# Properties

 Label 9680.2.a.cg.1.2 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.12489$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.484862 q^{3} +1.00000 q^{5} -0.484862 q^{7} -2.76491 q^{9} +O(q^{10})$$ $$q+0.484862 q^{3} +1.00000 q^{5} -0.484862 q^{7} -2.76491 q^{9} -5.28005 q^{13} +0.484862 q^{15} -2.48486 q^{17} -4.73463 q^{19} -0.235091 q^{21} -4.24977 q^{23} +1.00000 q^{25} -2.79518 q^{27} +3.76491 q^{29} +0.235091 q^{31} -0.484862 q^{35} +5.76491 q^{37} -2.56009 q^{39} +0.969724 q^{41} -0.249771 q^{43} -2.76491 q^{45} -3.28005 q^{47} -6.76491 q^{49} -1.20482 q^{51} +5.76491 q^{53} -2.29564 q^{57} +12.4995 q^{59} -9.70436 q^{61} +1.34060 q^{63} -5.28005 q^{65} +10.3103 q^{67} -2.06055 q^{69} +5.70436 q^{71} +1.75023 q^{73} +0.484862 q^{75} +12.0000 q^{79} +6.93945 q^{81} +16.7493 q^{83} -2.48486 q^{85} +1.82546 q^{87} +3.82546 q^{89} +2.56009 q^{91} +0.113987 q^{93} -4.73463 q^{95} +6.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 - q^7 + 8 * q^9 $$3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9} + q^{15} - 7 q^{17} + 3 q^{19} - 17 q^{21} + 4 q^{23} + 3 q^{25} + 7 q^{27} - 5 q^{29} + 17 q^{31} - q^{35} + q^{37} + 24 q^{39} + 2 q^{41} + 16 q^{43} + 8 q^{45} + 6 q^{47} - 4 q^{49} - 19 q^{51} + q^{53} - 25 q^{57} + 4 q^{59} - 11 q^{61} - 10 q^{63} + 16 q^{67} - 8 q^{69} - q^{71} + 22 q^{73} + q^{75} + 36 q^{79} + 19 q^{81} - 7 q^{85} - 9 q^{87} - 3 q^{89} - 24 q^{91} + 13 q^{93} + 3 q^{95} + 18 q^{97}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 - q^7 + 8 * q^9 + q^15 - 7 * q^17 + 3 * q^19 - 17 * q^21 + 4 * q^23 + 3 * q^25 + 7 * q^27 - 5 * q^29 + 17 * q^31 - q^35 + q^37 + 24 * q^39 + 2 * q^41 + 16 * q^43 + 8 * q^45 + 6 * q^47 - 4 * q^49 - 19 * q^51 + q^53 - 25 * q^57 + 4 * q^59 - 11 * q^61 - 10 * q^63 + 16 * q^67 - 8 * q^69 - q^71 + 22 * q^73 + q^75 + 36 * q^79 + 19 * q^81 - 7 * q^85 - 9 * q^87 - 3 * q^89 - 24 * q^91 + 13 * q^93 + 3 * q^95 + 18 * 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
$$3$$ 0.484862 0.279935 0.139968 0.990156i $$-0.455300\pi$$
0.139968 + 0.990156i $$0.455300\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −0.484862 −0.183261 −0.0916303 0.995793i $$-0.529208\pi$$
−0.0916303 + 0.995793i $$0.529208\pi$$
$$8$$ 0 0
$$9$$ −2.76491 −0.921636
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −5.28005 −1.46442 −0.732211 0.681078i $$-0.761511\pi$$
−0.732211 + 0.681078i $$0.761511\pi$$
$$14$$ 0 0
$$15$$ 0.484862 0.125191
$$16$$ 0 0
$$17$$ −2.48486 −0.602668 −0.301334 0.953519i $$-0.597432\pi$$
−0.301334 + 0.953519i $$0.597432\pi$$
$$18$$ 0 0
$$19$$ −4.73463 −1.08620 −0.543100 0.839668i $$-0.682749\pi$$
−0.543100 + 0.839668i $$0.682749\pi$$
$$20$$ 0 0
$$21$$ −0.235091 −0.0513011
$$22$$ 0 0
$$23$$ −4.24977 −0.886138 −0.443069 0.896487i $$-0.646110\pi$$
−0.443069 + 0.896487i $$0.646110\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.79518 −0.537934
$$28$$ 0 0
$$29$$ 3.76491 0.699126 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$30$$ 0 0
$$31$$ 0.235091 0.0422236 0.0211118 0.999777i $$-0.493279\pi$$
0.0211118 + 0.999777i $$0.493279\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.484862 −0.0819566
$$36$$ 0 0
$$37$$ 5.76491 0.947745 0.473873 0.880593i $$-0.342856\pi$$
0.473873 + 0.880593i $$0.342856\pi$$
$$38$$ 0 0
$$39$$ −2.56009 −0.409943
$$40$$ 0 0
$$41$$ 0.969724 0.151445 0.0757227 0.997129i $$-0.475874\pi$$
0.0757227 + 0.997129i $$0.475874\pi$$
$$42$$ 0 0
$$43$$ −0.249771 −0.0380897 −0.0190448 0.999819i $$-0.506063\pi$$
−0.0190448 + 0.999819i $$0.506063\pi$$
$$44$$ 0 0
$$45$$ −2.76491 −0.412168
$$46$$ 0 0
$$47$$ −3.28005 −0.478444 −0.239222 0.970965i $$-0.576892\pi$$
−0.239222 + 0.970965i $$0.576892\pi$$
$$48$$ 0 0
$$49$$ −6.76491 −0.966416
$$50$$ 0 0
$$51$$ −1.20482 −0.168708
$$52$$ 0 0
$$53$$ 5.76491 0.791871 0.395936 0.918278i $$-0.370420\pi$$
0.395936 + 0.918278i $$0.370420\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.29564 −0.304065
$$58$$ 0 0
$$59$$ 12.4995 1.62730 0.813651 0.581354i $$-0.197477\pi$$
0.813651 + 0.581354i $$0.197477\pi$$
$$60$$ 0 0
$$61$$ −9.70436 −1.24252 −0.621258 0.783606i $$-0.713378\pi$$
−0.621258 + 0.783606i $$0.713378\pi$$
$$62$$ 0 0
$$63$$ 1.34060 0.168900
$$64$$ 0 0
$$65$$ −5.28005 −0.654909
$$66$$ 0 0
$$67$$ 10.3103 1.25961 0.629803 0.776755i $$-0.283136\pi$$
0.629803 + 0.776755i $$0.283136\pi$$
$$68$$ 0 0
$$69$$ −2.06055 −0.248061
$$70$$ 0 0
$$71$$ 5.70436 0.676983 0.338491 0.940970i $$-0.390083\pi$$
0.338491 + 0.940970i $$0.390083\pi$$
$$72$$ 0 0
$$73$$ 1.75023 0.204849 0.102424 0.994741i $$-0.467340\pi$$
0.102424 + 0.994741i $$0.467340\pi$$
$$74$$ 0 0
$$75$$ 0.484862 0.0559870
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 6.93945 0.771050
$$82$$ 0 0
$$83$$ 16.7493 1.83848 0.919238 0.393702i $$-0.128806\pi$$
0.919238 + 0.393702i $$0.128806\pi$$
$$84$$ 0 0
$$85$$ −2.48486 −0.269521
$$86$$ 0 0
$$87$$ 1.82546 0.195710
$$88$$ 0 0
$$89$$ 3.82546 0.405498 0.202749 0.979231i $$-0.435012\pi$$
0.202749 + 0.979231i $$0.435012\pi$$
$$90$$ 0 0
$$91$$ 2.56009 0.268371
$$92$$ 0 0
$$93$$ 0.113987 0.0118199
$$94$$ 0 0
$$95$$ −4.73463 −0.485763
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.03028 −0.301524 −0.150762 0.988570i $$-0.548173\pi$$
−0.150762 + 0.988570i $$0.548173\pi$$
$$102$$ 0 0
$$103$$ 4.71995 0.465071 0.232535 0.972588i $$-0.425298\pi$$
0.232535 + 0.972588i $$0.425298\pi$$
$$104$$ 0 0
$$105$$ −0.235091 −0.0229425
$$106$$ 0 0
$$107$$ 1.21949 0.117893 0.0589465 0.998261i $$-0.481226\pi$$
0.0589465 + 0.998261i $$0.481226\pi$$
$$108$$ 0 0
$$109$$ −16.0294 −1.53533 −0.767667 0.640849i $$-0.778583\pi$$
−0.767667 + 0.640849i $$0.778583\pi$$
$$110$$ 0 0
$$111$$ 2.79518 0.265307
$$112$$ 0 0
$$113$$ −11.4693 −1.07894 −0.539469 0.842006i $$-0.681375\pi$$
−0.539469 + 0.842006i $$0.681375\pi$$
$$114$$ 0 0
$$115$$ −4.24977 −0.396293
$$116$$ 0 0
$$117$$ 14.5988 1.34966
$$118$$ 0 0
$$119$$ 1.20482 0.110445
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 0.470182 0.0423949
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.24977 0.377106 0.188553 0.982063i $$-0.439620\pi$$
0.188553 + 0.982063i $$0.439620\pi$$
$$128$$ 0 0
$$129$$ −0.121104 −0.0106626
$$130$$ 0 0
$$131$$ 15.7649 1.37739 0.688693 0.725053i $$-0.258185\pi$$
0.688693 + 0.725053i $$0.258185\pi$$
$$132$$ 0 0
$$133$$ 2.29564 0.199058
$$134$$ 0 0
$$135$$ −2.79518 −0.240571
$$136$$ 0 0
$$137$$ 13.5298 1.15593 0.577965 0.816061i $$-0.303847\pi$$
0.577965 + 0.816061i $$0.303847\pi$$
$$138$$ 0 0
$$139$$ −16.6206 −1.40974 −0.704872 0.709334i $$-0.748996\pi$$
−0.704872 + 0.709334i $$0.748996\pi$$
$$140$$ 0 0
$$141$$ −1.59037 −0.133933
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.76491 0.312659
$$146$$ 0 0
$$147$$ −3.28005 −0.270534
$$148$$ 0 0
$$149$$ −5.20482 −0.426395 −0.213198 0.977009i $$-0.568388\pi$$
−0.213198 + 0.977009i $$0.568388\pi$$
$$150$$ 0 0
$$151$$ 22.5601 1.83591 0.917957 0.396679i $$-0.129838\pi$$
0.917957 + 0.396679i $$0.129838\pi$$
$$152$$ 0 0
$$153$$ 6.87042 0.555440
$$154$$ 0 0
$$155$$ 0.235091 0.0188830
$$156$$ 0 0
$$157$$ 3.70436 0.295640 0.147820 0.989014i $$-0.452774\pi$$
0.147820 + 0.989014i $$0.452774\pi$$
$$158$$ 0 0
$$159$$ 2.79518 0.221673
$$160$$ 0 0
$$161$$ 2.06055 0.162394
$$162$$ 0 0
$$163$$ 15.5445 1.21754 0.608770 0.793347i $$-0.291663\pi$$
0.608770 + 0.793347i $$0.291663\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 22.0752 1.70823 0.854116 0.520082i $$-0.174099\pi$$
0.854116 + 0.520082i $$0.174099\pi$$
$$168$$ 0 0
$$169$$ 14.8789 1.14453
$$170$$ 0 0
$$171$$ 13.0908 1.00108
$$172$$ 0 0
$$173$$ −2.24977 −0.171047 −0.0855235 0.996336i $$-0.527256\pi$$
−0.0855235 + 0.996336i $$0.527256\pi$$
$$174$$ 0 0
$$175$$ −0.484862 −0.0366521
$$176$$ 0 0
$$177$$ 6.06055 0.455539
$$178$$ 0 0
$$179$$ −5.09083 −0.380506 −0.190253 0.981735i $$-0.560931\pi$$
−0.190253 + 0.981735i $$0.560931\pi$$
$$180$$ 0 0
$$181$$ 7.46927 0.555186 0.277593 0.960699i $$-0.410463\pi$$
0.277593 + 0.960699i $$0.410463\pi$$
$$182$$ 0 0
$$183$$ −4.70527 −0.347824
$$184$$ 0 0
$$185$$ 5.76491 0.423845
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.35528 0.0985820
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −2.48486 −0.178864 −0.0894321 0.995993i $$-0.528505\pi$$
−0.0894321 + 0.995993i $$0.528505\pi$$
$$194$$ 0 0
$$195$$ −2.56009 −0.183332
$$196$$ 0 0
$$197$$ 16.6888 1.18902 0.594512 0.804086i $$-0.297345\pi$$
0.594512 + 0.804086i $$0.297345\pi$$
$$198$$ 0 0
$$199$$ −4.73463 −0.335629 −0.167815 0.985819i $$-0.553671\pi$$
−0.167815 + 0.985819i $$0.553671\pi$$
$$200$$ 0 0
$$201$$ 4.99908 0.352608
$$202$$ 0 0
$$203$$ −1.82546 −0.128122
$$204$$ 0 0
$$205$$ 0.969724 0.0677285
$$206$$ 0 0
$$207$$ 11.7502 0.816697
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −0.235091 −0.0161843 −0.00809217 0.999967i $$-0.502576\pi$$
−0.00809217 + 0.999967i $$0.502576\pi$$
$$212$$ 0 0
$$213$$ 2.76583 0.189511
$$214$$ 0 0
$$215$$ −0.249771 −0.0170342
$$216$$ 0 0
$$217$$ −0.113987 −0.00773792
$$218$$ 0 0
$$219$$ 0.848620 0.0573444
$$220$$ 0 0
$$221$$ 13.1202 0.882559
$$222$$ 0 0
$$223$$ −19.7796 −1.32454 −0.662270 0.749266i $$-0.730407\pi$$
−0.662270 + 0.749266i $$0.730407\pi$$
$$224$$ 0 0
$$225$$ −2.76491 −0.184327
$$226$$ 0 0
$$227$$ −17.2195 −1.14290 −0.571449 0.820638i $$-0.693619\pi$$
−0.571449 + 0.820638i $$0.693619\pi$$
$$228$$ 0 0
$$229$$ 9.40871 0.621745 0.310873 0.950452i $$-0.399379\pi$$
0.310873 + 0.950452i $$0.399379\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.9239 −1.63282 −0.816408 0.577476i $$-0.804038\pi$$
−0.816408 + 0.577476i $$0.804038\pi$$
$$234$$ 0 0
$$235$$ −3.28005 −0.213967
$$236$$ 0 0
$$237$$ 5.81834 0.377942
$$238$$ 0 0
$$239$$ −11.5298 −0.745802 −0.372901 0.927871i $$-0.621637\pi$$
−0.372901 + 0.927871i $$0.621637\pi$$
$$240$$ 0 0
$$241$$ −15.0303 −0.968185 −0.484093 0.875017i $$-0.660850\pi$$
−0.484093 + 0.875017i $$0.660850\pi$$
$$242$$ 0 0
$$243$$ 11.7502 0.753778
$$244$$ 0 0
$$245$$ −6.76491 −0.432194
$$246$$ 0 0
$$247$$ 24.9991 1.59065
$$248$$ 0 0
$$249$$ 8.12110 0.514654
$$250$$ 0 0
$$251$$ −5.93945 −0.374895 −0.187447 0.982275i $$-0.560021\pi$$
−0.187447 + 0.982275i $$0.560021\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −1.20482 −0.0754484
$$256$$ 0 0
$$257$$ −3.59037 −0.223961 −0.111981 0.993710i $$-0.535719\pi$$
−0.111981 + 0.993710i $$0.535719\pi$$
$$258$$ 0 0
$$259$$ −2.79518 −0.173684
$$260$$ 0 0
$$261$$ −10.4096 −0.644340
$$262$$ 0 0
$$263$$ 16.9844 1.04730 0.523652 0.851933i $$-0.324569\pi$$
0.523652 + 0.851933i $$0.324569\pi$$
$$264$$ 0 0
$$265$$ 5.76491 0.354136
$$266$$ 0 0
$$267$$ 1.85482 0.113513
$$268$$ 0 0
$$269$$ 7.59037 0.462793 0.231397 0.972860i $$-0.425671\pi$$
0.231397 + 0.972860i $$0.425671\pi$$
$$270$$ 0 0
$$271$$ −14.5601 −0.884463 −0.442231 0.896901i $$-0.645813\pi$$
−0.442231 + 0.896901i $$0.645813\pi$$
$$272$$ 0 0
$$273$$ 1.24129 0.0751264
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −28.8099 −1.73102 −0.865509 0.500894i $$-0.833005\pi$$
−0.865509 + 0.500894i $$0.833005\pi$$
$$278$$ 0 0
$$279$$ −0.650006 −0.0389148
$$280$$ 0 0
$$281$$ −16.9697 −1.01233 −0.506164 0.862437i $$-0.668937\pi$$
−0.506164 + 0.862437i $$0.668937\pi$$
$$282$$ 0 0
$$283$$ 9.68968 0.575992 0.287996 0.957632i $$-0.407011\pi$$
0.287996 + 0.957632i $$0.407011\pi$$
$$284$$ 0 0
$$285$$ −2.29564 −0.135982
$$286$$ 0 0
$$287$$ −0.470182 −0.0277540
$$288$$ 0 0
$$289$$ −10.8255 −0.636792
$$290$$ 0 0
$$291$$ 2.90917 0.170539
$$292$$ 0 0
$$293$$ 6.37088 0.372191 0.186095 0.982532i $$-0.440417\pi$$
0.186095 + 0.982532i $$0.440417\pi$$
$$294$$ 0 0
$$295$$ 12.4995 0.727751
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 22.4390 1.29768
$$300$$ 0 0
$$301$$ 0.121104 0.00698034
$$302$$ 0 0
$$303$$ −1.46927 −0.0844071
$$304$$ 0 0
$$305$$ −9.70436 −0.555670
$$306$$ 0 0
$$307$$ −27.6585 −1.57855 −0.789277 0.614038i $$-0.789544\pi$$
−0.789277 + 0.614038i $$0.789544\pi$$
$$308$$ 0 0
$$309$$ 2.28853 0.130190
$$310$$ 0 0
$$311$$ −13.7044 −0.777103 −0.388551 0.921427i $$-0.627024\pi$$
−0.388551 + 0.921427i $$0.627024\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 0 0
$$315$$ 1.34060 0.0755342
$$316$$ 0 0
$$317$$ 12.1745 0.683790 0.341895 0.939738i $$-0.388931\pi$$
0.341895 + 0.939738i $$0.388931\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0.591287 0.0330024
$$322$$ 0 0
$$323$$ 11.7649 0.654617
$$324$$ 0 0
$$325$$ −5.28005 −0.292884
$$326$$ 0 0
$$327$$ −7.77203 −0.429794
$$328$$ 0 0
$$329$$ 1.59037 0.0876799
$$330$$ 0 0
$$331$$ 29.1202 1.60059 0.800295 0.599606i $$-0.204676\pi$$
0.800295 + 0.599606i $$0.204676\pi$$
$$332$$ 0 0
$$333$$ −15.9394 −0.873476
$$334$$ 0 0
$$335$$ 10.3103 0.563313
$$336$$ 0 0
$$337$$ 16.9239 0.921901 0.460950 0.887426i $$-0.347509\pi$$
0.460950 + 0.887426i $$0.347509\pi$$
$$338$$ 0 0
$$339$$ −5.56101 −0.302033
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 6.67408 0.360366
$$344$$ 0 0
$$345$$ −2.06055 −0.110936
$$346$$ 0 0
$$347$$ −7.77959 −0.417630 −0.208815 0.977955i $$-0.566961\pi$$
−0.208815 + 0.977955i $$0.566961\pi$$
$$348$$ 0 0
$$349$$ −7.52982 −0.403062 −0.201531 0.979482i $$-0.564592\pi$$
−0.201531 + 0.979482i $$0.564592\pi$$
$$350$$ 0 0
$$351$$ 14.7587 0.787762
$$352$$ 0 0
$$353$$ −22.5289 −1.19909 −0.599546 0.800340i $$-0.704652\pi$$
−0.599546 + 0.800340i $$0.704652\pi$$
$$354$$ 0 0
$$355$$ 5.70436 0.302756
$$356$$ 0 0
$$357$$ 0.584169 0.0309175
$$358$$ 0 0
$$359$$ 17.3482 0.915601 0.457800 0.889055i $$-0.348637\pi$$
0.457800 + 0.889055i $$0.348637\pi$$
$$360$$ 0 0
$$361$$ 3.41675 0.179829
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.75023 0.0916112
$$366$$ 0 0
$$367$$ 27.3094 1.42554 0.712770 0.701398i $$-0.247440\pi$$
0.712770 + 0.701398i $$0.247440\pi$$
$$368$$ 0 0
$$369$$ −2.68120 −0.139578
$$370$$ 0 0
$$371$$ −2.79518 −0.145119
$$372$$ 0 0
$$373$$ −2.24977 −0.116489 −0.0582444 0.998302i $$-0.518550\pi$$
−0.0582444 + 0.998302i $$0.518550\pi$$
$$374$$ 0 0
$$375$$ 0.484862 0.0250382
$$376$$ 0 0
$$377$$ −19.8789 −1.02382
$$378$$ 0 0
$$379$$ −22.4390 −1.15261 −0.576307 0.817233i $$-0.695507\pi$$
−0.576307 + 0.817233i $$0.695507\pi$$
$$380$$ 0 0
$$381$$ 2.06055 0.105565
$$382$$ 0 0
$$383$$ −5.21949 −0.266704 −0.133352 0.991069i $$-0.542574\pi$$
−0.133352 + 0.991069i $$0.542574\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.690594 0.0351048
$$388$$ 0 0
$$389$$ 27.5904 1.39889 0.699444 0.714688i $$-0.253431\pi$$
0.699444 + 0.714688i $$0.253431\pi$$
$$390$$ 0 0
$$391$$ 10.5601 0.534047
$$392$$ 0 0
$$393$$ 7.64380 0.385579
$$394$$ 0 0
$$395$$ 12.0000 0.603786
$$396$$ 0 0
$$397$$ 32.9385 1.65314 0.826569 0.562836i $$-0.190290\pi$$
0.826569 + 0.562836i $$0.190290\pi$$
$$398$$ 0 0
$$399$$ 1.11307 0.0557232
$$400$$ 0 0
$$401$$ 0.295643 0.0147637 0.00738186 0.999973i $$-0.497650\pi$$
0.00738186 + 0.999973i $$0.497650\pi$$
$$402$$ 0 0
$$403$$ −1.24129 −0.0618332
$$404$$ 0 0
$$405$$ 6.93945 0.344824
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −31.4087 −1.55306 −0.776530 0.630080i $$-0.783022\pi$$
−0.776530 + 0.630080i $$0.783022\pi$$
$$410$$ 0 0
$$411$$ 6.56009 0.323586
$$412$$ 0 0
$$413$$ −6.06055 −0.298220
$$414$$ 0 0
$$415$$ 16.7493 0.822191
$$416$$ 0 0
$$417$$ −8.05872 −0.394637
$$418$$ 0 0
$$419$$ 32.0294 1.56474 0.782368 0.622816i $$-0.214012\pi$$
0.782368 + 0.622816i $$0.214012\pi$$
$$420$$ 0 0
$$421$$ −28.9385 −1.41038 −0.705189 0.709020i $$-0.749138\pi$$
−0.705189 + 0.709020i $$0.749138\pi$$
$$422$$ 0 0
$$423$$ 9.06903 0.440951
$$424$$ 0 0
$$425$$ −2.48486 −0.120534
$$426$$ 0 0
$$427$$ 4.70527 0.227704
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.6197 −1.61941 −0.809703 0.586840i $$-0.800372\pi$$
−0.809703 + 0.586840i $$0.800372\pi$$
$$432$$ 0 0
$$433$$ 29.0596 1.39652 0.698258 0.715846i $$-0.253959\pi$$
0.698258 + 0.715846i $$0.253959\pi$$
$$434$$ 0 0
$$435$$ 1.82546 0.0875242
$$436$$ 0 0
$$437$$ 20.1211 0.962523
$$438$$ 0 0
$$439$$ −15.5298 −0.741198 −0.370599 0.928793i $$-0.620848\pi$$
−0.370599 + 0.928793i $$0.620848\pi$$
$$440$$ 0 0
$$441$$ 18.7044 0.890684
$$442$$ 0 0
$$443$$ −24.6282 −1.17012 −0.585061 0.810989i $$-0.698929\pi$$
−0.585061 + 0.810989i $$0.698929\pi$$
$$444$$ 0 0
$$445$$ 3.82546 0.181344
$$446$$ 0 0
$$447$$ −2.52362 −0.119363
$$448$$ 0 0
$$449$$ −8.06055 −0.380401 −0.190200 0.981745i $$-0.560914\pi$$
−0.190200 + 0.981745i $$0.560914\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 10.9385 0.513937
$$454$$ 0 0
$$455$$ 2.56009 0.120019
$$456$$ 0 0
$$457$$ 10.4849 0.490461 0.245231 0.969465i $$-0.421136\pi$$
0.245231 + 0.969465i $$0.421136\pi$$
$$458$$ 0 0
$$459$$ 6.94565 0.324195
$$460$$ 0 0
$$461$$ −36.3856 −1.69464 −0.847322 0.531079i $$-0.821787\pi$$
−0.847322 + 0.531079i $$0.821787\pi$$
$$462$$ 0 0
$$463$$ −27.3094 −1.26918 −0.634588 0.772851i $$-0.718830\pi$$
−0.634588 + 0.772851i $$0.718830\pi$$
$$464$$ 0 0
$$465$$ 0.113987 0.00528601
$$466$$ 0 0
$$467$$ 12.6060 0.583335 0.291667 0.956520i $$-0.405790\pi$$
0.291667 + 0.956520i $$0.405790\pi$$
$$468$$ 0 0
$$469$$ −4.99908 −0.230836
$$470$$ 0 0
$$471$$ 1.79610 0.0827600
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.73463 −0.217240
$$476$$ 0 0
$$477$$ −15.9394 −0.729817
$$478$$ 0 0
$$479$$ 20.5289 0.937989 0.468995 0.883201i $$-0.344616\pi$$
0.468995 + 0.883201i $$0.344616\pi$$
$$480$$ 0 0
$$481$$ −30.4390 −1.38790
$$482$$ 0 0
$$483$$ 0.999083 0.0454599
$$484$$ 0 0
$$485$$ 6.00000 0.272446
$$486$$ 0 0
$$487$$ −6.68876 −0.303097 −0.151548 0.988450i $$-0.548426\pi$$
−0.151548 + 0.988450i $$0.548426\pi$$
$$488$$ 0 0
$$489$$ 7.53694 0.340832
$$490$$ 0 0
$$491$$ 40.8851 1.84512 0.922559 0.385855i $$-0.126094\pi$$
0.922559 + 0.385855i $$0.126094\pi$$
$$492$$ 0 0
$$493$$ −9.35528 −0.421341
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −2.76583 −0.124064
$$498$$ 0 0
$$499$$ 20.0294 0.896637 0.448319 0.893874i $$-0.352023\pi$$
0.448319 + 0.893874i $$0.352023\pi$$
$$500$$ 0 0
$$501$$ 10.7034 0.478194
$$502$$ 0 0
$$503$$ −17.8108 −0.794143 −0.397072 0.917788i $$-0.629974\pi$$
−0.397072 + 0.917788i $$0.629974\pi$$
$$504$$ 0 0
$$505$$ −3.03028 −0.134846
$$506$$ 0 0
$$507$$ 7.21421 0.320394
$$508$$ 0 0
$$509$$ 8.93853 0.396193 0.198097 0.980182i $$-0.436524\pi$$
0.198097 + 0.980182i $$0.436524\pi$$
$$510$$ 0 0
$$511$$ −0.848620 −0.0375407
$$512$$ 0 0
$$513$$ 13.2342 0.584303
$$514$$ 0 0
$$515$$ 4.71995 0.207986
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −1.09083 −0.0478820
$$520$$ 0 0
$$521$$ 29.0596 1.27313 0.636563 0.771225i $$-0.280356\pi$$
0.636563 + 0.771225i $$0.280356\pi$$
$$522$$ 0 0
$$523$$ −32.2791 −1.41147 −0.705734 0.708477i $$-0.749383\pi$$
−0.705734 + 0.708477i $$0.749383\pi$$
$$524$$ 0 0
$$525$$ −0.235091 −0.0102602
$$526$$ 0 0
$$527$$ −0.584169 −0.0254468
$$528$$ 0 0
$$529$$ −4.93945 −0.214759
$$530$$ 0 0
$$531$$ −34.5601 −1.49978
$$532$$ 0 0
$$533$$ −5.12019 −0.221780
$$534$$ 0 0
$$535$$ 1.21949 0.0527234
$$536$$ 0 0
$$537$$ −2.46835 −0.106517
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.3250 1.64772 0.823860 0.566793i $$-0.191816\pi$$
0.823860 + 0.566793i $$0.191816\pi$$
$$542$$ 0 0
$$543$$ 3.62156 0.155416
$$544$$ 0 0
$$545$$ −16.0294 −0.686622
$$546$$ 0 0
$$547$$ 9.71904 0.415556 0.207778 0.978176i $$-0.433377\pi$$
0.207778 + 0.978176i $$0.433377\pi$$
$$548$$ 0 0
$$549$$ 26.8317 1.14515
$$550$$ 0 0
$$551$$ −17.8255 −0.759390
$$552$$ 0 0
$$553$$ −5.81834 −0.247421
$$554$$ 0 0
$$555$$ 2.79518 0.118649
$$556$$ 0 0
$$557$$ −16.7805 −0.711013 −0.355506 0.934674i $$-0.615692\pi$$
−0.355506 + 0.934674i $$0.615692\pi$$
$$558$$ 0 0
$$559$$ 1.31880 0.0557794
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 25.7190 1.08393 0.541964 0.840402i $$-0.317681\pi$$
0.541964 + 0.840402i $$0.317681\pi$$
$$564$$ 0 0
$$565$$ −11.4693 −0.482516
$$566$$ 0 0
$$567$$ −3.36467 −0.141303
$$568$$ 0 0
$$569$$ 39.4087 1.65210 0.826050 0.563597i $$-0.190583\pi$$
0.826050 + 0.563597i $$0.190583\pi$$
$$570$$ 0 0
$$571$$ 40.4149 1.69131 0.845656 0.533729i $$-0.179210\pi$$
0.845656 + 0.533729i $$0.179210\pi$$
$$572$$ 0 0
$$573$$ 5.81834 0.243065
$$574$$ 0 0
$$575$$ −4.24977 −0.177228
$$576$$ 0 0
$$577$$ −7.34816 −0.305908 −0.152954 0.988233i $$-0.548879\pi$$
−0.152954 + 0.988233i $$0.548879\pi$$
$$578$$ 0 0
$$579$$ −1.20482 −0.0500704
$$580$$ 0 0
$$581$$ −8.12110 −0.336920
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 14.5988 0.603588
$$586$$ 0 0
$$587$$ −41.9835 −1.73284 −0.866422 0.499312i $$-0.833586\pi$$
−0.866422 + 0.499312i $$0.833586\pi$$
$$588$$ 0 0
$$589$$ −1.11307 −0.0458633
$$590$$ 0 0
$$591$$ 8.09174 0.332850
$$592$$ 0 0
$$593$$ −19.3700 −0.795429 −0.397714 0.917509i $$-0.630197\pi$$
−0.397714 + 0.917509i $$0.630197\pi$$
$$594$$ 0 0
$$595$$ 1.20482 0.0493926
$$596$$ 0 0
$$597$$ −2.29564 −0.0939544
$$598$$ 0 0
$$599$$ −4.14335 −0.169293 −0.0846463 0.996411i $$-0.526976\pi$$
−0.0846463 + 0.996411i $$0.526976\pi$$
$$600$$ 0 0
$$601$$ −15.0303 −0.613098 −0.306549 0.951855i $$-0.599174\pi$$
−0.306549 + 0.951855i $$0.599174\pi$$
$$602$$ 0 0
$$603$$ −28.5071 −1.16090
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 12.6060 0.511660 0.255830 0.966722i $$-0.417651\pi$$
0.255830 + 0.966722i $$0.417651\pi$$
$$608$$ 0 0
$$609$$ −0.885097 −0.0358659
$$610$$ 0 0
$$611$$ 17.3188 0.700644
$$612$$ 0 0
$$613$$ 14.2791 0.576729 0.288364 0.957521i $$-0.406889\pi$$
0.288364 + 0.957521i $$0.406889\pi$$
$$614$$ 0 0
$$615$$ 0.470182 0.0189596
$$616$$ 0 0
$$617$$ 8.18166 0.329381 0.164691 0.986345i $$-0.447337\pi$$
0.164691 + 0.986345i $$0.447337\pi$$
$$618$$ 0 0
$$619$$ −39.0890 −1.57112 −0.785560 0.618786i $$-0.787625\pi$$
−0.785560 + 0.618786i $$0.787625\pi$$
$$620$$ 0 0
$$621$$ 11.8789 0.476684
$$622$$ 0 0
$$623$$ −1.85482 −0.0743118
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −14.3250 −0.571175
$$630$$ 0 0
$$631$$ 23.8860 0.950887 0.475444 0.879746i $$-0.342288\pi$$
0.475444 + 0.879746i $$0.342288\pi$$
$$632$$ 0 0
$$633$$ −0.113987 −0.00453057
$$634$$ 0 0
$$635$$ 4.24977 0.168647
$$636$$ 0 0
$$637$$ 35.7190 1.41524
$$638$$ 0 0
$$639$$ −15.7720 −0.623932
$$640$$ 0 0
$$641$$ −9.64380 −0.380907 −0.190454 0.981696i $$-0.560996\pi$$
−0.190454 + 0.981696i $$0.560996\pi$$
$$642$$ 0 0
$$643$$ −39.5151 −1.55832 −0.779162 0.626822i $$-0.784355\pi$$
−0.779162 + 0.626822i $$0.784355\pi$$
$$644$$ 0 0
$$645$$ −0.121104 −0.00476848
$$646$$ 0 0
$$647$$ 41.8695 1.64606 0.823030 0.567998i $$-0.192282\pi$$
0.823030 + 0.567998i $$0.192282\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −0.0552678 −0.00216612
$$652$$ 0 0
$$653$$ 9.41583 0.368470 0.184235 0.982882i $$-0.441019\pi$$
0.184235 + 0.982882i $$0.441019\pi$$
$$654$$ 0 0
$$655$$ 15.7649 0.615986
$$656$$ 0 0
$$657$$ −4.83922 −0.188796
$$658$$ 0 0
$$659$$ −12.1433 −0.473038 −0.236519 0.971627i $$-0.576006\pi$$
−0.236519 + 0.971627i $$0.576006\pi$$
$$660$$ 0 0
$$661$$ −26.0587 −1.01357 −0.506783 0.862073i $$-0.669166\pi$$
−0.506783 + 0.862073i $$0.669166\pi$$
$$662$$ 0 0
$$663$$ 6.36148 0.247059
$$664$$ 0 0
$$665$$ 2.29564 0.0890212
$$666$$ 0 0
$$667$$ −16.0000 −0.619522
$$668$$ 0 0
$$669$$ −9.59037 −0.370785
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 34.6353 1.33509 0.667547 0.744568i $$-0.267344\pi$$
0.667547 + 0.744568i $$0.267344\pi$$
$$674$$ 0 0
$$675$$ −2.79518 −0.107587
$$676$$ 0 0
$$677$$ 22.3709 0.859783 0.429891 0.902881i $$-0.358552\pi$$
0.429891 + 0.902881i $$0.358552\pi$$
$$678$$ 0 0
$$679$$ −2.90917 −0.111644
$$680$$ 0 0
$$681$$ −8.34908 −0.319937
$$682$$ 0 0
$$683$$ 28.6353 1.09570 0.547850 0.836576i $$-0.315446\pi$$
0.547850 + 0.836576i $$0.315446\pi$$
$$684$$ 0 0
$$685$$ 13.5298 0.516948
$$686$$ 0 0
$$687$$ 4.56193 0.174048
$$688$$ 0 0
$$689$$ −30.4390 −1.15963
$$690$$ 0 0
$$691$$ −7.65092 −0.291055 −0.145527 0.989354i $$-0.546488\pi$$
−0.145527 + 0.989354i $$0.546488\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −16.6206 −0.630457
$$696$$ 0 0
$$697$$ −2.40963 −0.0912712
$$698$$ 0 0
$$699$$ −12.0846 −0.457083
$$700$$ 0 0
$$701$$ −28.8851 −1.09098 −0.545488 0.838119i $$-0.683655\pi$$
−0.545488 + 0.838119i $$0.683655\pi$$
$$702$$ 0 0
$$703$$ −27.2947 −1.02944
$$704$$ 0 0
$$705$$ −1.59037 −0.0598968
$$706$$ 0 0
$$707$$ 1.46927 0.0552574
$$708$$ 0 0
$$709$$ −42.9991 −1.61486 −0.807432 0.589960i $$-0.799143\pi$$
−0.807432 + 0.589960i $$0.799143\pi$$
$$710$$ 0 0
$$711$$ −33.1789 −1.24431
$$712$$ 0 0
$$713$$ −0.999083 −0.0374160
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −5.59037 −0.208776
$$718$$ 0 0
$$719$$ 42.6741 1.59147 0.795737 0.605642i $$-0.207084\pi$$
0.795737 + 0.605642i $$0.207084\pi$$
$$720$$ 0 0
$$721$$ −2.28853 −0.0852291
$$722$$ 0 0
$$723$$ −7.28761 −0.271029
$$724$$ 0 0
$$725$$ 3.76491 0.139825
$$726$$ 0 0
$$727$$ 39.9301 1.48092 0.740462 0.672098i $$-0.234607\pi$$
0.740462 + 0.672098i $$0.234607\pi$$
$$728$$ 0 0
$$729$$ −15.1211 −0.560041
$$730$$ 0 0
$$731$$ 0.620646 0.0229554
$$732$$ 0 0
$$733$$ −17.7796 −0.656704 −0.328352 0.944555i $$-0.606493\pi$$
−0.328352 + 0.944555i $$0.606493\pi$$
$$734$$ 0 0
$$735$$ −3.28005 −0.120986
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −22.4390 −0.825432 −0.412716 0.910860i $$-0.635420\pi$$
−0.412716 + 0.910860i $$0.635420\pi$$
$$740$$ 0 0
$$741$$ 12.1211 0.445280
$$742$$ 0 0
$$743$$ −11.6656 −0.427969 −0.213985 0.976837i $$-0.568644\pi$$
−0.213985 + 0.976837i $$0.568644\pi$$
$$744$$ 0 0
$$745$$ −5.20482 −0.190690
$$746$$ 0 0
$$747$$ −46.3103 −1.69441
$$748$$ 0 0
$$749$$ −0.591287 −0.0216051
$$750$$ 0 0
$$751$$ 50.3544 1.83746 0.918728 0.394890i $$-0.129217\pi$$
0.918728 + 0.394890i $$0.129217\pi$$
$$752$$ 0 0
$$753$$ −2.87981 −0.104946
$$754$$ 0 0
$$755$$ 22.5601 0.821046
$$756$$ 0 0
$$757$$ 50.1798 1.82382 0.911908 0.410394i $$-0.134609\pi$$
0.911908 + 0.410394i $$0.134609\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 35.4087 1.28356 0.641782 0.766887i $$-0.278195\pi$$
0.641782 + 0.766887i $$0.278195\pi$$
$$762$$ 0 0
$$763$$ 7.77203 0.281366
$$764$$ 0 0
$$765$$ 6.87042 0.248400
$$766$$ 0 0
$$767$$ −65.9982 −2.38306
$$768$$ 0 0
$$769$$ 54.9679 1.98219 0.991096 0.133146i $$-0.0425080\pi$$
0.991096 + 0.133146i $$0.0425080\pi$$
$$770$$ 0 0
$$771$$ −1.74083 −0.0626946
$$772$$ 0 0
$$773$$ −25.2947 −0.909788 −0.454894 0.890546i $$-0.650323\pi$$
−0.454894 + 0.890546i $$0.650323\pi$$
$$774$$ 0 0
$$775$$ 0.235091 0.00844472
$$776$$ 0 0
$$777$$ −1.35528 −0.0486204
$$778$$ 0 0
$$779$$ −4.59129 −0.164500
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −10.5236 −0.376083
$$784$$ 0 0
$$785$$ 3.70436 0.132214
$$786$$ 0 0
$$787$$ −33.9612 −1.21059 −0.605294 0.796002i $$-0.706944\pi$$
−0.605294 + 0.796002i $$0.706944\pi$$
$$788$$ 0 0
$$789$$ 8.23509 0.293177
$$790$$ 0 0
$$791$$ 5.56101 0.197727
$$792$$ 0 0
$$793$$ 51.2395 1.81957
$$794$$ 0 0
$$795$$ 2.79518 0.0991350
$$796$$ 0 0
$$797$$ 48.9385 1.73349 0.866746 0.498750i $$-0.166207\pi$$
0.866746 + 0.498750i $$0.166207\pi$$
$$798$$ 0 0
$$799$$ 8.15046 0.288343
$$800$$ 0 0
$$801$$ −10.5771 −0.373722
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 2.06055 0.0726249
$$806$$ 0 0
$$807$$ 3.68028 0.129552
$$808$$ 0 0
$$809$$ 39.6803 1.39508 0.697542 0.716544i $$-0.254277\pi$$
0.697542 + 0.716544i $$0.254277\pi$$
$$810$$ 0 0
$$811$$ −6.70344 −0.235390 −0.117695 0.993050i $$-0.537550\pi$$
−0.117695 + 0.993050i $$0.537550\pi$$
$$812$$ 0 0
$$813$$ −7.05964 −0.247592
$$814$$ 0 0
$$815$$ 15.5445 0.544500
$$816$$ 0 0
$$817$$ 1.18257 0.0413730
$$818$$ 0 0
$$819$$ −7.07843 −0.247340
$$820$$ 0 0
$$821$$ −22.4096 −0.782101 −0.391051 0.920369i $$-0.627888\pi$$
−0.391051 + 0.920369i $$0.627888\pi$$
$$822$$ 0 0
$$823$$ −11.0378 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.4002 0.709386 0.354693 0.934983i $$-0.384585\pi$$
0.354693 + 0.934983i $$0.384585\pi$$
$$828$$ 0 0
$$829$$ 21.5298 0.747761 0.373881 0.927477i $$-0.378027\pi$$
0.373881 + 0.927477i $$0.378027\pi$$
$$830$$ 0 0
$$831$$ −13.9688 −0.484573
$$832$$ 0 0
$$833$$ 16.8099 0.582427
$$834$$ 0 0
$$835$$ 22.0752 0.763945
$$836$$ 0 0
$$837$$ −0.657123 −0.0227135
$$838$$ 0 0
$$839$$ 23.5592 0.813353 0.406677 0.913572i $$-0.366688\pi$$
0.406677 + 0.913572i $$0.366688\pi$$
$$840$$ 0 0
$$841$$ −14.8255 −0.511223
$$842$$ 0 0
$$843$$ −8.22797 −0.283386
$$844$$ 0 0
$$845$$ 14.8789 0.511850
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 4.69816 0.161240
$$850$$ 0 0
$$851$$ −24.4995 −0.839833
$$852$$ 0 0
$$853$$ 19.3406 0.662210 0.331105 0.943594i $$-0.392579\pi$$
0.331105 + 0.943594i $$0.392579\pi$$
$$854$$ 0 0
$$855$$ 13.0908 0.447697
$$856$$ 0 0
$$857$$ 37.5739 1.28350 0.641749 0.766915i $$-0.278209\pi$$
0.641749 + 0.766915i $$0.278209\pi$$
$$858$$ 0 0
$$859$$ 10.4390 0.356174 0.178087 0.984015i $$-0.443009\pi$$
0.178087 + 0.984015i $$0.443009\pi$$
$$860$$ 0 0
$$861$$ −0.227973 −0.00776932
$$862$$ 0 0
$$863$$ 43.7796 1.49027 0.745137 0.666911i $$-0.232384\pi$$
0.745137 + 0.666911i $$0.232384\pi$$
$$864$$ 0 0
$$865$$ −2.24977 −0.0764945
$$866$$ 0 0
$$867$$ −5.24885 −0.178260
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −54.4390 −1.84459
$$872$$ 0 0
$$873$$ −16.5895 −0.561468
$$874$$ 0 0
$$875$$ −0.484862 −0.0163913
$$876$$ 0 0
$$877$$ 35.6273 1.20305 0.601524 0.798855i $$-0.294560\pi$$
0.601524 + 0.798855i $$0.294560\pi$$
$$878$$ 0 0
$$879$$ 3.08899 0.104189
$$880$$ 0 0
$$881$$ −8.18166 −0.275647 −0.137824 0.990457i $$-0.544011\pi$$
−0.137824 + 0.990457i $$0.544011\pi$$
$$882$$ 0 0
$$883$$ −4.86330 −0.163663 −0.0818315 0.996646i $$-0.526077\pi$$
−0.0818315 + 0.996646i $$0.526077\pi$$
$$884$$ 0 0
$$885$$ 6.06055 0.203723
$$886$$ 0 0
$$887$$ 11.2507 0.377761 0.188881 0.982000i $$-0.439514\pi$$
0.188881 + 0.982000i $$0.439514\pi$$
$$888$$ 0 0
$$889$$ −2.06055 −0.0691087
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 15.5298 0.519686
$$894$$ 0 0
$$895$$ −5.09083 −0.170168
$$896$$ 0 0
$$897$$ 10.8798 0.363266
$$898$$ 0 0
$$899$$ 0.885097 0.0295196
$$900$$ 0 0
$$901$$ −14.3250 −0.477235
$$902$$ 0 0
$$903$$ 0.0587189 0.00195404
$$904$$ 0 0
$$905$$ 7.46927 0.248287
$$906$$ 0 0
$$907$$ −9.69679 −0.321977 −0.160988 0.986956i $$-0.551468\pi$$
−0.160988 + 0.986956i $$0.551468\pi$$
$$908$$ 0 0
$$909$$ 8.37844 0.277895
$$910$$ 0 0
$$911$$ 39.8236 1.31942 0.659708 0.751522i $$-0.270680\pi$$
0.659708 + 0.751522i $$0.270680\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −4.70527 −0.155552
$$916$$ 0 0
$$917$$ −7.64380 −0.252421
$$918$$ 0 0
$$919$$ 1.93945 0.0639765 0.0319882 0.999488i $$-0.489816\pi$$
0.0319882 + 0.999488i $$0.489816\pi$$
$$920$$ 0 0
$$921$$ −13.4105 −0.441893
$$922$$ 0 0
$$923$$ −30.1193 −0.991388
$$924$$ 0 0
$$925$$ 5.76491 0.189549
$$926$$ 0 0
$$927$$ −13.0502 −0.428626
$$928$$ 0 0
$$929$$ −48.5823 −1.59393 −0.796967 0.604022i $$-0.793564\pi$$
−0.796967 + 0.604022i $$0.793564\pi$$
$$930$$ 0 0
$$931$$ 32.0294 1.04972
$$932$$ 0 0
$$933$$ −6.64472 −0.217538
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −8.80986 −0.287806 −0.143903 0.989592i $$-0.545965\pi$$
−0.143903 + 0.989592i $$0.545965\pi$$
$$938$$ 0 0
$$939$$ 0.969724 0.0316457
$$940$$ 0 0
$$941$$ −44.9145 −1.46417 −0.732085 0.681214i $$-0.761452\pi$$
−0.732085 + 0.681214i $$0.761452\pi$$
$$942$$ 0 0
$$943$$ −4.12110 −0.134202
$$944$$ 0 0
$$945$$ 1.35528 0.0440872
$$946$$ 0 0
$$947$$ 13.0761 0.424918 0.212459 0.977170i $$-0.431853\pi$$
0.212459 + 0.977170i $$0.431853\pi$$
$$948$$ 0 0
$$949$$ −9.24129 −0.299985
$$950$$ 0 0
$$951$$ 5.90297 0.191417
$$952$$ 0 0
$$953$$ −25.6362 −0.830439 −0.415220 0.909721i $$-0.636295\pi$$
−0.415220 + 0.909721i $$0.636295\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −6.56009 −0.211836
$$960$$ 0 0
$$961$$ −30.9447 −0.998217
$$962$$ 0 0
$$963$$ −3.37179 −0.108654
$$964$$ 0 0
$$965$$ −2.48486 −0.0799905
$$966$$ 0 0
$$967$$ −41.4546 −1.33309 −0.666545 0.745465i $$-0.732227\pi$$
−0.666545 + 0.745465i $$0.732227\pi$$
$$968$$ 0 0
$$969$$ 5.70436 0.183250
$$970$$ 0 0
$$971$$ −36.4078 −1.16838 −0.584191 0.811616i $$-0.698588\pi$$
−0.584191 + 0.811616i $$0.698588\pi$$
$$972$$ 0 0
$$973$$ 8.05872 0.258351
$$974$$ 0 0
$$975$$ −2.56009 −0.0819886
$$976$$ 0 0
$$977$$ 30.0587 0.961664 0.480832 0.876813i $$-0.340335\pi$$
0.480832 + 0.876813i $$0.340335\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 44.3197 1.41502
$$982$$ 0 0
$$983$$ 26.3103 0.839169 0.419584 0.907716i $$-0.362176\pi$$
0.419584 + 0.907716i $$0.362176\pi$$
$$984$$ 0 0
$$985$$ 16.6888 0.531748
$$986$$ 0 0
$$987$$ 0.771110 0.0245447
$$988$$ 0 0
$$989$$ 1.06147 0.0337527
$$990$$ 0 0
$$991$$ −11.0596 −0.351321 −0.175660 0.984451i $$-0.556206\pi$$
−0.175660 + 0.984451i $$0.556206\pi$$
$$992$$ 0 0
$$993$$ 14.1193 0.448062
$$994$$ 0 0
$$995$$ −4.73463 −0.150098
$$996$$ 0 0
$$997$$ 24.4608 0.774681 0.387340 0.921937i $$-0.373394\pi$$
0.387340 + 0.921937i $$0.373394\pi$$
$$998$$ 0 0
$$999$$ −16.1140 −0.509824
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 9680.2.a.cg.1.2 3
4.3 odd 2 4840.2.a.r.1.2 yes 3
11.10 odd 2 9680.2.a.ch.1.2 3
44.43 even 2 4840.2.a.q.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.q.1.2 3 44.43 even 2
4840.2.a.r.1.2 yes 3 4.3 odd 2
9680.2.a.cg.1.2 3 1.1 even 1 trivial
9680.2.a.ch.1.2 3 11.10 odd 2