$$ 0 0
$$153$$ 10.9706 0.886917
$$154$$ 0 0
$$155$$ −5.65685 −0.454369
$$156$$ 0 0
$$157$$ 4.34315 0.346621 0.173310 0.984867i $$-0.444554\pi$$
0.173310 + 0.984867i $$0.444554\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ 13.1716 1.03168 0.515839 0.856686i $$-0.327480\pi$$
0.515839 + 0.856686i $$0.327480\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0.686292 0.0531068 0.0265534 0.999647i $$-0.491547\pi$$
0.0265534 + 0.999647i $$0.491547\pi$$
$$168$$ 0 0
$$169$$ −12.3137 −0.947208
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ 0 0
$$173$$ −16.1421 −1.22726 −0.613632 0.789592i $$-0.710292\pi$$
−0.613632 + 0.789592i $$0.710292\pi$$
$$174$$ 0 0
$$175$$ −2.82843 −0.213809
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 20.9706 1.56741 0.783707 0.621131i $$-0.213327\pi$$
0.783707 + 0.621131i $$0.213327\pi$$
$$180$$ 0 0
$$181$$ 9.31371 0.692283 0.346141 0.938182i $$-0.387492\pi$$
0.346141 + 0.938182i $$0.387492\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.4853 0.770893
$$186$$ 0 0
$$187$$ 14.6274 1.06966
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.9706 −1.51738 −0.758688 0.651454i $$-0.774159\pi$$
−0.758688 + 0.651454i $$0.774159\pi$$
$$192$$ 0 0
$$193$$ −3.17157 −0.228295 −0.114147 0.993464i $$-0.536414\pi$$
−0.114147 + 0.993464i $$0.536414\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 21.3137 1.51854 0.759269 0.650776i $$-0.225556\pi$$
0.759269 + 0.650776i $$0.225556\pi$$
$$198$$ 0 0
$$199$$ 15.3137 1.08556 0.542780 0.839875i $$-0.317372\pi$$
0.542780 + 0.839875i $$0.317372\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 21.6569 1.52001
$$204$$ 0 0
$$205$$ −7.65685 −0.534778
$$206$$ 0 0
$$207$$ 8.48528 0.589768
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −8.48528 −0.578691
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3.02944 −0.203782
$$222$$ 0 0
$$223$$ 6.34315 0.424768 0.212384 0.977186i $$-0.431877\pi$$
0.212384 + 0.977186i $$0.431877\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ 5.65685 0.375459 0.187729 0.982221i $$-0.439887\pi$$
0.187729 + 0.982221i $$0.439887\pi$$
$$228$$ 0 0
$$229$$ −25.3137 −1.67278 −0.836388 0.548137i $$-0.815337\pi$$
−0.836388 + 0.548137i $$0.815337\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.3137 1.39631 0.698154 0.715948i $$-0.254005\pi$$
0.698154 + 0.715948i $$0.254005\pi$$
$$234$$ 0 0
$$235$$ 10.8284 0.706369
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 26.9706 1.73733 0.868663 0.495403i $$-0.164980\pi$$
0.868663 + 0.495403i $$0.164980\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 0.828427 0.0527116
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.9706 −0.818695 −0.409347 0.912379i $$-0.634244\pi$$
−0.409347 + 0.912379i $$0.634244\pi$$
$$252$$ 0 0
$$253$$ 11.3137 0.711287
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −16.8284 −1.04973 −0.524864 0.851186i $$-0.675884\pi$$
−0.524864 + 0.851186i $$0.675884\pi$$
$$258$$ 0 0
$$259$$ 29.6569 1.84279
$$260$$ 0 0
$$261$$ 22.9706 1.42184
$$262$$ 0 0
$$263$$ 8.48528 0.523225 0.261612 0.965173i $$-0.415746\pi$$
0.261612 + 0.965173i $$0.415746\pi$$
$$264$$ 0 0
$$265$$ 12.8284 0.788044
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −14.9706 −0.899494 −0.449747 0.893156i $$-0.648486\pi$$
−0.449747 + 0.893156i $$0.648486\pi$$
$$278$$ 0 0
$$279$$ −16.9706 −1.01600
$$280$$ 0 0
$$281$$ 4.34315 0.259090 0.129545 0.991574i $$-0.458648\pi$$
0.129545 + 0.991574i $$0.458648\pi$$
$$282$$ 0 0
$$283$$ −24.4853 −1.45550 −0.727749 0.685843i $$-0.759434\pi$$
−0.727749 + 0.685843i $$0.759434\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −21.6569 −1.27836
$$288$$ 0 0
$$289$$ −3.62742 −0.213377
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 6.48528 0.378874 0.189437 0.981893i $$-0.439334\pi$$
0.189437 + 0.981893i $$0.439334\pi$$
$$294$$ 0 0
$$295$$ 1.65685 0.0964658
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.34315 −0.135508
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 6.00000 0.343559
$$306$$ 0 0
$$307$$ 5.65685 0.322854 0.161427 0.986885i $$-0.448390\pi$$
0.161427 + 0.986885i $$0.448390\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.0000 −1.13410 −0.567048 0.823685i $$-0.691915\pi$$
−0.567048 + 0.823685i $$0.691915\pi$$
$$312$$ 0 0
$$313$$ −28.6274 −1.61812 −0.809059 0.587728i $$-0.800023\pi$$
−0.809059 + 0.587728i $$0.800023\pi$$
$$314$$ 0 0
$$315$$ −8.48528 −0.478091
$$316$$ 0 0
$$317$$ −7.17157 −0.402796 −0.201398 0.979510i $$-0.564548\pi$$
−0.201398 + 0.979510i $$0.564548\pi$$
$$318$$ 0 0
$$319$$ 30.6274 1.71481
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.65685 −0.203473
$$324$$ 0 0
$$325$$ 0.828427 0.0459529
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 30.6274 1.68854
$$330$$ 0 0
$$331$$ 31.3137 1.72116 0.860579 0.509318i $$-0.170102\pi$$
0.860579 + 0.509318i $$0.170102\pi$$
$$332$$ 0 0
$$333$$ 31.4558 1.72377
$$334$$ 0 0
$$335$$ 11.3137 0.618134
$$336$$ 0 0
$$337$$ −11.1716 −0.608554 −0.304277 0.952584i $$-0.598415\pi$$
−0.304277 + 0.952584i $$0.598415\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −22.6274 −1.22534
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −34.8284 −1.86969 −0.934844 0.355059i $$-0.884461\pi$$
−0.934844 + 0.355059i $$0.884461\pi$$
$$348$$ 0 0
$$349$$ −31.9411 −1.70977 −0.854885 0.518818i $$-0.826372\pi$$
−0.854885 + 0.518818i $$0.826372\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −8.34315 −0.444061 −0.222030 0.975040i $$-0.571268\pi$$
−0.222030 + 0.975040i $$0.571268\pi$$
$$354$$ 0 0
$$355$$ −5.65685 −0.300235
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 14.3431 0.757002 0.378501 0.925601i $$-0.376440\pi$$
0.378501 + 0.925601i $$0.376440\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4.34315 −0.227331
$$366$$ 0 0
$$367$$ −17.4558 −0.911188 −0.455594 0.890188i $$-0.650573\pi$$
−0.455594 + 0.890188i $$0.650573\pi$$
$$368$$ 0 0
$$369$$ −22.9706 −1.19580
$$370$$ 0 0
$$371$$ 36.2843 1.88379
$$372$$ 0 0
$$373$$ −16.1421 −0.835808 −0.417904 0.908491i $$-0.637235\pi$$
−0.417904 + 0.908491i $$0.637235\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.34315 −0.326689
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 4.00000 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$384$$ 0 0
$$385$$ −11.3137 −0.576600
$$386$$ 0 0
$$387$$ −25.4558 −1.29399
$$388$$ 0 0
$$389$$ 32.6274 1.65428 0.827138 0.561999i $$-0.189968\pi$$
0.827138 + 0.561999i $$0.189968\pi$$
$$390$$ 0 0
$$391$$ 10.3431 0.523075
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 5.65685 0.284627
$$396$$ 0 0
$$397$$ −6.97056 −0.349843 −0.174921 0.984582i $$-0.555967\pi$$
−0.174921 + 0.984582i $$0.555967\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 0 0
$$403$$ 4.68629 0.233441
$$404$$ 0 0
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 41.9411 2.07894
$$408$$ 0 0
$$409$$ −9.31371 −0.460533 −0.230267 0.973128i $$-0.573960\pi$$
−0.230267 + 0.973128i $$0.573960\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 4.68629 0.230597
$$414$$ 0 0
$$415$$ −10.8284 −0.531547
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 23.3137 1.13895 0.569475 0.822009i $$-0.307147\pi$$
0.569475 + 0.822009i $$0.307147\pi$$
$$420$$ 0 0
$$421$$ −10.9706 −0.534673 −0.267336 0.963603i $$-0.586143\pi$$
−0.267336 + 0.963603i $$0.586143\pi$$
$$422$$ 0 0
$$423$$ 32.4853 1.57949
$$424$$ 0 0
$$425$$ −3.65685 −0.177383
$$426$$ 0 0
$$427$$ 16.9706 0.821263
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.6863 0.611077 0.305539 0.952180i $$-0.401164\pi$$
0.305539 + 0.952180i $$0.401164\pi$$
$$432$$ 0 0
$$433$$ −3.17157 −0.152416 −0.0762080 0.997092i $$-0.524281\pi$$
−0.0762080 + 0.997092i $$0.524281\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2.82843 −0.135302
$$438$$ 0 0
$$439$$ 8.97056 0.428142 0.214071 0.976818i $$-0.431328\pi$$
0.214071 + 0.976818i $$0.431328\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −16.4853 −0.783239 −0.391620 0.920127i $$-0.628085\pi$$
−0.391620 + 0.920127i $$0.628085\pi$$
$$444$$ 0 0
$$445$$ 0.343146 0.0162667
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10.6863 −0.504317 −0.252159 0.967686i $$-0.581140\pi$$
−0.252159 + 0.967686i $$0.581140\pi$$
$$450$$ 0 0
$$451$$ −30.6274 −1.44219
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.34315 0.109848
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ −22.1421 −1.02903 −0.514516 0.857481i $$-0.672028\pi$$
−0.514516 + 0.857481i $$0.672028\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −14.1421 −0.654420 −0.327210 0.944952i $$-0.606108\pi$$
−0.327210 + 0.944952i $$0.606108\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −33.9411 −1.56061
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 38.4853 1.76212
$$478$$ 0 0
$$479$$ 12.9706 0.592640 0.296320 0.955089i $$-0.404240\pi$$
0.296320 + 0.955089i $$0.404240\pi$$
$$480$$ 0 0
$$481$$ −8.68629 −0.396061
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −12.8284 −0.582509
$$486$$ 0 0
$$487$$ −26.6274 −1.20660 −0.603302 0.797513i $$-0.706149\pi$$
−0.603302 + 0.797513i $$0.706149\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.97056 −0.224318 −0.112159 0.993690i $$-0.535777\pi$$
−0.112159 + 0.993690i $$0.535777\pi$$
$$492$$ 0 0
$$493$$ 28.0000 1.26106
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ −16.0000 −0.717698
$$498$$ 0 0
$$499$$ −16.2843 −0.728984 −0.364492 0.931207i $$-0.618757\pi$$
−0.364492 + 0.931207i $$0.618757\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0.485281 0.0216376 0.0108188 0.999941i $$-0.496556\pi$$
0.0108188 + 0.999941i $$0.496556\pi$$
$$504$$ 0 0
$$505$$ 14.0000 0.622992
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 22.9706 1.01815 0.509076 0.860721i $$-0.329987\pi$$
0.509076 + 0.860721i $$0.329987\pi$$
$$510$$ 0 0
$$511$$ −12.2843 −0.543424
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −6.34315 −0.279512
$$516$$ 0 0
$$517$$ 43.3137 1.90493
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 29.3137 1.28426 0.642128 0.766597i $$-0.278052\pi$$
0.642128 + 0.766597i $$0.278052\pi$$
$$522$$ 0 0
$$523$$ 7.02944 0.307376 0.153688 0.988119i $$-0.450885\pi$$
0.153688 + 0.988119i $$0.450885\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −20.6863 −0.901109
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 4.97056 0.215704
$$532$$ 0 0
$$533$$ 6.34315 0.274752
$$534$$ 0 0
$$535$$ −2.34315 −0.101303
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −17.3137 −0.741638
$$546$$ 0 0
$$547$$ 20.2843 0.867293 0.433646 0.901083i $$-0.357227\pi$$
0.433646 + 0.901083i $$0.357227\pi$$
$$548$$ 0 0
$$549$$ 18.0000 0.768221
$$550$$ 0 0
$$551$$ −7.65685 −0.326193
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −22.9706 −0.973294 −0.486647 0.873599i $$-0.661780\pi$$
−0.486647 + 0.873599i $$0.661780\pi$$
$$558$$ 0 0
$$559$$ 7.02944 0.297314
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 10.3431 0.435912 0.217956 0.975959i $$-0.430061\pi$$
0.217956 + 0.975959i $$0.430061\pi$$
$$564$$ 0 0
$$565$$ 6.48528 0.272838
$$566$$ 0 0
$$567$$ −25.4558 −1.06904
$$568$$ 0 0
$$569$$ 35.9411 1.50673 0.753365 0.657602i $$-0.228429\pi$$
0.753365 + 0.657602i $$0.228429\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.82843 −0.117954
$$576$$ 0 0
$$577$$ −36.6274 −1.52482 −0.762410 0.647095i $$-0.775984\pi$$
−0.762410 + 0.647095i $$0.775984\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −30.6274 −1.27064
$$582$$ 0 0
$$583$$ 51.3137 2.12520
$$584$$ 0 0
$$585$$ 2.48528 0.102754
$$586$$ 0 0
$$587$$ −22.1421 −0.913904 −0.456952 0.889491i $$-0.651059\pi$$
−0.456952 + 0.889491i $$0.651059\pi$$
$$588$$ 0 0
$$589$$ 5.65685 0.233087
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 8.62742 0.354286 0.177143 0.984185i $$-0.443315\pi$$
0.177143 + 0.984185i $$0.443315\pi$$
$$594$$ 0 0
$$595$$ −10.3431 −0.424028
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −5.65685 −0.231133 −0.115566 0.993300i $$-0.536868\pi$$
−0.115566 + 0.993300i $$0.536868\pi$$
$$600$$ 0 0
$$601$$ −38.9706 −1.58964 −0.794821 0.606844i $$-0.792435\pi$$
−0.794821 + 0.606844i $$0.792435\pi$$
$$602$$ 0 0
$$603$$ 33.9411 1.38219
$$604$$ 0 0
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ −36.0000 −1.46119 −0.730597 0.682808i $$-0.760758\pi$$
−0.730597 + 0.682808i $$0.760758\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.97056 −0.362910
$$612$$ 0 0
$$613$$ 34.9706 1.41245 0.706224 0.707989i $$-0.250397\pi$$
0.706224 + 0.707989i $$0.250397\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −45.5980 −1.83571 −0.917853 0.396921i $$-0.870079\pi$$
−0.917853 + 0.396921i $$0.870079\pi$$
$$618$$ 0 0
$$619$$ 43.5980 1.75235 0.876175 0.481992i $$-0.160087\pi$$
0.876175 + 0.481992i $$0.160087\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0.970563 0.0388848
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 38.3431 1.52884
$$630$$ 0 0
$$631$$ −26.6274 −1.06002 −0.530010 0.847991i $$-0.677812\pi$$
−0.530010 + 0.847991i $$0.677812\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −20.9706 −0.832191
$$636$$ 0 0
$$637$$ 0.828427 0.0328235
$$638$$ 0 0
$$639$$ −16.9706 −0.671345
$$640$$ 0 0
$$641$$ 20.3431 0.803506 0.401753 0.915748i $$-0.368401\pi$$
0.401753 + 0.915748i $$0.368401\pi$$
$$642$$ 0 0
$$643$$ −15.1127 −0.595987 −0.297993 0.954568i $$-0.596317\pi$$
−0.297993 + 0.954568i $$0.596317\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6.14214 0.241472 0.120736 0.992685i $$-0.461475\pi$$
0.120736 + 0.992685i $$0.461475\pi$$
$$648$$ 0 0
$$649$$ 6.62742 0.260149
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11.6569 −0.456168 −0.228084 0.973641i $$-0.573246\pi$$
−0.228084 + 0.973641i $$0.573246\pi$$
$$654$$ 0 0
$$655$$ 1.65685 0.0647387
$$656$$ 0 0
$$657$$ −13.0294 −0.508327
$$658$$ 0 0
$$659$$ 9.65685 0.376178 0.188089 0.982152i $$-0.439771\pi$$
0.188089 + 0.982152i $$0.439771\pi$$
$$660$$ 0 0
$$661$$ −12.3431 −0.480093 −0.240046 0.970761i $$-0.577163\pi$$
−0.240046 + 0.970761i $$0.577163\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.82843 0.109682
$$666$$ 0 0
$$667$$ 21.6569 0.838557
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ 25.1127 0.968023 0.484012 0.875062i $$-0.339179\pi$$
0.484012 + 0.875062i $$0.339179\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 40.4264 1.55371 0.776857 0.629678i $$-0.216813\pi$$
0.776857 + 0.629678i $$0.216813\pi$$
$$678$$ 0 0
$$679$$ −36.2843 −1.39246
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.2843 0.470045 0.235022 0.971990i $$-0.424484\pi$$
0.235022 + 0.971990i $$0.424484\pi$$
$$684$$ 0 0
$$685$$ −4.34315 −0.165943
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −10.6274 −0.404872
$$690$$ 0 0
$$691$$ 39.3137 1.49556 0.747782 0.663944i $$-0.231119\pi$$
0.747782 + 0.663944i $$0.231119\pi$$
$$692$$ 0 0
$$693$$ −33.9411 −1.28932
$$694$$ 0 0
$$695$$ 20.9706 0.795459
$$696$$ 0 0
$$697$$ −28.0000 −1.06058
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.6863 −0.705771 −0.352886 0.935666i $$-0.614800\pi$$
−0.352886 + 0.935666i $$0.614800\pi$$
$$702$$ 0 0
$$703$$ −10.4853 −0.395460
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 39.5980 1.48924
$$708$$ 0 0
$$709$$ 43.9411 1.65024 0.825122 0.564955i $$-0.191106\pi$$
0.825122 + 0.564955i $$0.191106\pi$$
$$710$$ 0 0
$$711$$ 16.9706 0.636446
$$712$$ 0 0
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 3.31371 0.123926
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 16.2843 0.607301 0.303650 0.952784i $$-0.401795\pi$$
0.303650 + 0.952784i $$0.401795\pi$$
$$720$$ 0 0
$$721$$ −17.9411 −0.668162
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.65685 −0.284368
$$726$$ 0 0
$$727$$ 32.4853 1.20481 0.602406 0.798190i $$-0.294209\pi$$
0.602406 + 0.798190i $$0.294209\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −31.0294 −1.14767
$$732$$ 0 0
$$733$$ 29.3137 1.08273 0.541363 0.840789i $$-0.317908\pi$$
0.541363 + 0.840789i $$0.317908\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 45.2548 1.66698
$$738$$ 0 0
$$739$$ 31.3137 1.15189 0.575947 0.817487i $$-0.304634\pi$$
0.575947 + 0.817487i $$0.304634\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −50.6274 −1.85734 −0.928670 0.370907i $$-0.879047\pi$$
−0.928670 + 0.370907i $$0.879047\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 0 0
$$747$$ −32.4853 −1.18857
$$748$$ 0 0
$$749$$ −6.62742 −0.242161
$$750$$ 0 0
$$751$$ 22.6274 0.825686 0.412843 0.910802i $$-0.364536\pi$$
0.412843 + 0.910802i $$0.364536\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −19.3137 −0.702898
$$756$$ 0 0
$$757$$ 21.3137 0.774660 0.387330 0.921941i $$-0.373397\pi$$
0.387330 + 0.921941i $$0.373397\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −20.6274 −0.747743 −0.373872 0.927480i $$-0.621970\pi$$
−0.373872 + 0.927480i $$0.621970\pi$$
$$762$$ 0 0
$$763$$ −48.9706 −1.77285
$$764$$ 0 0
$$765$$ −10.9706 −0.396642
$$766$$ 0 0
$$767$$ −1.37258 −0.0495611
$$768$$ 0 0
$$769$$ 39.2548 1.41557 0.707783 0.706430i $$-0.249696\pi$$
0.707783 + 0.706430i $$0.249696\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −42.4853 −1.52809 −0.764045 0.645163i $$-0.776789\pi$$
−0.764045 + 0.645163i $$0.776789\pi$$
$$774$$ 0 0
$$775$$ 5.65685 0.203200
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 7.65685 0.274335
$$780$$ 0 0
$$781$$ −22.6274 −0.809673
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4.34315 −0.155014
$$786$$ 0 0
$$787$$ −12.2843 −0.437887 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.3431 0.652207
$$792$$ 0 0
$$793$$ −4.97056 −0.176510
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 11.1716 0.395717 0.197859 0.980231i $$-0.436601\pi$$
0.197859 + 0.980231i $$0.436601\pi$$
$$798$$ 0 0
$$799$$ 39.5980 1.40088
$$800$$ 0 0
$$801$$ 1.02944 0.0363734
$$802$$ 0 0
$$803$$ −17.3726 −0.613065
$$804$$ 0 0
$$805$$ −8.00000 −0.281963
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 3.37258 0.118574 0.0592869 0.998241i $$-0.481117\pi$$
0.0592869 + 0.998241i $$0.481117\pi$$
$$810$$ 0 0
$$811$$ −33.6569 −1.18185 −0.590926 0.806726i $$-0.701237\pi$$
−0.590926 + 0.806726i $$0.701237\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −13.1716 −0.461380
$$816$$ 0 0
$$817$$ 8.48528 0.296862
$$818$$ 0 0
$$819$$ 7.02944 0.245628
$$820$$ 0 0
$$821$$ 13.3137 0.464652 0.232326 0.972638i $$-0.425366\pi$$
0.232326 + 0.972638i $$0.425366\pi$$
$$822$$ 0 0
$$823$$ −3.79899 −0.132424 −0.0662122 0.997806i $$-0.521091\pi$$
−0.0662122 + 0.997806i $$0.521091\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −16.0000 −0.556375 −0.278187 0.960527i $$-0.589734\pi$$
−0.278187 + 0.960527i $$0.589734\pi$$
$$828$$ 0 0
$$829$$ −39.2548 −1.36338 −0.681688 0.731643i $$-0.738754\pi$$
−0.681688 + 0.731643i $$0.738754\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −3.65685 −0.126702
$$834$$ 0 0
$$835$$ −0.686292 −0.0237501
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 26.3431 0.909466 0.454733 0.890628i $$-0.349735\pi$$
0.454733 + 0.890628i $$0.349735\pi$$
$$840$$ 0 0
$$841$$ 29.6274 1.02164
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 12.3137 0.423604
$$846$$ 0 0
$$847$$ −14.1421 −0.485930
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 29.6569 1.01662
$$852$$ 0 0
$$853$$ 0.627417 0.0214823 0.0107412 0.999942i $$-0.496581\pi$$
0.0107412 + 0.999942i $$0.496581\pi$$
$$854$$ 0 0
$$855$$ 3.00000 0.102598
$$856$$ 0 0
$$857$$ 21.7990 0.744639 0.372320 0.928105i $$-0.378563\pi$$
0.372320 + 0.928105i $$0.378563\pi$$
$$858$$ 0 0
$$859$$ 0.686292 0.0234160 0.0117080 0.999931i $$-0.496273\pi$$
0.0117080 + 0.999931i $$0.496273\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −52.0000 −1.77010 −0.885050 0.465495i $$-0.845876\pi$$
−0.885050 + 0.465495i $$0.845876\pi$$
$$864$$ 0 0
$$865$$ 16.1421 0.548849
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 22.6274 0.767583
$$870$$ 0 0
$$871$$ −9.37258 −0.317578
$$872$$ 0 0
$$873$$ −38.4853 −1.30253
$$874$$ 0 0
$$875$$ 2.82843 0.0956183
$$876$$ 0 0
$$877$$ 9.79899 0.330888 0.165444 0.986219i $$-0.447094\pi$$
0.165444 + 0.986219i $$0.447094\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 29.3137 0.987604 0.493802 0.869574i $$-0.335607\pi$$
0.493802 + 0.869574i $$0.335607\pi$$
$$882$$ 0 0
$$883$$ −10.8284 −0.364406 −0.182203 0.983261i $$-0.558323\pi$$
−0.182203 + 0.983261i $$0.558323\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −25.6569 −0.861473 −0.430736 0.902478i $$-0.641746\pi$$
−0.430736 + 0.902478i $$0.641746\pi$$
$$888$$ 0 0
$$889$$ −59.3137 −1.98932
$$890$$ 0 0
$$891$$ −36.0000 −1.20605
$$892$$ 0 0
$$893$$ −10.8284 −0.362359
$$894$$ 0 0
$$895$$ −20.9706 −0.700969
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −43.3137 −1.44459
$$900$$ 0 0
$$901$$ 46.9117 1.56285
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −9.31371 −0.309598
$$906$$ 0 0
$$907$$ −12.2843 −0.407893 −0.203946 0.978982i $$-0.565377\pi$$
−0.203946 + 0.978982i $$0.565377\pi$$
$$908$$ 0 0
$$909$$ 42.0000 1.39305
$$910$$ 0 0
$$911$$ 45.2548 1.49936 0.749680 0.661801i $$-0.230208\pi$$
0.749680 + 0.661801i $$0.230208\pi$$
$$912$$ 0 0
$$913$$ −43.3137 −1.43347
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4.68629 0.154755
$$918$$ 0 0
$$919$$ 13.3726 0.441121 0.220560 0.975373i $$-0.429211\pi$$
0.220560 + 0.975373i $$0.429211\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 4.68629 0.154251
$$924$$ 0 0
$$925$$ −10.4853 −0.344754
$$926$$ 0 0
$$927$$ −19.0294 −0.625009
$$928$$ 0 0
$$929$$ −38.0000 −1.24674 −0.623370 0.781927i $$-0.714237\pi$$
−0.623370 + 0.781927i $$0.714237\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −14.6274 −0.478368
$$936$$ 0 0
$$937$$ −43.6569 −1.42621 −0.713104 0.701059i $$-0.752711\pi$$
−0.713104 + 0.701059i $$0.752711\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ −21.6569 −0.705244
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −42.8284 −1.39174 −0.695868 0.718169i $$-0.744980\pi$$
−0.695868 + 0.718169i $$0.744980\pi$$
$$948$$ 0 0
$$949$$ 3.59798 0.116795
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 37.7990 1.22443 0.612215 0.790692i $$-0.290279\pi$$
0.612215 + 0.790692i $$0.290279\pi$$
$$954$$ 0 0
$$955$$ 20.9706 0.678591
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.2843 −0.396680
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ −7.02944 −0.226520
$$964$$ 0 0
$$965$$ 3.17157 0.102097
$$966$$ 0 0
$$967$$ −23.1127 −0.743254 −0.371627 0.928382i $$-0.621200\pi$$
−0.371627 + 0.928382i $$0.621200\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −4.97056 −0.159513 −0.0797565 0.996814i $$-0.525414\pi$$
−0.0797565 + 0.996814i $$0.525414\pi$$
$$972$$ 0 0
$$973$$ 59.3137 1.90151
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 45.7990 1.46524 0.732620 0.680638i $$-0.238297\pi$$
0.732620 + 0.680638i $$0.238297\pi$$
$$978$$ 0 0
$$979$$ 1.37258 0.0438679
$$980$$ 0 0
$$981$$ −51.9411 −1.65835
$$982$$ 0 0
$$983$$ 7.31371 0.233271 0.116636 0.993175i $$-0.462789\pi$$
0.116636 + 0.993175i $$0.462789\pi$$
$$984$$ 0 0
$$985$$ −21.3137 −0.679111
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −15.3137 −0.485477
$$996$$ 0 0
$$997$$ 23.6569 0.749220 0.374610 0.927182i $$-0.377777\pi$$
0.374610 + 0.927182i $$0.377777\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6080.2.a.bd.1.1 2
4.3 odd 2 6080.2.a.be.1.2 2
8.3 odd 2 3040.2.a.f.1.2 2
8.5 even 2 3040.2.a.g.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.f.1.2 2 8.3 odd 2
3040.2.a.g.1.1 yes 2 8.5 even 2
6080.2.a.bd.1.1 2 1.1 even 1 trivial
6080.2.a.be.1.2 2 4.3 odd 2