# Properties

 Label 4840.2.a.bh.1.5 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100$$ x^8 - x^7 - 21*x^6 + 15*x^5 + 140*x^4 - 60*x^3 - 295*x^2 + 50*x + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$0.713352$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.713352 q^{3} +1.00000 q^{5} -0.376172 q^{7} -2.49113 q^{9} +O(q^{10})$$ $$q+0.713352 q^{3} +1.00000 q^{5} -0.376172 q^{7} -2.49113 q^{9} +2.82831 q^{13} +0.713352 q^{15} +4.83768 q^{17} -2.92990 q^{19} -0.268343 q^{21} +3.77226 q^{23} +1.00000 q^{25} -3.91711 q^{27} -8.44396 q^{29} +8.04011 q^{31} -0.376172 q^{35} +2.83990 q^{37} +2.01758 q^{39} +3.72034 q^{41} +6.48484 q^{43} -2.49113 q^{45} +2.58897 q^{47} -6.85849 q^{49} +3.45097 q^{51} -0.0487626 q^{53} -2.09005 q^{57} -1.64929 q^{59} -8.69705 q^{61} +0.937093 q^{63} +2.82831 q^{65} +11.4395 q^{67} +2.69095 q^{69} +14.1216 q^{71} -0.513225 q^{73} +0.713352 q^{75} -12.9823 q^{79} +4.67911 q^{81} -2.73687 q^{83} +4.83768 q^{85} -6.02351 q^{87} +12.0195 q^{89} -1.06393 q^{91} +5.73543 q^{93} -2.92990 q^{95} +16.0745 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9}+O(q^{10})$$ 8 * q + q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 $$8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} - 12 q^{13} + q^{15} - 2 q^{17} + 6 q^{19} - 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} - 8 q^{29} + 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} + 3 q^{41} + 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} + 7 q^{51} + 28 q^{53} - 25 q^{57} + 25 q^{59} - 10 q^{61} + 64 q^{63} - 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} - 38 q^{73} + q^{75} + 38 q^{79} + 32 q^{81} + 28 q^{83} - 2 q^{85} - 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} + 6 q^{95} + 21 q^{97}+O(q^{100})$$ 8 * q + q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 - 12 * q^13 + q^15 - 2 * q^17 + 6 * q^19 - 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 - 8 * q^29 + 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 + 3 * q^41 + 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 - 25 * q^57 + 25 * q^59 - 10 * q^61 + 64 * q^63 - 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 - 38 * q^73 + q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 - 2 * q^85 - 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * q^97

## 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
$$3$$ 0.713352 0.411854 0.205927 0.978567i $$-0.433979\pi$$
0.205927 + 0.978567i $$0.433979\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −0.376172 −0.142180 −0.0710899 0.997470i $$-0.522648\pi$$
−0.0710899 + 0.997470i $$0.522648\pi$$
$$8$$ 0 0
$$9$$ −2.49113 −0.830376
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 2.82831 0.784432 0.392216 0.919873i $$-0.371709\pi$$
0.392216 + 0.919873i $$0.371709\pi$$
$$14$$ 0 0
$$15$$ 0.713352 0.184187
$$16$$ 0 0
$$17$$ 4.83768 1.17331 0.586655 0.809837i $$-0.300444\pi$$
0.586655 + 0.809837i $$0.300444\pi$$
$$18$$ 0 0
$$19$$ −2.92990 −0.672165 −0.336082 0.941833i $$-0.609102\pi$$
−0.336082 + 0.941833i $$0.609102\pi$$
$$20$$ 0 0
$$21$$ −0.268343 −0.0585573
$$22$$ 0 0
$$23$$ 3.77226 0.786571 0.393285 0.919416i $$-0.371338\pi$$
0.393285 + 0.919416i $$0.371338\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −3.91711 −0.753848
$$28$$ 0 0
$$29$$ −8.44396 −1.56800 −0.784002 0.620759i $$-0.786825\pi$$
−0.784002 + 0.620759i $$0.786825\pi$$
$$30$$ 0 0
$$31$$ 8.04011 1.44405 0.722023 0.691869i $$-0.243213\pi$$
0.722023 + 0.691869i $$0.243213\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.376172 −0.0635847
$$36$$ 0 0
$$37$$ 2.83990 0.466877 0.233438 0.972372i $$-0.425002\pi$$
0.233438 + 0.972372i $$0.425002\pi$$
$$38$$ 0 0
$$39$$ 2.01758 0.323071
$$40$$ 0 0
$$41$$ 3.72034 0.581020 0.290510 0.956872i $$-0.406175\pi$$
0.290510 + 0.956872i $$0.406175\pi$$
$$42$$ 0 0
$$43$$ 6.48484 0.988929 0.494465 0.869198i $$-0.335364\pi$$
0.494465 + 0.869198i $$0.335364\pi$$
$$44$$ 0 0
$$45$$ −2.49113 −0.371356
$$46$$ 0 0
$$47$$ 2.58897 0.377639 0.188820 0.982012i $$-0.439534\pi$$
0.188820 + 0.982012i $$0.439534\pi$$
$$48$$ 0 0
$$49$$ −6.85849 −0.979785
$$50$$ 0 0
$$51$$ 3.45097 0.483233
$$52$$ 0 0
$$53$$ −0.0487626 −0.00669805 −0.00334903 0.999994i $$-0.501066\pi$$
−0.00334903 + 0.999994i $$0.501066\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.09005 −0.276834
$$58$$ 0 0
$$59$$ −1.64929 −0.214720 −0.107360 0.994220i $$-0.534240\pi$$
−0.107360 + 0.994220i $$0.534240\pi$$
$$60$$ 0 0
$$61$$ −8.69705 −1.11354 −0.556772 0.830666i $$-0.687960\pi$$
−0.556772 + 0.830666i $$0.687960\pi$$
$$62$$ 0 0
$$63$$ 0.937093 0.118063
$$64$$ 0 0
$$65$$ 2.82831 0.350809
$$66$$ 0 0
$$67$$ 11.4395 1.39756 0.698779 0.715338i $$-0.253727\pi$$
0.698779 + 0.715338i $$0.253727\pi$$
$$68$$ 0 0
$$69$$ 2.69095 0.323952
$$70$$ 0 0
$$71$$ 14.1216 1.67592 0.837962 0.545728i $$-0.183747\pi$$
0.837962 + 0.545728i $$0.183747\pi$$
$$72$$ 0 0
$$73$$ −0.513225 −0.0600684 −0.0300342 0.999549i $$-0.509562\pi$$
−0.0300342 + 0.999549i $$0.509562\pi$$
$$74$$ 0 0
$$75$$ 0.713352 0.0823708
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.9823 −1.46063 −0.730313 0.683113i $$-0.760626\pi$$
−0.730313 + 0.683113i $$0.760626\pi$$
$$80$$ 0 0
$$81$$ 4.67911 0.519901
$$82$$ 0 0
$$83$$ −2.73687 −0.300411 −0.150205 0.988655i $$-0.547993\pi$$
−0.150205 + 0.988655i $$0.547993\pi$$
$$84$$ 0 0
$$85$$ 4.83768 0.524720
$$86$$ 0 0
$$87$$ −6.02351 −0.645789
$$88$$ 0 0
$$89$$ 12.0195 1.27407 0.637034 0.770835i $$-0.280161\pi$$
0.637034 + 0.770835i $$0.280161\pi$$
$$90$$ 0 0
$$91$$ −1.06393 −0.111530
$$92$$ 0 0
$$93$$ 5.73543 0.594736
$$94$$ 0 0
$$95$$ −2.92990 −0.300601
$$96$$ 0 0
$$97$$ 16.0745 1.63212 0.816060 0.577967i $$-0.196154\pi$$
0.816060 + 0.577967i $$0.196154\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −15.5883 −1.55109 −0.775546 0.631291i $$-0.782525\pi$$
−0.775546 + 0.631291i $$0.782525\pi$$
$$102$$ 0 0
$$103$$ −9.25907 −0.912323 −0.456161 0.889897i $$-0.650776\pi$$
−0.456161 + 0.889897i $$0.650776\pi$$
$$104$$ 0 0
$$105$$ −0.268343 −0.0261876
$$106$$ 0 0
$$107$$ 2.15917 0.208735 0.104368 0.994539i $$-0.466718\pi$$
0.104368 + 0.994539i $$0.466718\pi$$
$$108$$ 0 0
$$109$$ 5.79052 0.554631 0.277315 0.960779i $$-0.410555\pi$$
0.277315 + 0.960779i $$0.410555\pi$$
$$110$$ 0 0
$$111$$ 2.02585 0.192285
$$112$$ 0 0
$$113$$ 19.0148 1.78876 0.894381 0.447306i $$-0.147616\pi$$
0.894381 + 0.447306i $$0.147616\pi$$
$$114$$ 0 0
$$115$$ 3.77226 0.351765
$$116$$ 0 0
$$117$$ −7.04568 −0.651373
$$118$$ 0 0
$$119$$ −1.81980 −0.166821
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 2.65391 0.239295
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 10.0973 0.895993 0.447996 0.894035i $$-0.352138\pi$$
0.447996 + 0.894035i $$0.352138\pi$$
$$128$$ 0 0
$$129$$ 4.62598 0.407294
$$130$$ 0 0
$$131$$ 6.29651 0.550128 0.275064 0.961426i $$-0.411301\pi$$
0.275064 + 0.961426i $$0.411301\pi$$
$$132$$ 0 0
$$133$$ 1.10215 0.0955682
$$134$$ 0 0
$$135$$ −3.91711 −0.337131
$$136$$ 0 0
$$137$$ −8.27329 −0.706835 −0.353418 0.935466i $$-0.614981\pi$$
−0.353418 + 0.935466i $$0.614981\pi$$
$$138$$ 0 0
$$139$$ 10.2671 0.870843 0.435422 0.900227i $$-0.356599\pi$$
0.435422 + 0.900227i $$0.356599\pi$$
$$140$$ 0 0
$$141$$ 1.84684 0.155532
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −8.44396 −0.701232
$$146$$ 0 0
$$147$$ −4.89252 −0.403528
$$148$$ 0 0
$$149$$ 4.90160 0.401554 0.200777 0.979637i $$-0.435653\pi$$
0.200777 + 0.979637i $$0.435653\pi$$
$$150$$ 0 0
$$151$$ 9.35625 0.761401 0.380701 0.924698i $$-0.375683\pi$$
0.380701 + 0.924698i $$0.375683\pi$$
$$152$$ 0 0
$$153$$ −12.0513 −0.974289
$$154$$ 0 0
$$155$$ 8.04011 0.645797
$$156$$ 0 0
$$157$$ 9.65817 0.770806 0.385403 0.922748i $$-0.374062\pi$$
0.385403 + 0.922748i $$0.374062\pi$$
$$158$$ 0 0
$$159$$ −0.0347849 −0.00275862
$$160$$ 0 0
$$161$$ −1.41902 −0.111834
$$162$$ 0 0
$$163$$ 4.48802 0.351529 0.175764 0.984432i $$-0.443760\pi$$
0.175764 + 0.984432i $$0.443760\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 23.4716 1.81628 0.908142 0.418662i $$-0.137501\pi$$
0.908142 + 0.418662i $$0.137501\pi$$
$$168$$ 0 0
$$169$$ −5.00067 −0.384667
$$170$$ 0 0
$$171$$ 7.29875 0.558150
$$172$$ 0 0
$$173$$ 12.9509 0.984637 0.492318 0.870415i $$-0.336150\pi$$
0.492318 + 0.870415i $$0.336150\pi$$
$$174$$ 0 0
$$175$$ −0.376172 −0.0284359
$$176$$ 0 0
$$177$$ −1.17653 −0.0884332
$$178$$ 0 0
$$179$$ −3.52933 −0.263795 −0.131897 0.991263i $$-0.542107\pi$$
−0.131897 + 0.991263i $$0.542107\pi$$
$$180$$ 0 0
$$181$$ 1.79178 0.133182 0.0665910 0.997780i $$-0.478788\pi$$
0.0665910 + 0.997780i $$0.478788\pi$$
$$182$$ 0 0
$$183$$ −6.20406 −0.458617
$$184$$ 0 0
$$185$$ 2.83990 0.208794
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.47351 0.107182
$$190$$ 0 0
$$191$$ 3.40113 0.246097 0.123048 0.992401i $$-0.460733\pi$$
0.123048 + 0.992401i $$0.460733\pi$$
$$192$$ 0 0
$$193$$ 7.77543 0.559688 0.279844 0.960046i $$-0.409717\pi$$
0.279844 + 0.960046i $$0.409717\pi$$
$$194$$ 0 0
$$195$$ 2.01758 0.144482
$$196$$ 0 0
$$197$$ 7.09410 0.505434 0.252717 0.967540i $$-0.418676\pi$$
0.252717 + 0.967540i $$0.418676\pi$$
$$198$$ 0 0
$$199$$ 12.7734 0.905481 0.452740 0.891642i $$-0.350446\pi$$
0.452740 + 0.891642i $$0.350446\pi$$
$$200$$ 0 0
$$201$$ 8.16039 0.575590
$$202$$ 0 0
$$203$$ 3.17638 0.222938
$$204$$ 0 0
$$205$$ 3.72034 0.259840
$$206$$ 0 0
$$207$$ −9.39719 −0.653150
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −7.34047 −0.505339 −0.252669 0.967553i $$-0.581308\pi$$
−0.252669 + 0.967553i $$0.581308\pi$$
$$212$$ 0 0
$$213$$ 10.0737 0.690236
$$214$$ 0 0
$$215$$ 6.48484 0.442263
$$216$$ 0 0
$$217$$ −3.02446 −0.205314
$$218$$ 0 0
$$219$$ −0.366110 −0.0247394
$$220$$ 0 0
$$221$$ 13.6825 0.920382
$$222$$ 0 0
$$223$$ 6.53875 0.437867 0.218934 0.975740i $$-0.429742\pi$$
0.218934 + 0.975740i $$0.429742\pi$$
$$224$$ 0 0
$$225$$ −2.49113 −0.166075
$$226$$ 0 0
$$227$$ −20.5461 −1.36369 −0.681847 0.731495i $$-0.738823\pi$$
−0.681847 + 0.731495i $$0.738823\pi$$
$$228$$ 0 0
$$229$$ −4.33094 −0.286197 −0.143098 0.989708i $$-0.545706\pi$$
−0.143098 + 0.989708i $$0.545706\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −28.6595 −1.87755 −0.938774 0.344533i $$-0.888037\pi$$
−0.938774 + 0.344533i $$0.888037\pi$$
$$234$$ 0 0
$$235$$ 2.58897 0.168886
$$236$$ 0 0
$$237$$ −9.26097 −0.601565
$$238$$ 0 0
$$239$$ 10.4766 0.677677 0.338838 0.940845i $$-0.389966\pi$$
0.338838 + 0.940845i $$0.389966\pi$$
$$240$$ 0 0
$$241$$ −15.5202 −0.999745 −0.499873 0.866099i $$-0.666620\pi$$
−0.499873 + 0.866099i $$0.666620\pi$$
$$242$$ 0 0
$$243$$ 15.0892 0.967971
$$244$$ 0 0
$$245$$ −6.85849 −0.438173
$$246$$ 0 0
$$247$$ −8.28666 −0.527267
$$248$$ 0 0
$$249$$ −1.95235 −0.123725
$$250$$ 0 0
$$251$$ −11.9701 −0.755547 −0.377774 0.925898i $$-0.623310\pi$$
−0.377774 + 0.925898i $$0.623310\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 3.45097 0.216108
$$256$$ 0 0
$$257$$ −8.47342 −0.528557 −0.264279 0.964446i $$-0.585134\pi$$
−0.264279 + 0.964446i $$0.585134\pi$$
$$258$$ 0 0
$$259$$ −1.06829 −0.0663804
$$260$$ 0 0
$$261$$ 21.0350 1.30203
$$262$$ 0 0
$$263$$ −12.5189 −0.771950 −0.385975 0.922509i $$-0.626135\pi$$
−0.385975 + 0.922509i $$0.626135\pi$$
$$264$$ 0 0
$$265$$ −0.0487626 −0.00299546
$$266$$ 0 0
$$267$$ 8.57416 0.524730
$$268$$ 0 0
$$269$$ −24.0649 −1.46726 −0.733630 0.679549i $$-0.762176\pi$$
−0.733630 + 0.679549i $$0.762176\pi$$
$$270$$ 0 0
$$271$$ 9.42221 0.572358 0.286179 0.958176i $$-0.407615\pi$$
0.286179 + 0.958176i $$0.407615\pi$$
$$272$$ 0 0
$$273$$ −0.758957 −0.0459342
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.7811 −1.42887 −0.714434 0.699703i $$-0.753316\pi$$
−0.714434 + 0.699703i $$0.753316\pi$$
$$278$$ 0 0
$$279$$ −20.0289 −1.19910
$$280$$ 0 0
$$281$$ −0.496904 −0.0296428 −0.0148214 0.999890i $$-0.504718\pi$$
−0.0148214 + 0.999890i $$0.504718\pi$$
$$282$$ 0 0
$$283$$ −4.68030 −0.278215 −0.139108 0.990277i $$-0.544423\pi$$
−0.139108 + 0.990277i $$0.544423\pi$$
$$284$$ 0 0
$$285$$ −2.09005 −0.123804
$$286$$ 0 0
$$287$$ −1.39949 −0.0826092
$$288$$ 0 0
$$289$$ 6.40318 0.376658
$$290$$ 0 0
$$291$$ 11.4668 0.672195
$$292$$ 0 0
$$293$$ −16.9203 −0.988493 −0.494246 0.869322i $$-0.664556\pi$$
−0.494246 + 0.869322i $$0.664556\pi$$
$$294$$ 0 0
$$295$$ −1.64929 −0.0960256
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 10.6691 0.617011
$$300$$ 0 0
$$301$$ −2.43942 −0.140606
$$302$$ 0 0
$$303$$ −11.1199 −0.638824
$$304$$ 0 0
$$305$$ −8.69705 −0.497992
$$306$$ 0 0
$$307$$ 34.3667 1.96141 0.980707 0.195486i $$-0.0626283\pi$$
0.980707 + 0.195486i $$0.0626283\pi$$
$$308$$ 0 0
$$309$$ −6.60497 −0.375744
$$310$$ 0 0
$$311$$ 7.36790 0.417796 0.208898 0.977937i $$-0.433012\pi$$
0.208898 + 0.977937i $$0.433012\pi$$
$$312$$ 0 0
$$313$$ −25.0606 −1.41651 −0.708253 0.705958i $$-0.750517\pi$$
−0.708253 + 0.705958i $$0.750517\pi$$
$$314$$ 0 0
$$315$$ 0.937093 0.0527992
$$316$$ 0 0
$$317$$ 28.0728 1.57673 0.788364 0.615209i $$-0.210929\pi$$
0.788364 + 0.615209i $$0.210929\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 1.54025 0.0859685
$$322$$ 0 0
$$323$$ −14.1739 −0.788658
$$324$$ 0 0
$$325$$ 2.82831 0.156886
$$326$$ 0 0
$$327$$ 4.13068 0.228427
$$328$$ 0 0
$$329$$ −0.973897 −0.0536927
$$330$$ 0 0
$$331$$ 21.9644 1.20727 0.603637 0.797259i $$-0.293718\pi$$
0.603637 + 0.797259i $$0.293718\pi$$
$$332$$ 0 0
$$333$$ −7.07456 −0.387683
$$334$$ 0 0
$$335$$ 11.4395 0.625007
$$336$$ 0 0
$$337$$ −22.3799 −1.21911 −0.609555 0.792744i $$-0.708652\pi$$
−0.609555 + 0.792744i $$0.708652\pi$$
$$338$$ 0 0
$$339$$ 13.5642 0.736709
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 5.21318 0.281485
$$344$$ 0 0
$$345$$ 2.69095 0.144876
$$346$$ 0 0
$$347$$ 24.7788 1.33020 0.665098 0.746756i $$-0.268390\pi$$
0.665098 + 0.746756i $$0.268390\pi$$
$$348$$ 0 0
$$349$$ 22.3282 1.19520 0.597600 0.801795i $$-0.296121\pi$$
0.597600 + 0.801795i $$0.296121\pi$$
$$350$$ 0 0
$$351$$ −11.0788 −0.591342
$$352$$ 0 0
$$353$$ −5.79232 −0.308294 −0.154147 0.988048i $$-0.549263\pi$$
−0.154147 + 0.988048i $$0.549263\pi$$
$$354$$ 0 0
$$355$$ 14.1216 0.749496
$$356$$ 0 0
$$357$$ −1.29816 −0.0687059
$$358$$ 0 0
$$359$$ −21.0367 −1.11027 −0.555137 0.831759i $$-0.687334\pi$$
−0.555137 + 0.831759i $$0.687334\pi$$
$$360$$ 0 0
$$361$$ −10.4157 −0.548195
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −0.513225 −0.0268634
$$366$$ 0 0
$$367$$ −23.6235 −1.23314 −0.616569 0.787301i $$-0.711478\pi$$
−0.616569 + 0.787301i $$0.711478\pi$$
$$368$$ 0 0
$$369$$ −9.26785 −0.482465
$$370$$ 0 0
$$371$$ 0.0183431 0.000952327 0
$$372$$ 0 0
$$373$$ 12.7740 0.661411 0.330705 0.943734i $$-0.392713\pi$$
0.330705 + 0.943734i $$0.392713\pi$$
$$374$$ 0 0
$$375$$ 0.713352 0.0368373
$$376$$ 0 0
$$377$$ −23.8821 −1.22999
$$378$$ 0 0
$$379$$ 24.7773 1.27273 0.636363 0.771390i $$-0.280438\pi$$
0.636363 + 0.771390i $$0.280438\pi$$
$$380$$ 0 0
$$381$$ 7.20295 0.369018
$$382$$ 0 0
$$383$$ 38.1710 1.95045 0.975223 0.221223i $$-0.0710048\pi$$
0.975223 + 0.221223i $$0.0710048\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −16.1546 −0.821183
$$388$$ 0 0
$$389$$ 2.40973 0.122178 0.0610891 0.998132i $$-0.480543\pi$$
0.0610891 + 0.998132i $$0.480543\pi$$
$$390$$ 0 0
$$391$$ 18.2490 0.922892
$$392$$ 0 0
$$393$$ 4.49163 0.226573
$$394$$ 0 0
$$395$$ −12.9823 −0.653212
$$396$$ 0 0
$$397$$ −11.5456 −0.579459 −0.289729 0.957109i $$-0.593565\pi$$
−0.289729 + 0.957109i $$0.593565\pi$$
$$398$$ 0 0
$$399$$ 0.786218 0.0393601
$$400$$ 0 0
$$401$$ 25.6006 1.27844 0.639218 0.769026i $$-0.279258\pi$$
0.639218 + 0.769026i $$0.279258\pi$$
$$402$$ 0 0
$$403$$ 22.7399 1.13276
$$404$$ 0 0
$$405$$ 4.67911 0.232507
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −9.36369 −0.463005 −0.231502 0.972834i $$-0.574364\pi$$
−0.231502 + 0.972834i $$0.574364\pi$$
$$410$$ 0 0
$$411$$ −5.90177 −0.291113
$$412$$ 0 0
$$413$$ 0.620419 0.0305288
$$414$$ 0 0
$$415$$ −2.73687 −0.134348
$$416$$ 0 0
$$417$$ 7.32405 0.358660
$$418$$ 0 0
$$419$$ −21.3716 −1.04407 −0.522035 0.852924i $$-0.674827\pi$$
−0.522035 + 0.852924i $$0.674827\pi$$
$$420$$ 0 0
$$421$$ 27.5205 1.34126 0.670632 0.741790i $$-0.266023\pi$$
0.670632 + 0.741790i $$0.266023\pi$$
$$422$$ 0 0
$$423$$ −6.44945 −0.313583
$$424$$ 0 0
$$425$$ 4.83768 0.234662
$$426$$ 0 0
$$427$$ 3.27159 0.158323
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.63236 −0.126796 −0.0633982 0.997988i $$-0.520194\pi$$
−0.0633982 + 0.997988i $$0.520194\pi$$
$$432$$ 0 0
$$433$$ 9.99044 0.480110 0.240055 0.970759i $$-0.422835\pi$$
0.240055 + 0.970759i $$0.422835\pi$$
$$434$$ 0 0
$$435$$ −6.02351 −0.288805
$$436$$ 0 0
$$437$$ −11.0523 −0.528705
$$438$$ 0 0
$$439$$ −2.85957 −0.136480 −0.0682400 0.997669i $$-0.521738\pi$$
−0.0682400 + 0.997669i $$0.521738\pi$$
$$440$$ 0 0
$$441$$ 17.0854 0.813590
$$442$$ 0 0
$$443$$ −40.3056 −1.91498 −0.957489 0.288470i $$-0.906853\pi$$
−0.957489 + 0.288470i $$0.906853\pi$$
$$444$$ 0 0
$$445$$ 12.0195 0.569781
$$446$$ 0 0
$$447$$ 3.49656 0.165382
$$448$$ 0 0
$$449$$ −1.95041 −0.0920457 −0.0460228 0.998940i $$-0.514655\pi$$
−0.0460228 + 0.998940i $$0.514655\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 6.67430 0.313586
$$454$$ 0 0
$$455$$ −1.06393 −0.0498779
$$456$$ 0 0
$$457$$ −35.7765 −1.67355 −0.836777 0.547543i $$-0.815563\pi$$
−0.836777 + 0.547543i $$0.815563\pi$$
$$458$$ 0 0
$$459$$ −18.9497 −0.884498
$$460$$ 0 0
$$461$$ −15.1433 −0.705295 −0.352648 0.935756i $$-0.614719\pi$$
−0.352648 + 0.935756i $$0.614719\pi$$
$$462$$ 0 0
$$463$$ −12.6301 −0.586972 −0.293486 0.955963i $$-0.594815\pi$$
−0.293486 + 0.955963i $$0.594815\pi$$
$$464$$ 0 0
$$465$$ 5.73543 0.265974
$$466$$ 0 0
$$467$$ 31.3546 1.45092 0.725460 0.688264i $$-0.241627\pi$$
0.725460 + 0.688264i $$0.241627\pi$$
$$468$$ 0 0
$$469$$ −4.30322 −0.198704
$$470$$ 0 0
$$471$$ 6.88968 0.317460
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −2.92990 −0.134433
$$476$$ 0 0
$$477$$ 0.121474 0.00556190
$$478$$ 0 0
$$479$$ 12.3355 0.563625 0.281812 0.959470i $$-0.409064\pi$$
0.281812 + 0.959470i $$0.409064\pi$$
$$480$$ 0 0
$$481$$ 8.03212 0.366233
$$482$$ 0 0
$$483$$ −1.01226 −0.0460595
$$484$$ 0 0
$$485$$ 16.0745 0.729906
$$486$$ 0 0
$$487$$ −21.9102 −0.992847 −0.496424 0.868080i $$-0.665354\pi$$
−0.496424 + 0.868080i $$0.665354\pi$$
$$488$$ 0 0
$$489$$ 3.20154 0.144779
$$490$$ 0 0
$$491$$ 27.3534 1.23444 0.617220 0.786790i $$-0.288259\pi$$
0.617220 + 0.786790i $$0.288259\pi$$
$$492$$ 0 0
$$493$$ −40.8492 −1.83976
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −5.31215 −0.238282
$$498$$ 0 0
$$499$$ −10.3980 −0.465478 −0.232739 0.972539i $$-0.574769\pi$$
−0.232739 + 0.972539i $$0.574769\pi$$
$$500$$ 0 0
$$501$$ 16.7435 0.748044
$$502$$ 0 0
$$503$$ 10.3868 0.463123 0.231561 0.972820i $$-0.425617\pi$$
0.231561 + 0.972820i $$0.425617\pi$$
$$504$$ 0 0
$$505$$ −15.5883 −0.693670
$$506$$ 0 0
$$507$$ −3.56724 −0.158427
$$508$$ 0 0
$$509$$ 23.0777 1.02290 0.511450 0.859313i $$-0.329109\pi$$
0.511450 + 0.859313i $$0.329109\pi$$
$$510$$ 0 0
$$511$$ 0.193061 0.00854051
$$512$$ 0 0
$$513$$ 11.4767 0.506710
$$514$$ 0 0
$$515$$ −9.25907 −0.408003
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 9.23854 0.405527
$$520$$ 0 0
$$521$$ 35.4191 1.55174 0.775868 0.630895i $$-0.217312\pi$$
0.775868 + 0.630895i $$0.217312\pi$$
$$522$$ 0 0
$$523$$ 18.1523 0.793745 0.396872 0.917874i $$-0.370096\pi$$
0.396872 + 0.917874i $$0.370096\pi$$
$$524$$ 0 0
$$525$$ −0.268343 −0.0117115
$$526$$ 0 0
$$527$$ 38.8955 1.69431
$$528$$ 0 0
$$529$$ −8.77004 −0.381306
$$530$$ 0 0
$$531$$ 4.10860 0.178298
$$532$$ 0 0
$$533$$ 10.5223 0.455770
$$534$$ 0 0
$$535$$ 2.15917 0.0933493
$$536$$ 0 0
$$537$$ −2.51766 −0.108645
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.2595 −0.699052 −0.349526 0.936927i $$-0.613657\pi$$
−0.349526 + 0.936927i $$0.613657\pi$$
$$542$$ 0 0
$$543$$ 1.27817 0.0548515
$$544$$ 0 0
$$545$$ 5.79052 0.248038
$$546$$ 0 0
$$547$$ −15.5972 −0.666886 −0.333443 0.942770i $$-0.608210\pi$$
−0.333443 + 0.942770i $$0.608210\pi$$
$$548$$ 0 0
$$549$$ 21.6655 0.924660
$$550$$ 0 0
$$551$$ 24.7399 1.05396
$$552$$ 0 0
$$553$$ 4.88359 0.207671
$$554$$ 0 0
$$555$$ 2.02585 0.0859925
$$556$$ 0 0
$$557$$ −10.5251 −0.445962 −0.222981 0.974823i $$-0.571579\pi$$
−0.222981 + 0.974823i $$0.571579\pi$$
$$558$$ 0 0
$$559$$ 18.3411 0.775747
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −11.7231 −0.494069 −0.247034 0.969007i $$-0.579456\pi$$
−0.247034 + 0.969007i $$0.579456\pi$$
$$564$$ 0 0
$$565$$ 19.0148 0.799959
$$566$$ 0 0
$$567$$ −1.76015 −0.0739194
$$568$$ 0 0
$$569$$ 9.57510 0.401409 0.200705 0.979652i $$-0.435677\pi$$
0.200705 + 0.979652i $$0.435677\pi$$
$$570$$ 0 0
$$571$$ −44.0439 −1.84318 −0.921589 0.388166i $$-0.873109\pi$$
−0.921589 + 0.388166i $$0.873109\pi$$
$$572$$ 0 0
$$573$$ 2.42620 0.101356
$$574$$ 0 0
$$575$$ 3.77226 0.157314
$$576$$ 0 0
$$577$$ −28.0466 −1.16760 −0.583798 0.811899i $$-0.698434\pi$$
−0.583798 + 0.811899i $$0.698434\pi$$
$$578$$ 0 0
$$579$$ 5.54662 0.230510
$$580$$ 0 0
$$581$$ 1.02953 0.0427123
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −7.04568 −0.291303
$$586$$ 0 0
$$587$$ −2.70939 −0.111828 −0.0559142 0.998436i $$-0.517807\pi$$
−0.0559142 + 0.998436i $$0.517807\pi$$
$$588$$ 0 0
$$589$$ −23.5567 −0.970637
$$590$$ 0 0
$$591$$ 5.06059 0.208165
$$592$$ 0 0
$$593$$ 35.1817 1.44474 0.722369 0.691508i $$-0.243053\pi$$
0.722369 + 0.691508i $$0.243053\pi$$
$$594$$ 0 0
$$595$$ −1.81980 −0.0746046
$$596$$ 0 0
$$597$$ 9.11192 0.372926
$$598$$ 0 0
$$599$$ −47.2602 −1.93100 −0.965500 0.260404i $$-0.916144\pi$$
−0.965500 + 0.260404i $$0.916144\pi$$
$$600$$ 0 0
$$601$$ −11.6431 −0.474932 −0.237466 0.971396i $$-0.576317\pi$$
−0.237466 + 0.971396i $$0.576317\pi$$
$$602$$ 0 0
$$603$$ −28.4973 −1.16050
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 45.9362 1.86449 0.932246 0.361825i $$-0.117846\pi$$
0.932246 + 0.361825i $$0.117846\pi$$
$$608$$ 0 0
$$609$$ 2.26588 0.0918180
$$610$$ 0 0
$$611$$ 7.32239 0.296232
$$612$$ 0 0
$$613$$ −3.35657 −0.135571 −0.0677854 0.997700i $$-0.521593\pi$$
−0.0677854 + 0.997700i $$0.521593\pi$$
$$614$$ 0 0
$$615$$ 2.65391 0.107016
$$616$$ 0 0
$$617$$ −16.4880 −0.663783 −0.331892 0.943318i $$-0.607687\pi$$
−0.331892 + 0.943318i $$0.607687\pi$$
$$618$$ 0 0
$$619$$ 2.05765 0.0827039 0.0413520 0.999145i $$-0.486834\pi$$
0.0413520 + 0.999145i $$0.486834\pi$$
$$620$$ 0 0
$$621$$ −14.7764 −0.592955
$$622$$ 0 0
$$623$$ −4.52142 −0.181147
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 13.7385 0.547791
$$630$$ 0 0
$$631$$ 35.2154 1.40190 0.700951 0.713209i $$-0.252759\pi$$
0.700951 + 0.713209i $$0.252759\pi$$
$$632$$ 0 0
$$633$$ −5.23634 −0.208126
$$634$$ 0 0
$$635$$ 10.0973 0.400700
$$636$$ 0 0
$$637$$ −19.3979 −0.768574
$$638$$ 0 0
$$639$$ −35.1787 −1.39165
$$640$$ 0 0
$$641$$ −33.6602 −1.32950 −0.664750 0.747066i $$-0.731462\pi$$
−0.664750 + 0.747066i $$0.731462\pi$$
$$642$$ 0 0
$$643$$ −2.01326 −0.0793951 −0.0396976 0.999212i $$-0.512639\pi$$
−0.0396976 + 0.999212i $$0.512639\pi$$
$$644$$ 0 0
$$645$$ 4.62598 0.182148
$$646$$ 0 0
$$647$$ 34.0410 1.33829 0.669145 0.743131i $$-0.266660\pi$$
0.669145 + 0.743131i $$0.266660\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −2.15751 −0.0845594
$$652$$ 0 0
$$653$$ −21.1626 −0.828156 −0.414078 0.910241i $$-0.635896\pi$$
−0.414078 + 0.910241i $$0.635896\pi$$
$$654$$ 0 0
$$655$$ 6.29651 0.246025
$$656$$ 0 0
$$657$$ 1.27851 0.0498794
$$658$$ 0 0
$$659$$ −21.3464 −0.831539 −0.415770 0.909470i $$-0.636488\pi$$
−0.415770 + 0.909470i $$0.636488\pi$$
$$660$$ 0 0
$$661$$ 11.3318 0.440757 0.220379 0.975414i $$-0.429271\pi$$
0.220379 + 0.975414i $$0.429271\pi$$
$$662$$ 0 0
$$663$$ 9.76041 0.379063
$$664$$ 0 0
$$665$$ 1.10215 0.0427394
$$666$$ 0 0
$$667$$ −31.8528 −1.23335
$$668$$ 0 0
$$669$$ 4.66443 0.180337
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 19.4852 0.751100 0.375550 0.926802i $$-0.377454\pi$$
0.375550 + 0.926802i $$0.377454\pi$$
$$674$$ 0 0
$$675$$ −3.91711 −0.150770
$$676$$ 0 0
$$677$$ 25.7596 0.990021 0.495010 0.868887i $$-0.335164\pi$$
0.495010 + 0.868887i $$0.335164\pi$$
$$678$$ 0 0
$$679$$ −6.04679 −0.232054
$$680$$ 0 0
$$681$$ −14.6566 −0.561643
$$682$$ 0 0
$$683$$ 13.0179 0.498117 0.249059 0.968488i $$-0.419879\pi$$
0.249059 + 0.968488i $$0.419879\pi$$
$$684$$ 0 0
$$685$$ −8.27329 −0.316106
$$686$$ 0 0
$$687$$ −3.08948 −0.117871
$$688$$ 0 0
$$689$$ −0.137916 −0.00525417
$$690$$ 0 0
$$691$$ −48.7463 −1.85440 −0.927198 0.374573i $$-0.877789\pi$$
−0.927198 + 0.374573i $$0.877789\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 10.2671 0.389453
$$696$$ 0 0
$$697$$ 17.9978 0.681717
$$698$$ 0 0
$$699$$ −20.4443 −0.773276
$$700$$ 0 0
$$701$$ −45.1532 −1.70541 −0.852706 0.522391i $$-0.825040\pi$$
−0.852706 + 0.522391i $$0.825040\pi$$
$$702$$ 0 0
$$703$$ −8.32062 −0.313818
$$704$$ 0 0
$$705$$ 1.84684 0.0695562
$$706$$ 0 0
$$707$$ 5.86388 0.220534
$$708$$ 0 0
$$709$$ −7.84526 −0.294635 −0.147318 0.989089i $$-0.547064\pi$$
−0.147318 + 0.989089i $$0.547064\pi$$
$$710$$ 0 0
$$711$$ 32.3406 1.21287
$$712$$ 0 0
$$713$$ 30.3294 1.13584
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 7.47352 0.279104
$$718$$ 0 0
$$719$$ 36.4371 1.35888 0.679438 0.733733i $$-0.262224\pi$$
0.679438 + 0.733733i $$0.262224\pi$$
$$720$$ 0 0
$$721$$ 3.48300 0.129714
$$722$$ 0 0
$$723$$ −11.0714 −0.411749
$$724$$ 0 0
$$725$$ −8.44396 −0.313601
$$726$$ 0 0
$$727$$ −15.4970 −0.574753 −0.287376 0.957818i $$-0.592783\pi$$
−0.287376 + 0.957818i $$0.592783\pi$$
$$728$$ 0 0
$$729$$ −3.27343 −0.121238
$$730$$ 0 0
$$731$$ 31.3716 1.16032
$$732$$ 0 0
$$733$$ −42.3920 −1.56579 −0.782893 0.622157i $$-0.786257\pi$$
−0.782893 + 0.622157i $$0.786257\pi$$
$$734$$ 0 0
$$735$$ −4.89252 −0.180463
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 30.3196 1.11532 0.557661 0.830069i $$-0.311699\pi$$
0.557661 + 0.830069i $$0.311699\pi$$
$$740$$ 0 0
$$741$$ −5.91130 −0.217157
$$742$$ 0 0
$$743$$ 20.8300 0.764179 0.382089 0.924125i $$-0.375205\pi$$
0.382089 + 0.924125i $$0.375205\pi$$
$$744$$ 0 0
$$745$$ 4.90160 0.179581
$$746$$ 0 0
$$747$$ 6.81790 0.249454
$$748$$ 0 0
$$749$$ −0.812221 −0.0296779
$$750$$ 0 0
$$751$$ −30.7746 −1.12298 −0.561491 0.827483i $$-0.689772\pi$$
−0.561491 + 0.827483i $$0.689772\pi$$
$$752$$ 0 0
$$753$$ −8.53891 −0.311175
$$754$$ 0 0
$$755$$ 9.35625 0.340509
$$756$$ 0 0
$$757$$ −25.3851 −0.922637 −0.461318 0.887235i $$-0.652623\pi$$
−0.461318 + 0.887235i $$0.652623\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33.6390 −1.21941 −0.609707 0.792627i $$-0.708713\pi$$
−0.609707 + 0.792627i $$0.708713\pi$$
$$762$$ 0 0
$$763$$ −2.17823 −0.0788573
$$764$$ 0 0
$$765$$ −12.0513 −0.435715
$$766$$ 0 0
$$767$$ −4.66471 −0.168433
$$768$$ 0 0
$$769$$ 29.4388 1.06159 0.530795 0.847500i $$-0.321893\pi$$
0.530795 + 0.847500i $$0.321893\pi$$
$$770$$ 0 0
$$771$$ −6.04453 −0.217688
$$772$$ 0 0
$$773$$ 24.7585 0.890501 0.445251 0.895406i $$-0.353115\pi$$
0.445251 + 0.895406i $$0.353115\pi$$
$$774$$ 0 0
$$775$$ 8.04011 0.288809
$$776$$ 0 0
$$777$$ −0.762068 −0.0273390
$$778$$ 0 0
$$779$$ −10.9002 −0.390541
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 33.0759 1.18204
$$784$$ 0 0
$$785$$ 9.65817 0.344715
$$786$$ 0 0
$$787$$ −37.4598 −1.33530 −0.667649 0.744476i $$-0.732699\pi$$
−0.667649 + 0.744476i $$0.732699\pi$$
$$788$$ 0 0
$$789$$ −8.93040 −0.317931
$$790$$ 0 0
$$791$$ −7.15284 −0.254326
$$792$$ 0 0
$$793$$ −24.5979 −0.873499
$$794$$ 0 0
$$795$$ −0.0347849 −0.00123369
$$796$$ 0 0
$$797$$ −48.1456 −1.70541 −0.852703 0.522397i $$-0.825038\pi$$
−0.852703 + 0.522397i $$0.825038\pi$$
$$798$$ 0 0
$$799$$ 12.5246 0.443088
$$800$$ 0 0
$$801$$ −29.9422 −1.05796
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −1.41902 −0.0500139
$$806$$ 0 0
$$807$$ −17.1667 −0.604297
$$808$$ 0 0
$$809$$ −37.2876 −1.31096 −0.655481 0.755211i $$-0.727534\pi$$
−0.655481 + 0.755211i $$0.727534\pi$$
$$810$$ 0 0
$$811$$ −8.87605 −0.311680 −0.155840 0.987782i $$-0.549808\pi$$
−0.155840 + 0.987782i $$0.549808\pi$$
$$812$$ 0 0
$$813$$ 6.72135 0.235728
$$814$$ 0 0
$$815$$ 4.48802 0.157208
$$816$$ 0 0
$$817$$ −18.9999 −0.664723
$$818$$ 0 0
$$819$$ 2.65039 0.0926121
$$820$$ 0 0
$$821$$ 31.1530 1.08725 0.543624 0.839329i $$-0.317052\pi$$
0.543624 + 0.839329i $$0.317052\pi$$
$$822$$ 0 0
$$823$$ −32.6349 −1.13758 −0.568790 0.822483i $$-0.692588\pi$$
−0.568790 + 0.822483i $$0.692588\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.7594 1.00006 0.500031 0.866008i $$-0.333322\pi$$
0.500031 + 0.866008i $$0.333322\pi$$
$$828$$ 0 0
$$829$$ −42.7970 −1.48640 −0.743200 0.669069i $$-0.766693\pi$$
−0.743200 + 0.669069i $$0.766693\pi$$
$$830$$ 0 0
$$831$$ −16.9643 −0.588485
$$832$$ 0 0
$$833$$ −33.1792 −1.14959
$$834$$ 0 0
$$835$$ 23.4716 0.812267
$$836$$ 0 0
$$837$$ −31.4940 −1.08859
$$838$$ 0 0
$$839$$ −28.5327 −0.985059 −0.492529 0.870296i $$-0.663928\pi$$
−0.492529 + 0.870296i $$0.663928\pi$$
$$840$$ 0 0
$$841$$ 42.3004 1.45863
$$842$$ 0 0
$$843$$ −0.354468 −0.0122085
$$844$$ 0 0
$$845$$ −5.00067 −0.172028
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −3.33870 −0.114584
$$850$$ 0 0
$$851$$ 10.7128 0.367232
$$852$$ 0 0
$$853$$ −33.4101 −1.14394 −0.571970 0.820275i $$-0.693821\pi$$
−0.571970 + 0.820275i $$0.693821\pi$$
$$854$$ 0 0
$$855$$ 7.29875 0.249612
$$856$$ 0 0
$$857$$ −11.9632 −0.408656 −0.204328 0.978902i $$-0.565501\pi$$
−0.204328 + 0.978902i $$0.565501\pi$$
$$858$$ 0 0
$$859$$ 23.4984 0.801755 0.400878 0.916132i $$-0.368705\pi$$
0.400878 + 0.916132i $$0.368705\pi$$
$$860$$ 0 0
$$861$$ −0.998328 −0.0340229
$$862$$ 0 0
$$863$$ 4.09414 0.139366 0.0696830 0.997569i $$-0.477801\pi$$
0.0696830 + 0.997569i $$0.477801\pi$$
$$864$$ 0 0
$$865$$ 12.9509 0.440343
$$866$$ 0 0
$$867$$ 4.56772 0.155128
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 32.3544 1.09629
$$872$$ 0 0
$$873$$ −40.0437 −1.35527
$$874$$ 0 0
$$875$$ −0.376172 −0.0127169
$$876$$ 0 0
$$877$$ −44.3667 −1.49816 −0.749079 0.662481i $$-0.769503\pi$$
−0.749079 + 0.662481i $$0.769503\pi$$
$$878$$ 0 0
$$879$$ −12.0701 −0.407115
$$880$$ 0 0
$$881$$ −43.0315 −1.44977 −0.724884 0.688871i $$-0.758107\pi$$
−0.724884 + 0.688871i $$0.758107\pi$$
$$882$$ 0 0
$$883$$ −36.5444 −1.22982 −0.614909 0.788598i $$-0.710807\pi$$
−0.614909 + 0.788598i $$0.710807\pi$$
$$884$$ 0 0
$$885$$ −1.17653 −0.0395485
$$886$$ 0 0
$$887$$ 1.79439 0.0602496 0.0301248 0.999546i $$-0.490410\pi$$
0.0301248 + 0.999546i $$0.490410\pi$$
$$888$$ 0 0
$$889$$ −3.79833 −0.127392
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −7.58541 −0.253836
$$894$$ 0 0
$$895$$ −3.52933 −0.117973
$$896$$ 0 0
$$897$$ 7.61084 0.254119
$$898$$ 0 0
$$899$$ −67.8903 −2.26427
$$900$$ 0 0
$$901$$ −0.235898 −0.00785890
$$902$$ 0 0
$$903$$ −1.74016 −0.0579090
$$904$$ 0 0
$$905$$ 1.79178 0.0595608
$$906$$ 0 0
$$907$$ −51.2671 −1.70230 −0.851149 0.524924i $$-0.824094\pi$$
−0.851149 + 0.524924i $$0.824094\pi$$
$$908$$ 0 0
$$909$$ 38.8324 1.28799
$$910$$ 0 0
$$911$$ 40.0803 1.32792 0.663960 0.747768i $$-0.268875\pi$$
0.663960 + 0.747768i $$0.268875\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −6.20406 −0.205100
$$916$$ 0 0
$$917$$ −2.36857 −0.0782171
$$918$$ 0 0
$$919$$ −1.08348 −0.0357406 −0.0178703 0.999840i $$-0.505689\pi$$
−0.0178703 + 0.999840i $$0.505689\pi$$
$$920$$ 0 0
$$921$$ 24.5156 0.807816
$$922$$ 0 0
$$923$$ 39.9402 1.31465
$$924$$ 0 0
$$925$$ 2.83990 0.0933754
$$926$$ 0 0
$$927$$ 23.0655 0.757571
$$928$$ 0 0
$$929$$ 14.2742 0.468320 0.234160 0.972198i $$-0.424766\pi$$
0.234160 + 0.972198i $$0.424766\pi$$
$$930$$ 0 0
$$931$$ 20.0947 0.658577
$$932$$ 0 0
$$933$$ 5.25591 0.172071
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 27.8901 0.911130 0.455565 0.890203i $$-0.349437\pi$$
0.455565 + 0.890203i $$0.349437\pi$$
$$938$$ 0 0
$$939$$ −17.8770 −0.583394
$$940$$ 0 0
$$941$$ −3.62440 −0.118152 −0.0590760 0.998253i $$-0.518815\pi$$
−0.0590760 + 0.998253i $$0.518815\pi$$
$$942$$ 0 0
$$943$$ 14.0341 0.457013
$$944$$ 0 0
$$945$$ 1.47351 0.0479332
$$946$$ 0 0
$$947$$ −54.9559 −1.78583 −0.892913 0.450230i $$-0.851342\pi$$
−0.892913 + 0.450230i $$0.851342\pi$$
$$948$$ 0 0
$$949$$ −1.45156 −0.0471196
$$950$$ 0 0
$$951$$ 20.0258 0.649382
$$952$$ 0 0
$$953$$ 9.03643 0.292719 0.146359 0.989231i $$-0.453244\pi$$
0.146359 + 0.989231i $$0.453244\pi$$
$$954$$ 0 0
$$955$$ 3.40113 0.110058
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3.11218 0.100498
$$960$$ 0 0
$$961$$ 33.6433 1.08527
$$962$$ 0 0
$$963$$ −5.37878 −0.173329
$$964$$ 0 0
$$965$$ 7.77543 0.250300
$$966$$ 0 0
$$967$$ 44.4619 1.42980 0.714900 0.699227i $$-0.246472\pi$$
0.714900 + 0.699227i $$0.246472\pi$$
$$968$$ 0 0
$$969$$ −10.1110 −0.324812
$$970$$ 0 0
$$971$$ 1.66867 0.0535502 0.0267751 0.999641i $$-0.491476\pi$$
0.0267751 + 0.999641i $$0.491476\pi$$
$$972$$ 0 0
$$973$$ −3.86219 −0.123816
$$974$$ 0 0
$$975$$ 2.01758 0.0646143
$$976$$ 0 0
$$977$$ 22.6537 0.724755 0.362377 0.932031i $$-0.381965\pi$$
0.362377 + 0.932031i $$0.381965\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −14.4249 −0.460552
$$982$$ 0 0
$$983$$ 46.9510 1.49750 0.748752 0.662850i $$-0.230653\pi$$
0.748752 + 0.662850i $$0.230653\pi$$
$$984$$ 0 0
$$985$$ 7.09410 0.226037
$$986$$ 0 0
$$987$$ −0.694731 −0.0221135
$$988$$ 0 0
$$989$$ 24.4625 0.777863
$$990$$ 0 0
$$991$$ −0.122612 −0.00389489 −0.00194745 0.999998i $$-0.500620\pi$$
−0.00194745 + 0.999998i $$0.500620\pi$$
$$992$$ 0 0
$$993$$ 15.6684 0.497221
$$994$$ 0 0
$$995$$ 12.7734 0.404943
$$996$$ 0 0
$$997$$ 20.1831 0.639205 0.319602 0.947552i $$-0.396451\pi$$
0.319602 + 0.947552i $$0.396451\pi$$
$$998$$ 0 0
$$999$$ −11.1242 −0.351954
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 4840.2.a.bh.1.5 8
4.3 odd 2 9680.2.a.de.1.4 8
11.2 odd 10 440.2.y.d.81.3 16
11.6 odd 10 440.2.y.d.201.3 yes 16
11.10 odd 2 4840.2.a.bg.1.5 8
44.35 even 10 880.2.bo.k.81.2 16
44.39 even 10 880.2.bo.k.641.2 16
44.43 even 2 9680.2.a.df.1.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.3 16 11.2 odd 10
440.2.y.d.201.3 yes 16 11.6 odd 10
880.2.bo.k.81.2 16 44.35 even 10
880.2.bo.k.641.2 16 44.39 even 10
4840.2.a.bg.1.5 8 11.10 odd 2
4840.2.a.bh.1.5 8 1.1 even 1 trivial
9680.2.a.de.1.4 8 4.3 odd 2
9680.2.a.df.1.4 8 44.43 even 2